Explanation statuses — Whether the record believes the ACCOUNT, which is not what the same word means on a practice. An explanation can be Rejected while the thing it explained goes on working exactly as well as it ever did (ADR-031).
The current account — the best explanation the record holds for why this happens.
737 of 1352 THEORY documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| THEORY-003 v3 | Maximizing HSIC between representations and image identity maximizes the average squared MMD between the images’ view distributions, whose kernel mean embeddings exist by the Riesz representation theorem | Li, Pogodin, Sutherland & Gretton (2021), LIT-248, eq. 5 and App. B.2; Muandet et al. (2017), LIT-260, Lemma 3.1 — the one place in this family where Riesz is invoked explicitly. InfoNCE is not HSIC: Li et al.’s eq. 28 decomposes it exactly as −HSIC plus a log-mean-exp term, and Theorem B.1 bounds it with no variance assumption. | Active |
| THEORY-004 v8 | A representation is determined by its centred kernel up to a translation and an orthogonal transformation, and the rotation-aligned distance between two representations equals the Bures distance between their kernels | Harvey, Larsen & Williams (2023), LIT-250, Thm 1; Jorgensen & Tian, LIT-259, Remark 1.34 and Cor. 1.35, which name the kernel-to-Hilbert-space construction as GNS and state its minimal realizations unique up to a unitary (proof left as an exercise). This is the GNS connection the owner asked about, stated. | Active |
| THEORY-006 v4 | InfoNCE is a lower bound on mutual information for every critic and can never exceed the log of the batch size | Poole et al. (2019), LIT-246, §2.3, Eqs. 8–10 — an exact proof (CPC’s own derivation, ANTH-LIT-589, passes through an approximation). Tschannen et al. (LIT-247) show that tightening the bound does not explain what makes the representations good. | Active |
| THEORY-007 v4 | Kernel PCA under the positive-pair density ratio recovers the eigenfunctions of the positive-pair Markov chain, and their top span is minimax-optimal for linear prediction of approximately view-invariant targets | Johnson, El Hanchi & Maddison (2023), LIT-258, Thm 3.2 and Thm 4.1 — proved for finite augmentation spaces at the population level, Thm 4.1 for d at least the number of communicating classes. HaoChen et al.’s spectral loss is the same factorization of the normalized positive-pair adjacency. Whether trained networks land on these eigenfunctions is not shown. | Active |
| THEORY-008 v5 | On a finite set of stimuli, what a regularized linear readout can decode from a representation is a function of its normalized kernel, and CKA, CCA and GULP are averages of readout agreement | Harvey, Lipshutz & Williams (2024), LIT-257, eqs. 2, 7–8, Prop. 2, Cor. 1 and Table 1 — proved identities. This is the finite-dimensional content of “a probe is a functional”: the probe’s output is fixed by the kernel, whatever orthonormal basis the representation is written in, for G = aC_X + bI; under any linear change of basis only for the CCA member, G = C_X. | Active |
| THEORY-009 v7 | In the positive-pair spectral self-supervised papers, self-adjointness comes from the symmetry of the positive-pair distribution, not from the Riesz representation theorem; elsewhere in the family it comes from a symmetrized loss, a precondition on the kernel, or composing the conditional-expectation operator with its adjoint | An inference from how the spectral SSL papers (LIT-258, LIT-249, LIT-252, LIT-242) obtain their eigenbases: each uses a symmetric pair matrix or kernel, and none invokes Riesz. In all but Simon et al., whose loss is symmetrized by hand, and Pfau et al., whose SpIN takes symmetry as a precondition on the kernel k, that symmetry is the symmetry of the positive-pair distribution. Answers the owner’s “do we even need Riesz?” for this family of results. | Active |
| THEORY-011 v4 | Generalized contextuality is strictly broader than Kochen–Specker contextuality: a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two | Spekkens (2005), LIT-266, §III and §IV, read in full in NOTE-748; restated in Budroni et al. (LIT-263, §IV.E, read in NOTE-844). Corrects the reading that contextuality needs dimension three. The dimension-two limit holds for sharp measurements only. | Active |
| THEORY-012 v8 | An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality | Abramsky & Brandenburger (2011), LIT-016, Thm 8.1 with Prop 3.1, and Thms 5.4 and 5.9 — proved for finite measurement scenarios with no-signalling built into the definition. The graded version, the contextual fraction (LIT-265), is read (NOTE-236): CF = 0 exactly when this criterion holds. The review that places this formulation among its equivalents is read (NOTE-844); it asserts the equivalence with citations and does not show it. | Active |
| THEORY-014 v3 | In both formalisms of contextuality, classicality is the existence of a nonnegative version of a linear representation that always exists over the reals, so negativity is universal and only its unavoidability is nonclassical | An inference joining two proved results: LIT-016 Thms 5.4 and 5.9 with Thm 8.1, and LIT-003 Thm 2.8 with Cor 3.5. Each half is proved and read; the parallel is the record’s own, and LIT-003 attempts no comparison with LIT-016. | Active |
| THEORY-015 v3 | Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism, so every Kochen–Specker proof refutes universal generalized noncontextuality while measurement noncontextuality alone is consistent with quantum theory | Spekkens (2005), LIT-266, §III, §VIII.A (Eqs. 77–88), §VIII.B, §IX — the bridge between the record’s two contextuality threads, which no held work states. Read in full in NOTE-748, which confirmed the §VIII.A proof and its need for the full preparation set. Joining it to the sheaf form (LIT-016) is the record’s inference. | Active |
| THEORY-016 v3 | An operational theory admits a generalized-noncontextual model exactly when its GPT admits a positive quasiprobability representation, and for a tomographically local theory any diagram-preserving such model is an exact frame with exactly as many ontic states as the GPT’s dimension | Schmid et al. (2020), LIT-003, Prop 3.2, Prop 3.4, Cor 3.5, Thm 4.1, Cor 4.2, Prop 4.3 — proved. Spekkens’s odd-d stabilizer Wigner representation (LIT-007) is a worked instance. The theorem gives no test for whether a positive representation exists. | Active |
| THEORY-017 v6 | In a Hilbert-space model only unitarily invariant structure is intrinsic; a basis or tensor factorisation, and so any parts or features, is extra data supplied from outside the space, and the record’s Hilbert-space works differ in where they get it | Carroll (2021), LIT-123: a quantum theory is fixed up to unitary equivalence by the Hamiltonian’s spectrum, and subsystems come from a factorisation that the dynamics must select. Van Rijsbergen (LIT-262) computes only traces and inner products, and treats each observable’s eigenbasis as a “point of view”. DisCoCat’s ε/η composition (LIT-273) is orthogonally covariant, while its Frobenius sequel (LIT-272) stipulates a basis and composes by elementwise product in it. THEORY-004 is the same point for learned representations. This is a synthesis across the four, not a result any one of them states. It does not say that basis-dependent models are wrong; it says what they have assumed. | Active |
| THEORY-018 v4 | Softmax training identifies a language model’s final-layer representation only up to an invertible linear map, so no inner product on it is intrinsic, and Park, Choe and Veitch’s causal inner product is fixed by a stipulation | Park, Choe & Veitch (2023), LIT-304, prove the invariance γ ↦ Aγ + β, λ ↦ A⁻ᵀλ (eq. 3.1). They choose M = Cov(γ)⁻¹ by setting a free diagonal D = I, and admit they have no principle for choosing D. NOTE-302 derives, and the filer re-checked, that Cov(γ) alone is compatible with every positive-definite M. So the paper’s remark that most inner products are ruled out holds only once the true concept directions are known. Under superposition (LIT-323) no inner product can make all features orthogonal. | Active |
| THEORY-019 v4 | Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting with a group has eigenspaces built from its real irreducibles, so abelian rotation groups force pairs over ℝ, and the spectrum fixes neither the group nor a basis inside a multiplet | Peter & Weyl (1927), LIT-329 §3, build representations as eigenspaces of an invariant kernel, and Kondor & Trivedi, LIT-305 Lemma 6, show that intertwiners preserve isotypic components. Neither states the degeneracy corollary or its converse. This document puts that together from several readings, and that assembly, with its proof, is now the record’s own claim, CLAIM-385; what stays here is the reading. It adds the real-scalar correction in NOTE-282, found in the reading of LIT-362, and the counting in the reading of LIT-346. It does not support “degeneracy detects symmetry”: a spectrum is consistent with several groups and says nothing about the characters inside a multiplet. | Active |
| THEORY-020 | Every Boolean algebra is the clopen algebra of its Stone space of prime ideals, and a finite one is the power set of its atoms, so a single Boolean context always has two-valued valuations and the Kochen–Specker obstruction lies in how contexts overlap | Stone (1936, 1937), LIT-353 and LIT-358: every Boolean ring is isomorphic to the compact open sets of its prime-ideal space, a totally disconnected, locally compact Hausdorff space that is compact exactly when there is a unit, and every such space arises this way. Kochen & Specker (LIT-298, Thm 0) turn this into the criterion that a partial Boolean algebra embeds in a Boolean one iff its two-valued homomorphisms separate points. The joining of the two, that non-classicality needs more than one context, is the record’s. Neither Stone paper uses “ultrafilter” or states a duality of categories. | Active |
| THEORY-021 v2 | In the exact configurations of uniform superposition (regular polygons, the tetrahedron, antipodal pairs, and tegum products of these that span the hidden space, with equal norms within each factor) the features form a rank-one POVM compressed from the n-feature basis: their dimensionality-weighted projectors sum to the identity | Elhage et al. (2022), LIT-323, found that features in uniform superposition settle into polytopes and tegum products, with feature dimensionalities clustering at 3/4, 2/3, 1/2, 2/5 and 3/8. The tight-frame and POVM reading is the reader’s inference in NOTE-278, checked numerically there. It was refined when filing: a tegum product is not globally tight, but the Dᵢ-weighted POVM still sums to the identity. None of this is in the paper. It does not make superposition contextual or quantum, and it says nothing yet about trained real models. | Active |
| THEORY-026 v4 | For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient | Still, Sivak, Bell & Crooks (2012), LIT-327: β⟨W_diss⟩ for the switch x_t → x_{t+1} equals I[s_t;x_t] − I[s_t;x_{t+1}] (Eq. 14, an identity). Under a Markov drive this nostalgia is nonnegative, and summed it lower- bounds dissipation when each kernel keeps its equilibrium stationary; both conditions are unstated in the paper and derived here. Fiderer et al. (LIT-041) show that with feedback, in one binary channel, predictive, max-entropy-action and efficient agents are disjoint. The claim is an ensemble average, not a per-operation floor. Whether the bound fails without the two conditions is QUESTION-191. | Active |
| THEORY-027 | Two individually decisive people, weak Pareto and an unrestricted domain cannot all hold, even when choice need only yield a best element; unlike Arrow’s theorem, the conflict is not removed by interpersonal comparability | Sen (1970), LIT-285: no social decision function satisfies U, weak Pareto and minimal liberalism. The proof was checked in this record, including the unproved one-way strengthening. Sen (1998), LIT-282: the paradox, unlike Arrow’s, is not resolved by interpersonal comparisons. The theorem is settled. Whether pairwise decisiveness is what liberty is remains open, and the record reads the rights-as-game-forms literature only through Sen’s reply to it. | Active |
| THEORY-028 v2 | The expected bias of any data-dependent choice, and the expected generalisation error of any learning algorithm, is at most √(2σ²·I) for σ-subgaussian losses, where I is the information the output carries about the data | Russo & Zou (2015), LIT-356, Prop. 1: |E[φ_T − µ_T]| ≤ σ√(2·I(T;φ)) for any selection T among σ-sub-Gaussian statistics. Xu & Raginsky (2017), LIT-347, Thm 1: |gen| ≤ √(2σ²·I(S;W)/n) for any algorithm P_{W|S}. Both proofs rest on Donsker–Varadhan and were checked by the readers. The bound is in expectation only. It is vacuous for deterministic learners on continuous hypotheses, where I(S;W) is infinite, and for any real network when I is bounded by counting bits. It is tight only for Gaussian argmax and threshold selection. | Active |
| THEORY-030 v3 | For a memory evolving reversibly with its environment, Landauer’s bound prices only steps that lose logical distinguishability, at least k ln 2 of entropy per bit lost, exported to the environment and not necessarily as heat, so copying onto a blank register, measuring or evaluating a bit has no minimum cost; learning a label has a floor only in total entropy production, at least the information learnt, which in Goldt and Seifert’s perceptron can be paid in the weights’ own entropy with vanishing heat | Bennett (2003), LIT-360: the cost attaches to many-to-one maps of the logical state, at least k ln 2 of entropy per bit exported to non-information-bearing degrees of freedom, which “need not” be heat. The record derives it for a memory and environment evolving reversibly from an uncorrelated start: the environment’s entropy rises by at least the logical entropy lost, H(X | f(X)), and an injective step has no floor, attained exactly by a permutation. Goldt & Seifert (LIT-308) bound information learnt by total entropy production, payable with vanishing heat in their toy model. It does not license a k_BT ln 2 heat floor per judgment, per gradient step or per bit learnt. Whether the bound survives without reversible dynamics is QUESTION-215. | Active |
| THEORY-031 | The likelihood-ratio detector is optimal under the payoff, Neyman–Pearson and posterior criteria, which differ only in its threshold, so performance is an ROC and the criterion a point on it; d′ is criterion-free only for equal-variance normal signal and noise | Peterson & Birdsall (1953), LIT-364 — the threshold on ℓ = f_SN/f_N is optimal for expected value and at fixed false-alarm rate, and the two optimum families coincide (Part I Thms 1–7). The ROC’s slope equals the threshold (Thm 8). Tanner & Swets (1954), LIT-368, carry this over to the human observer as d′ plus a criterion. Stanislaw & Todorov (1999), LIT-367, state that d′ varies with the criterion unless the variances are equal. The separation of sensitivity from bias is a property of the model, not of the data. | Active |
| THEORY-032 | The Yoneda lemma determines an object only up to isomorphism, from its whole hom-functor, among objects the category already has; as a formal version of “an object is determined by its relations” it presupposes the relata | nLab, Yoneda lemma (rev. 2026-08-17), LIT-221 — y is fully faithful and y(c) ≅ y(d) iff c ≅ d, and pointwise bijections of hom-sets without naturality do not suffice. Lam & Wüthrich (LIT-222) state two of its consequences without the name. The philosophical reading, a relational criterion of identity up to isomorphism rather than a construction of objects from relations, is the record’s own. It does not say that the lemma supports eliminating objects. | Active |
| THEORY-034 v3 | Chakravartty’s dilemma reaches only structural realisms that keep relata but deny them every intrinsic identity-fixing feature, and of the seven structural-realist texts read here against it, two are of that kind: Floridi’s informational structural realism and Esfeld and Lam’s moderate structural realism | Chakravartty (2012), LIT-196 — non-eliminative OSR must choose eliminativism or a role for intrinsic features in identity, and the no-priority variant fares no better. The placements of the record’s other OSR texts on that map are the record’s own synthesis, from NOTE-192, NOTE-095, NOTE-099, NOTE-189 and NOTE-142. The account does not say the dilemma is sound, and it does not say eliminative OSR survives; the chapter itself leaves eliminative OSR untouched. Whether the founding texts are of the target kind is QUESTION-336. | Active |
| THEORY-037 v3 | In the topos programme quantum propositions form a distributive Heyting algebra, not an orthocomplemented lattice, and its negation is a pseudo-complement under which excluded middle can fail | Döring & Isham (2007, II), LIT-335 — daseinisation maps each projector injectively to a clopen sub-object of the spectral presheaf, with δ(P∨Q) = δ(P)∨δ(Q) but only δ(P∧Q) ⪯ δ(P)∧δ(Q). The orthocomplement is carried to inner daseinisation, not to ¬δ(P). That Sub_cl(Σ) is Heyting (Thm 2.5) is sketched in the source, and the record re-derives it: each Σ_V is extremally disconnected, so Sub_cl(Σ) is a frame. Paper I (LIT-343) chooses intuitionistic deduction. Isham–Butterfield (LIT-325) state that the Heyting algebra is fixed by the base category. | Active |
| THEORY-041 | Birkhoff and von Neumann proposed the modular law, not orthomodularity, and proved it fails for closed subspaces in infinite dimension; Husimi used the orthomodular identity only as an unnamed proof step | Birkhoff & von Neumann (1936), LIT-306 — the lattice of closed subspaces is orthocomplemented and not distributive. They propose modularity (L5), motivated by a dimension function, and give an explicit infinite-dimensional pentagon that breaks it. The word “orthomodular” does not occur. Husimi (1937), LIT-357, derives modularity as the parallelogram law under a finite-chain assumption and uses the orthomodular identity as a step without naming it. Citing either paper for “the orthomodular lattice of quantum logic” misattributes it. | Active |
| THEORY-042 | Under a superselection rule a coherent superposition across sectors is operationally a mixture, because no observable connects the sectors; Wick, Wightman and Wigner proved such a rule for integer versus half-integer spin and only postulated it for charge | Wick, Wightman & Wigner (1952), LIT-339 — they define a superselection rule and show that a superposition across sectors “is not a pure state, but a statistical mixture”. They prove the rule from time reversal for integer versus half-integer angular momentum, and postulate it for charge with “no conclusive evidence”. What follows for Birkhoff–von Neumann’s lattice (LIT-306), a product of sector lattices with central sector projections, is the record’s inference. The centre-of-the-algebra formulation belongs to Haag, who is not read. | Active |
| THEORY-046 v2 | Expected tangible rewards for merely engaging in an interesting task lower later free-choice engagement with it | Deci, Koestner and Ryan (1999, LIT-557): across 55 experiments, expected tangible rewards given for working on an interesting task lowered free-choice engagement after the reward stopped (d = −0.40; children −0.43, college students −0.21), and lowered self-reported interest less (−0.15). By their study-by-study account, Cameron and Pierce’s rival analysis found the same for this contingency (performance-independent rewards, −0.29), while disputing nearly everything else. Their 2001 reply (LIT-562), read here, re-codes 145 studies and finds it again: rewards offered for doing an interesting task lowered free choice, −0.35 across 57 studies. Active: both sides of the dispute find this effect. | Active |
| THEORY-053 | A treatment’s efficacy is not evidence for its theory of change: only change in the theorised process that is specific and comes before the outcome bears on it, and moving the process without moving the outcome counts against the theory | Salkovskis, Sighvatsson and Sigurdsson (LIT-538): “The effectiveness of a treatment is not evidence for the validity of the theory on which it is based”, with one exception, that changing the theory’s crucial factor without changing the problem disconfirms it. Hofmann and Hayes (LIT-544), from the other wing of CBT: a model whose process cannot be shown to apply “is incorrect and needs to be set aside” even if the intervention works, and a procedure that fails to move the process leaves the model untested. Active because it is close to a logical point and both state it. What it does not say is that mediation proves a mechanism; both sources deny that. | Active |
| THEORY-066 | A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system | The formal core of Bruineberg et al.’s critique (LIT-603, C4 in NOTE-447). In a directed acyclic graph the blanket of a node set A is its parents, children and the children’s other parents, and A is conditionally independent of everything outside A and its blanket. That holds for every A, so any set whose blanket leaves something outside gets a valid internal, blanket and external partition. Having a blanket therefore picks out no set. In Friston’s soup (LIT-526) the set was picked by spectral clustering with k = 8, not by the blanket. Active because it is a short derivation that anyone can check. It covers graphical (Pearl) blankets and the 2013 procedure. It does not cover the later sparse-coupling definitions at nonequilibrium steady state, which the record has not read. | Active |
| THEORY-068 v2 | In a two-alternative choice made by two competing leaky accumulators, the decision is optimal only when leak and mutual inhibition balance, so that the network integrates the difference of the evidence without loss; imbalance bounds accuracy at any fixed time and weights early or late evidence | Bogacz et al. (LIT-569) prove that the pure drift diffusion model is the continuum limit of the sequential probability ratio test, and so the optimal two-choice decider, and that the linearised Usher–McClelland network reduces to it exactly when leak equals inhibition (and, for free response, when both are also large). Usher and McClelland (LIT-565) derived the difference of the two units as an Ornstein–Uhlenbeck process with net leak K = k − β: K ≠ 0 caps accuracy, leak-dominant weights late evidence and inhibition-dominant early. Active because the core is derivation, checkable on paper. What it does not say: that brains are balanced, that the result carries to more than two options or to non-stationary evidence, or that the primacy and recency seen in six participants are explained by it. | Active |
| THEORY-073 v2 | In Friston’s 2013 formulation and the applications the record holds, a Markov blanket does not individuate a system: where it falls is fixed by modelling choices made before it is traced (the coupling graph, the internal set and the time at which steady state is assumed), and a blanket read from a fixed graph is the same whether the system produced the boundary or it was imposed | Bruineberg et al. (LIT-603) and Raja et al. (LIT-598) argue, by different routes, that the Markov blanket of the free-energy literature is placed and not found. The formal core is THEORY-066: the partition takes the internal set as input. This account adds what the critiques show about practice. In Friston’s soup (LIT-526) the graph was built from one kind of coupling and steady state was assumed when the run stopped. In published applications the blanket sits wherever the model needs it: in the receptors of a brain, the membrane of a bacillus, the beam joining two pendulums. The formalism also gives the same description to a membrane the cell makes and a beam nobody’s states make. So the blanket presupposes the boundary it is offered as the criterion of. Active since 2026-10-27, narrowed to the 2013 formulation and the applications held. The later steady-state definitions have not been read, and the unrestricted claim is THEORY-1097, Proposed. | Active |
| THEORY-081 v2 | The evidence of any model formed from a full model by changing only its prior is the full evidence times the full posterior expectation of the prior ratio, so the Savage–Dickey ratio is its point-mass case and choosing which parameters to keep is evidence maximisation over the prior | Friston & Penny (2011), LIT-620, Eq. 3: if models share a likelihood and differ only in priors nested inside a full prior, then p(y|m_i) = p(y|m_F)·E_{p(θ|y,m_F)}[p_i(θ)/p_F(θ)]. Their Eq. 6 is the point-mass case, the Savage–Dickey ratio that Wagenmakers et al. (2010), LIT-609, derive from the other end, and that Dickey (1971), LIT-610, stated in this form for any continuous likelihood. Friston, Parr & Zeidman, LIT-614, restate it for any new prior and give conjugate closed forms. Active because the identity is three lines of Bayes’ rule that anyone can check. The two Savage–Dickey conditions turn out to be the same condition. The identity is exact. Its Laplace and variational forms are only as good as the full posterior, and with no constraint on the prior family the optimum is maximum likelihood. | Active |
| THEORY-082 | Steepest descent is defined only relative to a metric, and Amari’s proof that natural-gradient learning is Fisher efficient needs the Fisher information at the optimum to be invertible, the condition whose failure WBIC calls singular; in a multilayer perceptron it fails wherever an output weight is zero | Amari (1998), LIT-607. Theorem 1 is a Lagrangian derivation: under |dw|² = Σg_ij dw_i dw_j the steepest-descent direction is −G⁻¹∇L, so it changes with G. Theorem 2 shows that online learning with step 1/t along the Fisher natural gradient attains the Cramér–Rao bound G⁻¹/t, and its proof uses G(w*)⁻¹. Watanabe, LIT-616, Eq. 14, calls a truth regular when the optimum is one point and its Fisher is positive definite, and singular otherwise. From Amari’s own Eq. 6.12, the score for hidden unit i’s input weights carries the factor v_i. So the Fisher of a multilayer perceptron loses rank wherever an output weight is zero. That set is part of the optimum whenever the teacher needs fewer hidden units than the student has. Active because each step is checkable in the two texts. The claim that the Fisher is the only invariant metric is Chentsov’s, cited and not held. | Active |
| THEORY-083 v3 | In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet | Papyan, Han & Donoho (2020), LIT-618, measured 480 trained classifiers on seven datasets with VGG, ResNet and DenseNet. Through the terminal phase, within-class variability of last-layer features falls (NC1), and the centred class means become equinorm and equiangular with cosine −1/(C − 1) (NC2). The classifier aligns with them (NC3), and the decision becomes nearest class mean (NC4). They prove that NC1 and NC2 imply NC3 and NC4 for the MSE-optimal and max-margin classifiers, and that the simplex ETF is the unique error-exponent-optimal codebook. Read from the plots, collapse is close to complete on the four easier datasets and partial on ImageNet at the last epoch. Papyan (2020), LIT-613, sees class means separate and grow more orthogonal with depth. The measurements cover one protocol, and the dynamics are unexplained outside free-features models. Within them, THEORY-278 (LIT-974) derives collapse for MSE under a re-whitened flow, and THEORY-286 (LIT-994) gives the cross-entropy endpoint under class imbalance. The paper’s further claim, that the Hessian’s class outliers are a symptom of collapse, is argued and not measured, and is left to THEORY-078. | Active |
| THEORY-086 v2 | In the kernel regime gradient descent fits the target eigenspace by eigenspace of the neural tangent kernel, at rates set by their eigenvalues; on uniform spherical data the eigenspaces are the harmonic degrees, and for fully connected ReLU networks the non-zero eigenvalues decay as k^(−d) at every depth | Three derived results that compose. Cao et al. (2019), LIT-624, Theorem 4.2: for a wide two-layer ReLU network, the training residual’s component in the top-r_k NTK eigenspace falls as (1 − λ_{r_k})^T, for any target. Basri et al. (2019), LIT-612: on uniform spherical data the Gram matrix is a convolution, so its eigenvectors are harmonics, with closed-form eigenvalues per frequency. The bias-free kernel is zero on odd frequencies k ≥ 3, and a bias restores them. Bietti & Bach (2021), LIT-608, Corollary 3: the L-layer ReLU NTK decays as k^(−d) on S^(d−1), the same exponent at every depth. Active for the kernel regime on uniform spherical data, where every step is proved. It says nothing about networks whose kernel moves. | Active |
| THEORY-087 v3 | The Bayesian complexity penalty grows as (d/2) log n in a regular model, where the curvature at the optimum sets the Occam factor, and as λ log n in a singular one, where λ is the real log canonical threshold; with the prior positive on the optimal set, λ falls below d/2 when that set is infinite or has a point of degenerate curvature, and equals d/2 when it is finitely many points of non-degenerate curvature | MacKay (1992), LIT-623, writes the evidence as best-fit likelihood times an Occam factor, the ratio of posterior to prior accessible volume, which under a Gaussian approximation is P(w_MP)(2π)^{k/2} det^{-1/2}A. When the likelihood dominates, det A grows as n^d, so the penalty is (d/2) log n. Watanabe’s WBIC paper, LIT-616, recovers that case (Lemma 3, Theorem 5) and states what replaces it when the Fisher information is degenerate at the optimum: λ log n − (m − 1) log log n, with λ the RLCT (Theorem 2). The regular half is re-derived by Laplace’s method with its conditions. The singular half is Watanabe’s cited theorem, checked by exact free energies in two singular models of known λ. The volume reading of λ is in Watanabe’s book, which is unread. | Active |
| THEORY-102 v3 | In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration | Vishne, Gerber, Knight & Deouell (2023), LIT-644: in 430 visually responsive subdural electrodes from 10 patients viewing images for 300–1,500 ms with no report on analysed trials, high-frequency activity and single-electrode category selectivity fell about 80% from peak by 800–900 ms, while the pattern across ventral temporal electrodes kept near-ceiling, time-invariant category and exemplar information that ended a fixed delay after the image went off. Prefrontal and parietal information was confined to about 150–600 ms and ignored duration. This met the predictions both IIT’s and the global workspace’s proponents registered for the paradigm, so it did not discriminate them. Above- threshold images only, sparse right-lateralised frontal coverage, one frequency band; it does not show the code tracks experience rather than input. | Active |
| THEORY-105 v2 | For RoBERTa-base fine-tuned on English classification data, with loss measured by a linear probe refitted on the frozen encoder, points sampled from the convex hull of models fine-tuned on one dataset, one task family or twelve GLUE and SuperGLUE datasets lost no more than any model fine-tuned elsewhere and beat the group’s own members in 88% to 96.7% of comparisons, and the centroid of the general-level models was a better starting point for BitFit than the pretrained model | Gueta et al. (2023), LIT-658: RoBERTa-base models fine-tuned on the same dataset, the same task family, or any of 12 GLUE and SuperGLUE datasets cluster in weight space. Sampled points in the convex hull of each group beat the group’s own members in 88% to 96.7% of comparisons. Extrapolating past the members raises the loss quickly, so the members sit near the region’s edge and averaging moves inward. The paper calls the region convex, but shows only that the hull lies inside it, by a probe-based loss. Its reading of model soups and SWA as picking points inside the region is argued in its §8, not tested. Moving from the centroid towards the origin stays low-loss far past the members, which the authors cannot explain. Promoted 2026-10-20 on the probe-loss comparisons measured (ADR-047 v3); the geometric thesis is THEORY-828. | Active |
| THEORY-111 v3 | In image classifiers, between two networks linearly connected by spawning from a shared checkpoint or by permutation alignment (MLPs, and VGG-16, ResNet-20 and ResNet-50 on CIFAR-10 and Tiny-ImageNet), nearly every layer’s features at three points on the line were proportional to the interpolation of the endpoints’ features, and pairs that were not linearly connected were far from it | Zhou et al. (2023), LIT-674: whenever two image classifiers are linearly mode connected, by spawning or by permutation, the features of nearly every layer of the interpolated network are proportional to the interpolation of the endpoints’ features. This is layerwise linear feature connectivity, and two conditions, weak additivity of ReLU and a commutativity of weights with features, imply it exactly. Lubana et al.’s Lemma 2, LIT-669, gives the exact one-hidden-layer case: linear connectivity between interpolating minima forces shared activation patterns, and with shared patterns the hidden features are linear along the path. Measured on pairs at three interpolation points, image classification only. The regularity outside image classification, and the step to many-model averages such as soups and SWA, are QUESTION-318. | Active |
| THEORY-112 v2 | Aligning independently trained solutions over the architecture’s symmetries removed the linear barrier between them for MNIST MLPs and 32×-wide ResNet-20s on CIFAR-10, two-thirds of it for an ImageNet ResNet-50, and not all of it for VGG-16 or ResNet-20 at 1× width, the barrier after weight matching falling with width; for 6- to 8-layer ViTs and GPT-2s, permutations alone removed 63–90% of it, and adding an orthogonal map on the residual stream, learned against the loss at the path’s midpoint, took it to zero or near zero except on BookCorpus (0.42) | Entezari et al. (2022), LIT-652, conjectured that SGD solutions are one basin modulo permutation. Git Re-Basin (2023), LIT-661, aligned MNIST MLPs and 32×-wide ResNet-20s to zero barrier, but not 1×-width models or ImageNet ResNet-50. Ferbach et al. (2024), LIT-680, proved it for wide nets with independent neurons, with a width requirement exponential in the previous layer’s width. Theus et al. (2025), LIT-664, took transformers to zero only with an orthogonal map on the residual stream added; permutations alone left 0.29–1.60. Bo Zhao et al. (2025), LIT-655, place permutations inside the full symmetry group. Lubana et al.’s barriers between models that compute differently are the boundary. Whether most of the barrier goes at standard width on ImageNet or a language corpus under data-free alignment, and whether the barrier left rises with depth, is QUESTION-317. | Active |
| THEORY-113 v2 | Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model | Stoica et al. (2023), LIT-665, merge ResNets trained from different initialisations on disjoint label sets. Permute-then-average fails, which they explain by the permuted model lying in a basin for its own task only. Mean cross-model feature correlation falls with depth (0.50, 0.37, 0.27 at three depths). Merging only the early layers and keeping later layers as task heads recovers most of the ensemble’s accuracy. Merging features within one model as well as across both helps once the models are wide enough to hold redundant features. With full merging on ImageNet-1k, ZipIt! is no better than permutation (8.6%). No same-task control separates the effect of task difference from that of depth; that is QUESTION-316. | Active |
| THEORY-114 v2 | In image classification, linear mode connectivity appeared during training and not at initialization: copies of ResNet-20 and VGG-16 on CIFAR-10 and of ResNet-50 and Inception-v3 on ImageNet, trained from a shared state, became stable to SGD noise in test error only after 1.5–20% of training (LeNet on MNIST was stable from the start), and between independently initialized MLPs on MNIST and CIFAR-10 the barrier after weight-matching alignment was high at initialization and fell gradually through training | Frankle et al. (2020), LIT-654: two copies trained from step k under different SGD noise are linearly connected only once k is a little way in, at 1.5–3% of training for VGG-16 and ResNet-20 on CIFAR-10 and 16–20% for Inception-v3 and ResNet-50 on ImageNet. Git Re-Basin (2023), LIT-661: between independently trained MLPs, the barrier after permutation alignment is high at initialization and falls gradually over training. Entezari et al.’s theorem proves alignment-LMC at random initialization; Ferbach et al. (2024), LIT-680, show the width that needs grows recursively exponentially with depth, far past practical widths, which is why the two are consistent. | Active |
| THEORY-117 v3 | In Uselis, Dittadi and Oh’s two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair | Uselis, Dittadi & Oh (2025), LIT-667. In an n × n grid of two concepts with k combinations seen per value, ResNet-50s trained from scratch fit seen pairs and fail on unseen ones; four times the data of the same combinations leaves 60–80% drops, while more values and more combinations help. Features pass from spurious, to decodable but entangled, to linearly factored as coverage grows. Model selection was on the test set, by design, so the results bound what is possible. The record re-derived that exactly factored embeddings of every unseen pair are fixed by seen pairs connecting every value, as the paper’s k = 2 cycle does. Honest selection, other architectures and more concepts are QUESTION-207. | Active |
| THEORY-121 v4 | Moral responsibility requires guidance control, moderate reasons-responsiveness of the agent’s own actual-sequence mechanism, and not regulative control over alternative possibilities, so responsibility is compatible with causal determinism | Fischer and Ravizza, in their own précis (LIT-728, read): moral responsibility requires guidance control, not regulative control over alternatives. Guidance control is acting from a mechanism one has made one’s own by taking responsibility for it, which is moderately reasons-responsive: regularly receptive to reasons, some moral, and at least weakly reactive. Fischer’s 1987 paper (LIT-687, read) is the weak-responsiveness origin. The view is compatible with causal determinism. | Active |
| THEORY-123 v2 | In ad hoc groups of students negotiating a bonus, a confederate’s randomly assigned pleasant or unpleasant mood spread to the other members, by their self-reports and by blind coders, and the more pleasant mood a group caught, the more cooperative and the less conflicted coders rated it, a correlation with measured contagion rather than a manipulated effect; members’ moods also converged without a confederate, and members did not credit their mood for how effective they judged themselves; the numbers are from the 2001 manuscript | Barsade (2002), LIT-706, read from the August 2001 manuscript, so every number is the manuscript’s. Study 1 (94 undergraduates, 29 groups): a confederate randomly assigned to show pleasant or unpleasant mood moved members’ self-reported mood by +.41 or −.26 (χ²(1) = 9.97), and blind video-coders rated members’ mood 2.75 against 2.33 (χ²(1) = 10.30). Members who caught more pleasant mood were rated more cooperative, and at group level coder-rated contagion correlated with cooperativeness (r = .44) and against conflict (r = −.42). Study 2 (113 MBA students, 26 groups, no confederate): the ICC of members’ mood rose from .00 to .24, and contagion correlated with members’ ratings of cooperation (.49) and conflict (−.49). Members related how pleasantly they behaved to others’ conduct but showed “absolutely no awareness” of mood bearing on their judged effectiveness. Valence and energy hypotheses failed or were marginal. Limits: one experiment, one actor, retrospective mood, correlational process effects. Promoted 2026-10-20 on what the manuscript measured (ADR-047 v3); the wider claim is THEORY-826. | Active |
| THEORY-124 v2 | In Posani et al.’s analysis of the IBL Brainwide Map, recorded while trained mice performed one visual decision task, the eight-variable selectivity profiles of neurons within one cortical area clustered beyond a covariance-matched Gaussian null only in VISp, AUDp and SSp-ul of about 20 areas tested, while profiles pooled across areas clustered along anatomical lines; and over the task conditions each area distinguishes, at least 95% of random balanced dichotomies were linearly decodable above a label-shuffled null in 15 of 16 areas, a lenient criterion | Posani et al. (2026), LIT-708: in about 14,000 cortical units of the IBL Brainwide Map, the 8-variable selectivity profiles of 4,617 selective neurons beat a covariance-matched Gaussian null for clustering within a region only in VISp, AUDp and SSp-ul. Elsewhere the profiles form an elongated, unclustered cloud. After merging conditions a region cannot tell apart (leaving 5 to 16 of 16), ≥ 95% of random balanced dichotomies are decodable above a shuffle null in 15 of 16 regions, GU being 0.82. Types do appear when regions are pooled, aligned with anatomy. One task, overtrained mice, a lenient separability criterion. | Active |
| THEORY-127 v4 | On the court agenda, and on any agenda in which every proposition reaches every other by conditional entailment and some minimally inconsistent triple stays consistent with two of its members negated, a group whose judgments are robustly rational under universal domain and respect unanimity cannot form its judgment on each proposition from its members’ judgments on that proposition alone unless one member dictates; its judgments must supervene on the members’ whole sets of judgments, and so can depart from what the members judge on the same proposition while being wholly fixed by them | List & Pettit (2006), LIT-705: under universal domain, robust group rationality (complete, consistent, deductively closed group judgments for every profile of rational member judgments) is inconsistent with majoritarian, uniform proposition-wise and proposition-wise supervenience (Propositions 1–3; the last two proved elsewhere and cited), and consistent with set-wise supervenience, which the premise-based and distributed premise-based procedures achieve (Propositions 4–5). Members’ judgments on a conclusion can then be insufficient, or even unnecessary, to fix the group’s judgment on it. List (2014), LIT-700, makes this the mark of a third kind of collective attitude: corporate attitudes, a group agent’s, supervene “holistically”, while aggregate and common attitudes supervene proposition by proposition. Active since 2026-10-19 for categorical judgments on the court agenda and on agendas with two stated connectedness conditions: the record re-derived Proposition 3 (and Proposition 2 as a corollary) itself. The extension to corporate attitudes in general is QUESTION-152. | Active |
| THEORY-132 | Stability, a system’s speed of return to equilibrium after disturbance, and resilience, its capacity to absorb change and persist within its domain of attraction, are distinct properties that can trade off, so a system can be highly unstable and highly resilient, and managing it for constancy can erode its resilience | Holling (1973), LIT-741. Stability is return to equilibrium. Resilience is persistence under change, set by the extent of the domain of attraction and the system’s distance from its edge. Budworm forests and insect populations in harsh climates are unstable and resilient; homogeneous lakes are stable and fragile. Active for the distinction, which has become standard. The claim that management for constancy erodes resilience rests on suggestive cases. | Active |
| THEORY-133 v2 | In List and Menzies’s account of causes as difference-makers, a higher-level property that makes a difference to an effect is its cause, and its realizer is not, exactly when in some nearest situation without that realizer the property is realized differently and the effect still occurs; which holds depends on each causal system’s counterfactual structure, not on levels a priori | List & Menzies (2009), LIT-722. Take causes to be difference-makers. Then Kim’s exclusion principle fails, and a revised one holds or fails system by system. Given that a higher-level property makes a difference to an effect, its realizer is not a cause exactly when some nearest situation without the realizer has the property realized otherwise and the effect present; the record re-derived this. The claim that higher-level relations are typically insensitive to their realizers is an observation about scientific practice, and Pernu disputes the paper’s main example. Which systems meet the condition is QUESTION-213. | Active |
| THEORY-134 v2 | Whether a system can persist is a question about a constraint set fixed independently of its dynamics: a set is viable when from each of its states some evolution can stay in it, and when it is not, its viability kernel is the largest part that is, from the rest of which every evolution leaves in finite time | Aubin’s viability theory, read in his 1999 mini-course notes LIT-714. Given dynamics and a set K of required states, K is viable if some evolution from each state stays in K and invariant if every one does. Nagumo’s theorem reduces viability to a tangency condition. The viability kernel is the largest viable subset of K; from outside it every evolution exits K. The notes do not compare viability with stability; that contrast is a further step. | Active |
| THEORY-136 v3 | A persistent or abrupt change is not evidence that a system has alternative stable states: field patterns such as jumps, multimodality and dual relationships also arise from a stepwise driver or a simple threshold, and only hysteresis, initial-state dependence or a lasting shift after a temporary disturbance, with slow return excluded, comes close to showing one | Scheffer & Carpenter (2003), LIT-726. Alternative attractors predict sudden shifts, multimodal states and dual responses to a driver, and so do discontinuous drivers and threshold responses. Discriminating evidence is experimental: different starting states reaching different end states, temporary disturbances causing lasting shifts, or hysteresis. Slow recovery can mimic all three. Positive feedback must be strong to produce alternative states. Active as a methodological standard. | Active |
| THEORY-138 | X is a cause of Y exactly when some intervention on X with respect to Y would change Y, where an intervention is an exogenous causal process that alone sets X (M1), affects Y only through X (M2), is not caused by anything that affects Y by another route (M3), and leaves Y’s causes off the path through X unchanged (M4); so defined, causation is not reduced to non-causal facts but is not viciously circular, and needs no human agent | Woodward, “Causation and Manipulability” (SEP, 2001; revised 2023), LIT-713. A cause is what changes the effect under a surgical, exogenous intervention, defined relative to the effect by M1–M4. The account separates total from direct causes by combining interventions, and treats natural experiments as interventions. Active as the standard the record uses for what a causal claim asserts. This statement says nothing about invariance, proportionality or levels. | Active |
| THEORY-140 v2 | The Columbia Accident Investigation Board found that the institutional causes of the Challenger accident had not been fixed and recurred in Columbia’s: in both, evidence that the design was not performing as expected was repeatedly reinterpreted as acceptable, each decision seeming correct in itself and each survived anomaly justifying the next flight, while budget and schedule pressure, consensus sign-off, a hierarchy that kept dissent from rising and a safety organization dependent on the program it oversaw sustained the pattern for twenty years and undid post-Challenger reforms | Columbia Accident Investigation Board (2003), LIT-734, applying Vaughan’s normalization of deviance (LIT-715, unread). Foam strikes and O-ring erosion followed the same trajectory of incremental acceptance. Organizational culture, structure and political constraint reproduced it, so “history is cause”. Two cases in one agency, reconstructed after the fact. The title states the Board’s finding; whether the pattern is general and causal, which is Vaughan’s theory in a book the record has not read, is QUESTION-334. | Active |
| THEORY-146 v2 | In Holman and Bruner’s bandit model of honest Bayesian researchers whose beliefs funding cannot touch, who differ in method and whose influence and training of newcomers follow productivity, industry funding of those whose methods already favour its drug lowers, in the runs their figures report, the share of the community on the better treatment, and has no effect without methodological diversity; at the one setting reported, an independent funder that counts industry-funded trials toward its merit threshold lowers that share further, and one that excludes them raises it | Holman & Bruner (2017), LIT-744. In a Zollman bandit model with methodological diversity, productivity-weighted influence and turnover, industry funding of agents whose methods favour the drug lowers the chance of converging on the better action, though no agent’s beliefs respond to funding. A merit-based independent funder worsens it; one ignoring industry-funded trials counters it. Held as what the model shows; whether real fields work this way is QUESTION-315. | Active |
| THEORY-149 | Users can govern a shared resource durably without either privatisation or central control, and the institutions that last are marked not by particular rules but by structural regularities: user-made and user-monitored rules, graduated sanctions, cheap local conflict resolution, recognised rights to self-organise, and nested layers of governance | Ostrom’s account, read in her Nobel lecture (LIT-719). Coded case meta-analyses, laboratory and field experiments and multi-country field studies agree that resource users often self-govern successfully. Long-surviving institutions share eight design principles that failed ones lack. The principles are correlates of survival, not tested causes, and their reach beyond common-pool resources is not shown. | Active |
| THEORY-155 v2 | Iterated learning by Bayesian agents who sample from the posterior converges, in an ergodic chain, to the shared prior | Griffiths & Kalish (2007), LIT-769, read in NOTE-599: a chain of learners who share a prior and each sample a hypothesis from their posterior is a Gibbs sampler, so the probability a learner holds a hypothesis converges to its prior probability, whatever the amount of data passed on; with MAP learners the outcome centres on the prior’s mode but depends on data amount and noise, and can amplify weak biases. It is a theorem about an idealized chain. It does not say human chains reach the prior, and Kirby, Cornish & Smith’s laboratory chains, LIT-771, are consistent with it without testing it. Whether human chains do is QUESTION-137. | Active |
| THEORY-156 v3 | For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation | Blackwell (1953), LIT-781, Theorem 8 with Theorems 1–3: for experiments with finitely many states and decision problems with bounded losses, α attains every loss vector β attains, in every problem, exactly when a Markov kernel turns α’s observation into β’s under every state. The direction from simulation to informativeness is Blackwell (1951); the finite converse is Sherman and Stein (1951); the general converse is proved in 1953. It is a qualitative order and says nothing about how much worse an incomparable experiment is. The record re-derived the converse for standard Borel outcome spaces (note of 2026-10-29); arbitrary spaces, which the paper claims through a quoted 1951 result, are QUESTION-321. | Active |
| THEORY-158 v2 | For a Markov process on a finite state space, an observable commutes with the generator exactly when its mean and variance are both conserved in every state, which is exactly when it is constant on each connected component of the transition graph, so a conserved mean alone does not give a symmetry | Baez & Fong (2013), LIT-773, read in NOTE-598: Theorem 1 (finite state space, four equivalent conditions), which the record re-derived. Theorems 2–3 (σ-finite measure spaces, without the component form) are reported as the source’s (QUESTION-144). Unlike quantum mechanics, where a mean conserved in every state already implies commutation, a Markov process can conserve the mean of an observable it does not commute with. The theorem covers diagonal observables only; it says nothing about symmetries that permute states. | Active |
| THEORY-160 v2 | In the algebraic Hamiltonian setting, symmetry and conservation correspond because the bracket is antisymmetric, which for a bilinear bracket is each observable conserving itself; the theorem’s content lies in identifying observables with generators | Baez (2020), LIT-819, read in NOTE-630, Theorems 3, 4, 8 and 10, with Alfsen and Shultz’s theorem quoted as Theorem 12 and not checked here (QUESTION-142). With unique solutions to the flow equations, “a generates symmetries of b iff b generates symmetries of a” is {a, b} = 0 ⇔ {b, a} = 0. It assumes reversible one-parameter groups and an observable–generator map; it says nothing about dissipative or stochastic dynamics, nor about the Lagrangian form of Noether’s theorem. | Active |
| THEORY-161 v2 | In Zhao et al.’s (2025) and Chai et al.’s (2023) semantic rate–distortion frameworks, meaning is a latent variable with a known joint law with the observation, so their limits are indirect source-coding limits; Zhao et al.’s posterior-matching semantic distortion is, with KL divergence, exactly the information bottleneck, and with total variation a bound on the expected lost value of every bounded-loss decision | From the readings of Zhao et al. (2025), LIT-793, Chai et al. (2023), LIT-812, and the survey of Xin, Fan and Letaief (2024), LIT-823. These frameworks fix meaning as a hidden S with a known p(S, X); their theorems are indirect rate–distortion results that do not depend on what S is. Zhao et al.’s posterior distortion reduces, for KL, to I(S;X) − I(S;Y), making the function the information bottleneck (LIT-338), and for TV bounds the extra Bayes risk of every bounded loss. The two identities are this record’s derivations, not the papers’. | Active |
| THEORY-163 v3 | In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data | Zaslavsky, Kemp, Regier & Tishby (2018), LIT-810, read in NOTE-631: treating a colour lexicon as an encoder from Gaussian perceptual meanings to words, the World Color Survey languages and English lie near the IB curve of LIT-338 at β ≈ 1.03, beat hue-rotated variants of themselves, and their soft, partly inconsistent naming is what IB optima look like. It does not show that languages are driven to the bound by any process, and the need distribution that makes the fit best is estimated from the naming data themselves. | Active |
| THEORY-164 v3 | As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated | Raviv, Meyer & Lev-Ari (2019), LIT-825, read in NOTE-628: fully connected groups of eight built more systematic languages than groups of four, faster and more uniformly, with total interaction and network structure fixed; input variability, larger in bigger groups, predicted the gains. One laboratory study, two group sizes, and a mechanism supported by lagged association, not manipulation. It says nothing yet about natural communities, where size travels with network density and second-language learners. | Active |
| THEORY-165 v2 | Within a fixed measurement scenario, the non-contextual fraction is the value of a linear programme whose constraint vector is the empirical model’s probability table, and the contextual fraction is convex under mixing | From Abramsky, Barbosa & Mansfield (2017), LIT-265: the non-contextual fraction is the value of a linear programme whose constraint vector is the empirical model (Eq. 3), and convexity of CF under mixing is their Theorem 2. Piecewise linearity and the Lipschitz bound are the record’s own derivation, now CLAIM-372. The Lipschitz constant depends on the scenario and is not bounded here. Nothing here applies to signalling data, where CF is undefined. | Active |
| THEORY-166 v3 | In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability | Wang, Solloway, Shiffrin & Busemeyer (2014), LIT-795, on the QQ equality derived in LIT-834: across the 66 Pew surveys of 2001-2011 that varied the order of two questions, question order shifts the answers (p = 0.0004) but not the probability that the two answers agree (p = 0.4625). The Lüders projection model predicts this without parameters, by a one-line identity. A classical model with order events can be built to satisfy the equality, and LIT-264 shows that data meeting it are noncontextual, so it is evidence for a constraint on order effects, not for nonclassical probability or contextuality. A preregistered test with a constrained classical rival is QUESTION-210. | Active |
| THEORY-168 v3 | In Contextuality-by-Default, whether a fixed set of measurements is contextual depends on how the system is represented: which couplings are imposed, and which dichotomizations of the variables are included | Dzhafarov, Cervantes & Kujala (2017), LIT-790, with Dzhafarov & Kujala’s CbD 2.0 (LIT-831) and CbD 1.0 (LIT-777). The same measurements can receive different verdicts. Maximal and multimaximal couplings can disagree when a content appears in three or more contexts (four, for binary variables), and the record’s worked case shows it. Multimaximal couplings of non-binary variables can be destroyed by coarse-graining. With every dichotomization kept, two content-sharing variables are contextual on their own unless one nominally dominates the other. The authors present the representation as the analyst’s choice, not as a defect. What the record does not get from these papers is any rule for choosing it. | Active |
| THEORY-169 v2 | For unnormalized causal linear attention and for softmax attention, conditioning on a context is exactly training a learner on it and predicting: the first is one batch gradient step of a linear inner model from zero, the second kernel regression that stores the context, and one linear self-attention layer with constructed weights carries one gradient step of in-context linear regression; for these layers conditioning and per-sequence weight updates differ in what state is kept, not in kind | Sun et al. (2024), LIT-801, Theorems 1–2, with von Oswald et al. (2022), LIT-792, Proposition 1. Linear attention equals a layer whose state is a linear model’s weights, updated by batch gradient descent on the context from zero; softmax attention equals a Nadaraya–Watson learner whose “training” appends the context to a list; and, with constructed weights, one linear attention layer reproduces one gradient step of a linear regression on in-context pairs. These are identities of output for specific layers. They do not say that trained transformers do gradient descent (the anthology rejects that, ANTH-THEORY-068), nor anything about weight updates that persist across sequences. | Active |
| THEORY-171 v3 | When the causal constraints on events depend on context, deterministic assignments that are causal in each context can fail to glue into one causal assignment, so causal structure is itself a source of contextuality | From Gogioso & Pinzani’s trilogy: The Topology of Causality, LIT-808, read in NOTE-612, with the definitions of The Combinatorics of Causality, LIT-800 (NOTE-635). Over a space of input histories with the lowerset topology, causal functions form a separated presheaf, and on the space Θ₃ it is not a sheaf: two causal assignments on disjoint opens give one event two outputs on their common extension. That example was checked here, and it settles the claim. The source’s Theorem 2.34, that the presheaf is a sheaf exactly when no two contexts impose incompatible causal pasts on a common event, is reported as the source’s. Its proof was not read, and it is QUESTION-143. It does not say any physical or quantum data display the effect. | Active |
| THEORY-172 v4 | In Frank and Goodman’s one-shot reference games, listeners’ mean bets matched a parameter-free listener inverting a speaker who chooses among true utterances by informativeness, with a separately measured salience prior, and not salience alone; and in Bergen, Goodman and Levy’s signalling games with novel symbols, listeners read a signal as excluding what the available alternative would have conveyed | Frank & Goodman (2012), LIT-821, read in NOTE-634: a parameter-free Bayesian listener inverting a size-principle speaker, with a separately measured salience prior, matches mean listener bets over seven context types at r = .99 (.87 without the 0 and 100 predictions), while salience alone correlates at r = .19. Bergen, Goodman & Levy (2012), LIT-809, read in NOTE-613: with novel signals and no conventions, listeners read the signal as the object its alternative did not name, and lexical uncertainty derives Horn’s mapping. Goodman & Frank (2016), LIT-816, states the framework. Shown for artificial games with given alternatives, by fits of condition means with no rival fitted; a test against named rivals on trial-level data is QUESTION-209. | Active |
| THEORY-174 v3 | Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios’ model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it | Karvonen (2018), LIT-846; Abramsky, Barbosa, Karvonen & Mansfield (2019), LIT-847; Barbosa, Karvonen & Mansfield (2021, v2 2024), LIT-845. A simulation answers each measurement of the target by a jointly measurable set of the source’s, with shared classical randomness, so the cover may change. Along any simulation the non-contextual fraction cannot fall; contextual models cannot be cloned; and which maps between model sets are classical non-adaptive procedures is a non-contextuality question on a hom scenario, re-derived here. No-signalling models only, and nothing about information preserved. | Active |
| THEORY-175 v3 | Cosine similarities to matched anchor points are unchanged by exactly the maps that preserve each sample’s cosines to the anchors, among them every orthogonal map times a positive scalar and every per-sample positive rescaling but not translations; in Moschella et al.’s experiments independently trained encoders agreed in them only approximately, enough for a decoder reading them to transfer untrained between seeds, architectures and languages with loss, while their nearest neighbours mostly differed | Moschella et al. (2022; ICLR 2023), LIT-849, as read in NOTE-652. The invariance is by construction; its class (the maps preserving each sample’s cosines to the anchors, which include every orthogonal map times a positive scalar and every per-sample positive rescaling, but not translations) is the reader’s statement, re-derived in a dated note. The approximate agreement is the paper’s evidence: relative vectors match at cosine 0.86–0.97 across encoders, while ten-nearest-neighbour overlap is 0.10–0.39. It does not say the maps between trained spaces are isometries, which the source never measures (QUESTION-211), nor that the transfer needs no paired data: the anchors are paired data. | Active |
| THEORY-176 v2 | A task-loss stitching penalty compares representations in one decision problem, through a fixed top network and a restricted map class, so it is not Blackwell’s order: representations that are functions of each other can stitch differently, and a negative penalty shows the fitted map did not reproduce the layers it replaced | Bansal, Nakkiran & Barak (2021), LIT-850, define the stitching penalty L_ℓ(r; A) − L(A), with L_ℓ(r; A) the least task loss of A’s top layers fed r through a map from a simple family S, and use it to call one representation better than another. Read against Blackwell (1953), LIT-781, and THEORY-156, that ordering is one coordinate of the comparison of experiments with the decision rules cut down to A’s head composed with S. It is not invariant under Blackwell equivalence, which the paper intends, and it meets Blackwell’s order only where the fitted map reproduces the replaced layers exactly. The relations are this record’s derivations from the paper’s definitions, not the paper’s. | Active |
| THEORY-177 v2 | In a binary cyclic system, the Contextuality-by-Default measure of signalling is twice the sum, over contents, of the least direct influence of context that a canonical causal model of the data must place on each content, and each content’s least influence is the total-variation distance between its two marginals | Wang, Sadrzadeh, Abramsky & Cervantes (2021), LIT-851, Proposition 1. It joins CbD’s maximal couplings (LIT-777, Theorem 3.3) to Jones’s 2019 correspondence between couplings and canonical models, which the paper quotes and the record derives independently. The result measures signalling. Its minima are per content, each reached in its own model; one model reaches them all only if the system is noncontextual. It says nothing about contextuality beyond the bound that a rank-2 system with Δ ≥ 2 cannot be contextual, and identifies no particular causal model. | Active |
| THEORY-178 v3 | In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit | Zhao, Wytock, Reynolds and Motter (2024), LIT-854. In an ensemble of sign-consistent Boolean rules on the 87-gene core of the E. coli regulatory network, 51 genes admit a transient knockout or overexpression that moves the network to a different attractor, and every such gene reaches a strongly connected component with a positive circuit. So one genotype may hold several heritable expression states without epigenetic marks, in the model. The evidence is a model ensemble, read in full in NOTE-655, and the title claims no more. No transient perturbation has been shown to have a lasting effect in cells; that experiment is QUESTION-172. | Active |
| THEORY-179 v2 | In Bhattacharyya et al.’s single-cell swarm assays of E. coli MG1655, the daughters of one mother swarmed alike at the fourth generation sampled and varied again at the seventh; swarm size correlated with a plasmid fepA-promoter reporter of Fur derepression by low iron (Spearman r = 0.71), and an iron chelator or added iron shifted swarming and held the shared state longer; a three-class switching model fitted to two time points matched the lineages only with a delay of three to four generations before switching, which the authors read as ruling out a bistable switch | Bhattacharyya et al. (2023), LIT-853. Single-cell swarm assays show sibling homogeneity in swarming at four generations and its loss by seven, a correlation (Spearman r = 0.71) between a Fur-repressed reporter and swarming, and longer persistence when iron is clamped. The iron link is read through a promoter reporter and is correlational, the inheritance rests on few founders and three time points, and the paper’s “memory”, “conditioning” and “decision-making” are functional labels, not evidence of learning. | Active |
| THEORY-181 v3 | In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely | Liu et al. (2022), LIT-858, in a toy built for it: same-answer training pairs force equal embedding sums, equal sums carry answers to unseen pairs, and for addition with p = 10 the fraction at which the forced equations first pin the embedding to a + kb (about 0.4) matches the onset of generalisation and Power et al.’s critical fraction (LIT-341). It is shown for one size, 1D embeddings and an operation built into the architecture. It is not shown that larger networks generalise by this mechanism, and for S₃ the networks generalise beyond it. | Active |
| THEORY-184 v2 | Under a permutation-equivariant update whose map is K-Lipschitz, coincident neurons stay coincident at every step size, and at step size below 1/K distinct neurons cannot merge in finitely many steps | Yang, Poggio, Chuang and Ziyin (2025), LIT-861, read in NOTE-663. If an update commutes with permuting neurons (GD, SGD or Adam on a permutation-symmetric loss) and its map is K-Lipschitz, one step of size η changes every pairwise neuron distance by a factor in [1 − ηK, 1 + ηK]. This is a proved constraint on trajectories. It does not say that training simplifies above 1/K, that neurons never merge in the limit, or that any scale-dependent shape of the neuron cloud is preserved. | Active |
| THEORY-185 v4 | When each binary attribute of a word affects its co-occurrence independently and multiplicatively, the PMI is affine in the attributes with rank at most d + 1, so a spectral PMI embedding is a linear image of the attribute hypercube and parallelogram analogies hold exactly; over the full hypercube of words, each attribute’s two values equally frequent, the raw co-occurrence ratio mixes in products of attributes, and keeps exact analogies only when the strengths are alike enough that every attribute mode outranks every product mode | Korchinski, Karkada, Bahri and Wyart (2025), LIT-863: proved for a generative model of co-occurrence in which words are bundles of independent binary attributes, with robustness to noise, pruning and deleted pairs argued from eigenvalue bounds and simulated. The embeddings are eigendecompositions, not trained models; that real co-occurrence is close to this form is inferred from spectral and accuracy-curve shapes, not measured. | Active |
| THEORY-187 v3 | In a deep linear network trained by gradient descent on whitened inputs from decoupled, balanced initial weights, depth slows learning through the initial end-to-end strength of each mode: counting iterations as learning time over the stable step size, from strength of order one the number of iterations stays bounded as depth grows, once the step size falls with depth as stability requires, while from small strength ε the plateau lengthens from ln(1/ε) with one hidden layer towards 1/ε at great depth | Saxe, McClelland and Ganguli (2014), LIT-864, Eqs. 13–17 and Appendix B: exact for decoupled initial weights and whitened inputs, with the step size bounded by the Hessian at the optimum, and supported on MNIST from decoupled starts up to 100 layers. It does not show that random orthogonal initialisation is such a start. The paper’s account of that, isometry of the product of layers, is argued and simulated, not derived. It says nothing about nonlinear networks’ learning time, and it is not advice on how to initialise. | Active |
| THEORY-189 v2 | Nested angular cones in the Poincaré ball, symmetric about the outward ray with an aperture depending only on the apex’s radius, are transitive only if the aperture is at most π/2 and r sin ψ(r)/(1 − r²) does not increase outward; so no such cone can be defined near the origin, the widest admissible family is sin ψ = K(1 − r²)/r, and every point a cone contains lies farther from the origin than its apex | Ganea, Bécigneul and Hofmann (2018), LIT-866: Lemma 2 and Theorem 3 prove the necessity half under four stated requirements on the cones; the transitivity of the closed-form family (Theorem 4), which the source asserts without proof, is proved in this record’s note of 2026-10-19; the radial monotonicity is this record’s derivation from the hyperbolic law of cosines. It says what a cone order on the ball can be, not which posets embed in it, and nothing about meets, joins or learned embeddings. | Active |
| THEORY-190 v3 | Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes’ link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form | García-Pérez, Boguñá and Serrano (2017), LIT-869: derived to first order, checked on one synthetic network over eight layers and on six real ones. Supernode hidden degree is the ℓ^β sum of its members’; β is invariant. The record proves the closure at large separation, with relative error of order β times the block width over the separation, and computes that between neighbouring supernodes the union of member links exceeds the S¹ form (by 32–86% in a worked example with two-node blocks). The source’s flow ⟨k⟩′ = r^ν⟨k⟩, with ν = 2/(γ−1) − 1 or 2/β − 1, rests on the closure at every separation and is QUESTION-203. It does not show that real networks are generated by a hidden metric space, or that the nested layers are a hierarchy found in the data. | Active |
| THEORY-191 v2 | In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball | Zhou, Smith & Sharpee (2018), LIT-872: for strawberry, tomato, blueberry and mouse-urine volatiles, and for Dravnieks’ perceptual ratings of 127 odorants, clique-topology Betti curves match a 3D hyperbolic model and reject uniform Euclidean cubes of every dimension tried. It does not show a tree in the data, and because the fitted odour model is a thin shell, it does not separate a hyperbolic metric from a sphere in flat space. | Active |
| THEORY-192 v2 | In twelve sessions of rat dorsal CA1 from three data sets, pairwise correlations among place cells have the Betti-curve signature of points sampled uniformly from a three-dimensional hyperbolic ball, not of a low-dimensional Euclidean cube, and the fitted radius rose with the logarithm of exploration time | Zhang, Rich, Lee & Sharpee (2022), LIT-876: in twelve sessions from three data sets, the rank order of CA1 spike correlations matched uniform samples from a 3D hyperbolic ball and not uniform samples from low-dimensional Euclidean cubes or shuffled data, and the ball’s radius rose with log exploration time from seconds to days. It does not show that the metric, as opposed to the distribution of field scales, is hyperbolic, nor that a tree is represented. | Active |
| THEORY-193 v2 | A simplicial complex grown by gluing each new d-simplex to one (d − 1)-face, by a rule that uses no geometry, has a 1-skeleton whose geodesic triangles are 2-thin at every size, so hyperbolic network geometry in Gromov’s sense is an outcome of the combinatorial growth; the equal-link drawing as ideal simplices is a convention, a tiling of the hyperbolic plane for d = 2 in the manifold flavor but not a tiling of hyperbolic d-space for d ≥ 3 | Bianconi and Rahmede (2017), LIT-867, read in NOTE-678, for the model, the degree law and the ideal-simplex drawing. The coarse hyperbolicity of the graph is the record’s: the source asserts the hyperbolic geometry through its drawing and measures nothing on the graph. Every such complex has a 1-skeleton with 2-thin geodesic triangles, whatever the flavor, so its hyperbolicity is a property of the network. The constant-curvature picture, ideal simplices in the Poincaré ball, is a choice of drawing. It tiles H² for d = 2 in the manifold flavor, and for d ≥ 3 it is not a tiling. This says nothing about whether real networks grow this way. | Active |
| THEORY-194 v5 | On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead | Koch-Janusz and Ringel (2017), LIT-873: shown numerically on two 2D lattice models whose relevant variables were known (Ising block spins, dimer electric fields), with an iterated Ising flow giving T_c to about 1% and ν ≈ 1.0 ± 0.15, and analytically only for 1D and quasi-1D chains. The general equivalence of the objective with a short-ranged effective Hamiltonian is argued, not proved, and no system with unknown relevant variables was tried. | Active |
| THEORY-195 v3 | On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class | Cagnetta, Petrini, Tomasini, Favero and Wyart (2023), LIT-877: measured for depth-(L + 1) CNNs and deep fully connected networks on the Random Hierarchy Model with s = 2 and L ≤ 4, with a large-v scaling argument for the correlation threshold and an exact one-step-gradient model of a two-layer network on one patch. The law n_c m^L is fitted, not proved for trained deep networks, and in this model the correlation threshold and the grammar’s size are the same expression, so the data cannot say which of the two sets it. | Active |
| THEORY-197 v4 | Mehta and Schwab’s correspondence between Kadanoff’s variational renormalization group and restricted Boltzmann machines is an identity of parametrizations, under which an exact RG step is a perfect fit of the data distribution; away from that point it does not make training by distribution fitting a renormalization | Mehta and Schwab (2014), LIT-882, Eqs. 18–22: setting the RG kernel T = −E + H makes the RG coarse Hamiltonian equal the RBM’s hidden-marginal Hamiltonian for every parameter value, and Kadanoff’s exactness condition equal to zero KL divergence. Short of exactness the paper itself says the two use different variational schemes, and Koch-Janusz and Ringel (LIT-873) give a case where distribution fitting keeps the wrong variables. The free-energy part of the argument was derived independently here; Lin, Tegmark and Rolnick’s v2 appendix makes it in substance (LIT-887). | Active |
| THEORY-198 v2 | For a finite-range lattice Hamiltonian, a block coarse-graining that keeps all the mutual information the block shares with the system beyond a buffer makes the coarse measure factorize across the block, in 1D and, in D dimensions, when capture by a hyperplane of blocks amounts to capture by each block; barring exact cancellation of existing couplings, the renormalized Hamiltonian then gains no range, and in 1D product disorder gains no correlations across the block. The coarse-grainings actually optimized fall short of that condition, and for them the link is computed only for two-spin blocks of 1D Ising chains | Lenggenhager, Gökmen, Ringel, Huber and Koch-Janusz (2018), LIT-881: proved in 1D, and in D dimensions under an extra per-block assumption and barring fine-tuned cancellations. The full-capture condition is not met in the paper’s own worked case (decimation of two-spin Ising blocks keeps about half the information), and the decay of longer-range, many-spin and disorder-correlation terms as more information is kept is shown perturbatively for two-spin blocks of 1D chains only, and is monotonic only locally. | Active |
| THEORY-202 v3 | In variational renormalization over general Boltzmann machines, neither a matched partition function nor Kadanoff’s exact trace condition makes a coarse-graining a renormalization: the first holds for joint Hamiltonians whose marginal is not the data, the second for hidden variables that do not interact with the system, so the variables a renormalization keeps must be fixed by a further stated condition | Lin and Tegmark (2016), LIT-887, arXiv v1–v2 appendix, a part dropped from the journal text: a family of joint Hamiltonians with Z_tot = Z and the wrong marginal, and the remark that any non-interacting pair meets the trace condition. Under Mehta and Schwab’s identification (LIT-882) the trace condition is a perfect fit of the data, so even an exact variational RG step leaves the coarse variables’ meaning free. The restricted Boltzmann machine proper needs visible–visible terms for the second construction; that narrows the example, not the claim. Promoted on 2026-10-20 with both constructions re-derived. Whether the trace condition suffices inside Kadanoff’s kernel families is QUESTION-205. | Active |
| THEORY-204 v3 | On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses | Cagnetta, Raventós, Ganguli and Wyart (2026), LIT-888, building on LIT-883’s effective context window. Under assumed scaling forms for the conditional entropy and for each lag’s learning curve, the exponent is min(δ, γ/(2β)) (re-derived 2026-10-20). Measured on TinyStories and WikiText-103, with transformers up to about 10^8 tokens and horizons of tens of tokens, the fitted exponent overlaps γ/(2β), best near the start of the envelope; γ is estimated from trained models’ own n-gram losses. Whether the horizon is why is open (QUESTION-233). | Active |
| THEORY-205 v2 | For a short-range lattice model on a cylinder with a symmetric transfer matrix whose leading excited eigenvalue is non-degenerate, with block and environment as slabs L_B transfer steps apart, the uninformative information-bottleneck encoder first loses stability at β = (λ0/λ1)^(2L_B), in the direction of the boundary weak value of the leading excited eigenvector, which by the cited conformal finite-size relation is the lowest-dimension primary at large circumference; the uninformative encoder can stop being optimal at a smaller β, and nothing is shown for blocks inside a buffer shell or for the fixed-alphabet RSMI optimum | Gordon, Banerjee, Koch-Janusz and Ringel (2020), LIT-885, Eqs. 3–5 and B2–C11, with one numerical check on a three-site critical Ising cylinder, reproduced exactly in NOTE-690. The theorem holds for slabs of a cylinder, at the point where the uninformative encoder first loses stability. The global first transition can come earlier, and when the two coincide is QUESTION-185. The paper’s “equivalence” of IB and RG relevance, and LIT-878’s use of it for RSMI filters on planar blocks, go beyond it. | Active |
| THEORY-206 v2 | For networks of smooth units without biases, one hidden layer needs exactly ∏(rᵢ+1) neurons to match the monomial x₁^r₁⋯xₙ^rₙ’s Taylor polynomial at the origin, and a deep network O(Σ log rᵢ), with the upper bounds holding for uniform approximation too, so the cost of flattening a polynomial grows exponentially with its degree in distinct variables and not with the number of inputs: products of many inputs separate depths, bounded-degree polynomials do not | Rolnick and Tegmark (2017), LIT-890, Theorems 4.1 to 4.3, extending Lin, Tegmark and Rolnick’s 2ⁿ for a product (LIT-887). Proved for Taylor matching at the origin with bias-free units; the uniform version’s lower bound rests on an unproved step. The flip side, that a polynomial of degree d with c monomials costs one layer at most c·2ᵈ however many inputs it has, is the edge: the separation does not reach the low-order polynomials LIT-887 calls natural. | Active |
| THEORY-211 v3 | In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise | Ziyin, Li & Ueda (LIT-1012, read in NOTE-714), with the gradient flow conservation law and the Itô term from Neural Mechanics (LIT-1013). If ℓ(u, w) = ℓ(λu, w/λ) for every sample, then under the SDE model of SGD d(‖u‖² − ‖w‖²)/dt = −T(uᵀC₁u − wᵀC₂w), with T = η/S. The noise has no component along the symmetry, but its dependence on the parameters produces a deterministic drift through Itô’s lemma. The drift points to the noise-balanced point, and weight decay adds to its rate. In a diagonal linear network the stationary distribution follows exactly: it is non-ergodic, has a memory of initialization, and collapses to zero above a critical temperature. The law is re-derived here (2026-10-19) within the SDE model; the tests are small synthetic networks, and whether it holds in deep networks on real data is QUESTION-164. | Active |
| THEORY-213 v3 | Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons | Ziyin (2024), LIT-1014, read in NOTE-716. If a per-sample loss is invariant under the reflection I − 2OOᵀ, then Oᵀθ = 0 is a stationary condition for GD and SGD. The Hessian is block diagonal there. Above a critical weight decay all minima satisfy the constraint. Under SGD with strong noise the subspace is attractive even at a strict saddle, by a Lyapunov-exponent argument. Rescaling, rotation and permutation symmetries imply mirrors, so they yield sparsity, low rank and merged neurons, with collapse and loss of plasticity as consequences. The SGD half is approximate and the network evidence is limited; the title holds only the exact half. Promoted on 2026-10-20 on the exact half, re-derived. The SGD half is QUESTION-206. | Active |
| THEORY-214 v2 | On the 2022 r/place canvas, the abrupt overthrow of a crowd-built image from a stable period is predicted up to hours ahead, on held-out images and on the 2023 edition, by a combination of many image-level series far better than by any generic slowing-down indicator; the series the predictor relies on include low past restoring of attacked pixels as well as attack outrunning restoration, and about a quarter of overthrows give no warning at the 20-minute range | Falmagne, Stephenson & Levin (2026), LIT-1025, read in NOTE-731. On about 10,900 compositions of 2022, a predictor of time-to-overthrow built from 19 composition-level series with a 7-hour memory catches half the overthrows within 20 minutes at 3.6% false positives, beats variance and autocorrelation trends, and transfers to 2023. Its precursors are many: attack outrunning defence, a turbulent history, little accepted change, uncoordinated activity, simple images, low past defence; slowing down and speeding up both occur. About a quarter of overthrows are unpredictable at the 20-minute range. The social readings of the precursors are the authors’ conjecture from SHAP curves; nothing here measures plans. | Active |
| THEORY-215 | A Hawkes process with a Lipschitz rate function and signed kernels has a unique stationary regime, which suitably started versions approach, when the Lipschitz-weighted absolute kernel mass is below one (a spectral radius below one, for a network) or when the rate is bounded | Brémaud and Massoulié (1996), LIT-1035, read in NOTE-728: Theorems 1, 2, 7 and 8, proved. Inhibition is allowed because the conditions use |h|. The conditions are sufficient, not necessary; the kernels are fixed in time; uniqueness is among solutions with finite mean intensity. In the linear excitatory case the condition is the branching ratio below one, but in the nonlinear signed case the spectral-radius quantity is a bound, not a branching ratio, and a bounded rate needs no bound on the kernels at all. Nothing is proved about cascade sizes or about what happens past the threshold. | Active |
| THEORY-216 v3 | On the 2017 r/place log, with coalitions recovered from co-placement, which coalitions survive is barely predictable early from their size, area, start time and colour entropy, the size distribution of recovered coalitions shifts toward large ones as the event proceeds, and some lasting artworks were held by coalitions that kept working under sustained overwriting | Wu & Silva (2024; v3 2026), LIT-1032, read in NOTE-724. On the 2017 r/place log, coalitions are recovered, failed ones included, from co-placement and image segmentation. Under 7% of updates survive to the end. Recovered coalition sizes shift toward large over the event. A decision tree on four state features predicts survival at F1 0.09 in the first 48 hours and 0.67 later. Illustrated successful coalitions keep working under overwriting throughout, and some failures look like successes until a late collapse. The recovery is validated only against final artworks, the lifecycles are ten examples, and nothing here observes plans. | Active |
| THEORY-218 v2 | In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated | Litherland & Mørch (2021), LIT-1037, read in NOTE-733. For the 2017 Mona Lisa, 76,774 accounts placed tiles in the region while a forum of 161 negotiated the design through linked blueprints. Central members admitted a Swiss flag and a rainbow frame and erased r/trees objects, and a tag on the image recruited newcomers. Whether an addition was vandalism or part of the image was settled by negotiation. The authors argue placement data cannot reveal cooperation, conflict or affiliation. One artwork; the link from forum power to placements is “unclear” by the authors’ account; their “instruction” is Reddit’s rules, not planners’ orders. | Active |
| THEORY-222 v5 | With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level | Hoel, Albantakis & Tononi (2013), LIT-1038, read in NOTE-726. Effective information is effectiveness (determinism minus degeneracy) times the log of the state-space size, measured under uniform interventions. Coarse-graining always loses size and can gain effectiveness; when the gain wins, causal emergence CE = EI(macro) − EI(micro) is positive. Four toy systems give 0.40 to 2.92 bits. The arithmetic is exact. The reading that the EI peak marks where causation happens rests on imposing a uniform distribution at each level, which the paper adopts and does not test; the title does not claim it. Promoted on 2026-10-20 on a construction. Whether the gain survives one intervention distribution at both levels is QUESTION-204. | Active |
| THEORY-227 v2 | Mutual information between an input and its representation does not fix the representation’s quality: invertible encoders all preserve it exactly, yet under a linear probe some do better than the raw input and some worse | Tschannen, Djolonga, Rubenstein, Gelly & Lucic (2020), LIT-247, read in NOTE-749. Invertible RealNVP encoders on MNIST halves keep the true MI constant, yet trained with an MI estimator they gain linear-probe accuracy, and one trained adversarially against a linear probe ends “significantly worse than using raw pixels”. At matched bound values the encoder architecture decides quality. What makes MI-trained representations good must therefore come from somewhere other than MI. The paper’s alternative, that InfoNCE works as metric learning, is argued, not tested. | Active |
| THEORY-228 v2 | On a finite augmentation space, at the population optimum of the spectral contrastive loss with k ≥ 2r, linear-probe error is bounded, up to constant and logarithmic factors, by the error of recovering the label from an augmentation divided by the squared sparsest-partition conductance of the augmentation graph | HaoChen, Wei, Gaidon & Ma (2021), LIT-249, read in NOTE-740. On a finite augmentation space, with labels recoverable from augmentations up to error α and k ≥ 2r, every population minimizer of the spectral contrastive loss has linear-probe error Õ(α/ρ²_⌊k/2⌋), where ρ is the sparsest-partition conductance of the augmentation graph (Thm 3.8). The guarantee needs no conditional independence of the views given the label. It is a statement about the population optimum of one loss: it says nothing about what training reaches, nothing about InfoNCE, and its infinite-space extension is asserted, not proved. | Active |
| THEORY-229 v3 | For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs’ symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ | Simon et al. (2023), LIT-242, read in NOTE-741. For L = ‖C − I‖²_F with λ = 1, no batch norm and C symmetrized by hand, a linear model from aligned initialization learns mode j at τ_j ≈ −log(s_j(0)²γ_j)/(8γ_j) (Prop. 4.1, proved), the top spectral solutions are exactly the minimum-norm zero-loss ones (Prop. 4.3), and the kernel version is kernel PCA on K_Γ (Prop. 5.1). Generic small initialization is covered by an informal derivation only. The operator is built from the model’s kernel and the sample pairs, not from the positive-pair density ratio, and the stepwise growth seen in ResNets under SimCLR and VICReg is an experiment, not a consequence. | Active |
| THEORY-230 v2 | The Fisher–Rao geometry of a language model’s next-token laws over a set of contexts is a function of those laws alone, so models with identical laws share it, while activation geometry changes under invertible reparameterizations of the hidden layer that leave outputs unchanged; in Picozzi’s ten transformer, state-space and recurrent models it agreed at a mean rank correlation of 0.88 against 0.62 for activation geometry, and about 9% of the top eigenvector overlap exceeded a null matched to the shared laws | Picozzi (2026), LIT-936, read in NOTE-755, a single-author v1 preprint with code on request. The matrix of Fisher–Rao distances between a model’s next-token laws on a context battery is invariant to every output-preserving reparameterization, while activation geometry has a GL(d) gauge (Prop. 1); two models with identical outputs can have hidden geometries that rank-correlate at −4/17 (Prop. 3). Over ten transformer, state-space and recurrent models the mean rank agreement is 0.88 against 0.62 for mid-layer activations. Convergence is proved only as a consequence of each model’s excess risk to a common law, and that bound is vacuous on real models. Against a law-matched null, about 9% of the top-six eigenvector overlap is shared beyond the laws. Promoted on 2026-10-20. Replication and a test that varies the law are QUESTION-216. | Active |
| THEORY-231 v3 | Within classical realism, a generalized-noncontextuality no-go refutes Leibnizian representations, not realist representations as such; it rests on a stronger assumption than a Bell-like no-go and so applies where none exists, as in prepare-measure scenarios, but only Leibnizianity implying noncontextuality is proved | Schmid, Selby and Spekkens (2020), LIT-945, §VI.C, §VIII (Eqs. 200, 214, 215), read in part in NOTE-737, which is Skimmed: §§II–IV and the proofs in Appendices C–G were not read. A Leibnizian classical realist representation implies a generalized-noncontextual one, so every noncontextuality no-go is a no-go for Leibnizian representations. The converse is Conjecture 1′, so a Leibnizianity failure is not shown to be a noncontextuality failure. The prepare-measure claim is stated, not derived, in the text read. | Active |
| THEORY-233 v2 | CKA and normalized Bures similarity are the Euclidean and the Bures comparison of normalized kernels, and each bounds the other only within a factor set by the kernels’ ranks, so agreement in one does not carry to the other with a dimension-free constant | Harvey, Larsen & Williams (2023), LIT-250, §5 and App. A.3, read in NOTE-753. CKA = 1 − ½‖K_X/‖K_X‖_F − K_Y/‖K_Y‖_F‖²_F, while NBS = 1 − ½B²(K_X/Tr K_X, K_Y/Tr K_Y), with B the Bures distance. They induce the same topology, but CKA/√(r(K_X)r(K_Y)) ≤ NBS² ≤ min[r(K_X), r(K_Y)]·CKA, saturated at rank 1. NBS bounds CKA of the square-root kernels, with equality for commuting kernels. Linear kernels only, and the discrepancy is illustrated on Wishart samples, not trained networks. | Active |
| THEORY-234 v2 | Two softmax next-token predictors induce the same distribution exactly when their embeddings and unembeddings are related by an invertible linear map on the subspaces that enter the logits, with arbitrary distortion off them, so a linear relational property is shared by all such models or none when it lies in those subspaces, and can fail to be shared otherwise | Marconato et al. (2024), LIT-950, read in NOTE-746. For p(y|s) ∝ exp(f(s)ᵀg(y)) on a shared vocabulary, p_{f,g} = p_{f̃,g̃} iff the models are extended-linearly equivalent: invertible linear maps between the subspaces M and N that enter the logits, at any dimensions and without the diversity condition (Thm 5). The dimension of M is then a property of the distribution. Relational linearity, linear subspaces and linear probing hold in all equivalent models or none when Γ ⊆ N and Γ_q ⊆ M (Thm 14), and parallelism is shared only within N (Thm 15). Exact equality of distributions only, and no experiments. | Active |
| THEORY-236 v2 | A representational similarity index that is invariant to every invertible linear map takes the same value on all representations whose rank equals the number of examples | Kornblith, Norouzi, Lee & Hinton (2019), LIT-254, Theorem 1, read in NOTE-743. If s(X, Z) = s(XA, Z) for every invertible A, and rank(X) = rank(Y) = n, then s(X, Z) = s(Y, Z): any two full-row-rank representations of the same n examples are related by an invertible linear map. So when width reaches the number of examples, an index with that invariance, such as CCA, says nothing. The theorem fixes no group; the paper’s case for orthogonal invariance plus isotropic scaling is an argument from training dynamics, not a consequence of it. | Active |
| THEORY-237 v3 | The critic-based variational lower bounds on mutual information, Donsker–Varadhan, NWJ, MINE’s and InfoNCE, are relaxations of one unnormalized Barber–Agakov bound, and, when the joint and the product of the marginals are mutually absolutely continuous, each attains its supremum over critics only at a critic equal to the log density ratio up to a term its normalization leaves free, where every bound but InfoNCE equals the mutual information | Poole, Ozair, van den Oord, Alemi & Tucker (2019), LIT-246, read in NOTE-739. Writing q(x|y) = p(x)e^f/Z(y) in Barber and Agakov’s bound gives I_UBA, tight at f = log p(y|x) + c(y). Jensen gives Donsker–Varadhan, a tangent bound on the log gives TUBA and NWJ (unique optimal critic 1 + log p(x|y)/p(x)), and NWJ applied to a K-sample problem with the batch mean as normalizer gives InfoNCE. MINE’s bias-corrected gradient is the exact gradient of TUBA. A Monte-Carlo DV estimate is neither an upper nor a lower bound. The InfoNCE optimum is cited, not derived, in the source (the record derives it, note of 2026-10-19), and the bias–variance comparisons are on toys. | Active |
| THEORY-238 v3 | The expected generalization error of an encoder–decoder learner is bounded by the encoder alone, through the KL divergence of its train-and-ghost latents to a prior symmetric under label-preserving permutations, for any decoder; and under lossy compression the bound can stay finite for deterministic continuous encoders, whose lossless bound is infinite | Sefidgaran, Zaidi & Krasnowski (2023), LIT-245, read in NOTE-742. Theorem 4: E[gen] ≤ inf 2√((2·MDL + K + 2)/n) + ε, where MDL is the expected KL of the joint law of the latents of the training set and an independent ghost set to any prior invariant under label-preserving permutations, and ε is the distortion allowed by lossy compression. It holds for any decoder. With a product prior it reduces to sample-wise mutual information. The bound is never evaluated, the prior may depend on unseen ghost data, and a 2√((K+2)/n) floor remains. It does not prove that I(X;U) cannot bound generalization. | Active |
| THEORY-239 v3 | For any joint law of an input and a context, symmetric or not, the top-d left singular functions of the conditional-expectation operator attain the minimum, over d-dimensional spans, of worst-case linear-probe approximation error over tasks of compatibility at least 1 − ε, namely (s₁² − (1 − ε)²)/(s₁² − s²_{d+1}) when s_{d+1} < 1 − ε ≤ s₁, though other spans can tie | Zhai (2025), LIT-975, read in NOTE-770: Lemma 1.4, Theorem 3.2 and Theorem 3.4 with App. B. The operator g ↦ E[g(A)|X] runs from L²(P_A) to L²(P_X) and is never assumed self-adjoint; its singular value decomposition comes from composing it with its adjoint. The guarantee is for linear probes and approximation error at the population level only. The thesis never works a non-reversible kernel on one space; the note of 2026-10-19 works a three-state one. It says nothing of what a trained network reaches. | Active |
| THEORY-242 v3 | A spiral of silence can make a small, fixed minority opinion look far more widespread than it is without any member changing belief, and in the model that shows it the network’s direct distortion of what members see is small | Vilone and Polizzi (2024), LIT-939, read in full in NOTE-810. In agent-based models on scale-free networks with fixed private beliefs, a hard-core minority drives 55–60% of agents, nearly all of them in the majority, into silence, so that expressed and perceived opinion lean far toward the minority. The friendship-paradox distortion is small and of no fixed sign. This is simulation only. Beliefs never change, so nothing here shows a community coming to believe the minority view, and the majority is overturned only when the minority exceeds about 30%. | Active |
| THEORY-243 v3 | In the three-unit identity network x ↦ x(1, 0, 1) ↦ (0, 2, 1)·h, aligning the input with the third unit and aligning it with the direction (h₁ + h₂)/√2, two alignments related by an orthogonal change of basis, pass every interchange intervention on the input equally, so interchange interventions on that variable alone do not fix which internal direction mediates it | Wu et al. (2024), LIT-961, Appendix A, and Makelov et al. (2023), LIT-968, Appendix A.3 and §8, read in NOTE-809 and NOTE-782. Both rotate the same three-unit identity network. The input aligns with h₃ in one basis and with (h₁ + h₂)/√2 in the other, equally well under interchange interventions on the input (re-derived 2026-10-20). The reply says intervening on further variables separates them; that was not checked. Makelov et al. rest their verdict on component boundaries; the reply rests its on the interventions performed. Shown exactly for one constructed network only; whether trained networks have such ties is open (QUESTION-234). | Active |
| THEORY-244 v2 | For a computable source distribution f, Shannon entropy and mutual information are the expectations of their Kolmogorov counterparts only to within the complexity of f: 0 ≤ E K(x) − H(f) ≤ K(f) + O(1), and I(X;Y) − K(f) − O(1) ≤ E I(x:y) ≤ I(X;Y) + 2K(f) + O(1); Grünwald and Vitányi’s matching statement for the distortion-rate function rests on a step the record has not verified | Grünwald & Vitányi (2004), LIT-224, read in NOTE-776. For computable f: 0 ≤ E_f K(x) − H(f) ≤ K(f) + O(1) (Thm 2.10), and I(X;Y) − K(f) < E I(x:y) < I(X;Y) + 2K(f) to O(1) (Thm 3.6), both re-derived by the record from domination of f by the universal distribution and the coding theorem; the entropy gap is attained by a point mass. The expected structure function is stated to bracket D_m(R) and to equal D(R) per symbol in the limit (Thm 6.27), but its right-hand step is asserted, so that bridge is not part of this account’s claim (QUESTION-208). Sufficiency links only one way (Thm 5.13). None of this holds for a complex or unknown source, or for one string. | Active |
| THEORY-245 v3 | In a preregistered multimodal test with clearly seen images, prefrontal cortex carried a seen image’s category but not its identity, and a face’s orientation at most weakly, while posterior cortex carried category, face orientation and the identity of objects, letters and false fonts, though not of faces; no prefrontal site showed the preregistered onset-and-offset response, and one, found in an exploratory analysis, responded at offset earlier than predicted | Cogitate Consortium, Ferrante et al. (2025), LIT-973, read in NOTE-812. In iEEG and fMRI, with Bayesian support for the nulls, and against a positive control, prefrontal cortex held category but no identity or orientation, and none of 655 prefrontal electrodes showed the onset-and-offset response GNWT predicted. It does not show these dimensions were unaccessed, or that prefrontal cortex lacks them in other signals: high gamma and BOLD only, no single units, and every image was seen. | Active |
| THEORY-246 v2 | Among ResNet-50s trained on ImageNet with nine objectives and stopped early, seven of which beat softmax cross-entropy’s top-1 by up to 0.9 points, fixed penultimate features with more class separation transfer worse by linear probe to datasets whose classes are mostly not ImageNet’s (Spearman ρ = −0.93 across the nine; mean accuracy 75.9% for softmax to 48.3% for squared error), and within cosine softmax a higher temperature raises separation and lowers that accuracy; the objectives’ representations differ mainly in the last third of the network, and full fine-tuning brings mean transfer to within 1.3 points across them | Kornblith, Chen, Lee & Norouzi (2021), LIT-1044, read in NOTE-762. Across nine objectives for ResNet-50 on ImageNet, penultimate class separation R² runs from 0.349 (softmax) to 0.845 (squared error), Papyan’s Tr(Σ_W Σ_B†)/C from 21.2 to 0.33, and mean linear transfer to eight new-class datasets falls with R² (Spearman ρ = −0.93), from 75.9% to 48.3%. Cosine-softmax temperature raises R² from 0.24 to 0.77 and lowers transfer from 76.0% to 58.9%. CKA puts the differences in the last third of the network; fine-tuning erases them. The cost is for new classes: relearning the training classes, more separation helps. Networks early-stopped near 92% training accuracy; one dataset; correlational across losses. Promoted 2026-10-20 on this one study (ADR-047 v3), narrowed to what it measured. | Active |
| THEORY-247 v2 | For finite action-free labelled Markov chains, the discounted bisimulation metric, defined as the unique fixed point of Ferns et al.’s equation, is a bicausal optimal-transport distance, and it equals the value of a finite linear program over occupancy couplings | Calo, Jonsson, Neu, Schwartz & Segovia-Aguas (2024), LIT-962, read in NOTE-757. For finite action-free chains with real labels r and discount γ, the bisimulation metric’s fixed-point equation is the Bellman equation of transport over bicausal couplings with cost (1−γ)|r(x)−r(y)|, so the two coincide. The transport cost equals 1/(1−γ) times the minimum of a linear objective over occupancy couplings, under flow and coherence constraints (Lemma 1, Theorem 1). It says nothing of systems with actions. Its link to trace distance is one-way and second-hand. | Active |
| THEORY-248 v2 | The expected generalisation error of a learning algorithm is at most √(2σ²R/n) + ε, where R is the least information about the sample carried by any surrogate output whose expected generalisation error is within ε; I(S;W) is the lossless case, and for an L-Lipschitz loss R is at most the rate-distortion function of the output’s law at distortion ε/(2L), so the bound is finite wherever that function is, including where I(S;W) is infinite | Sefidgaran, Gohari, Richard & Şimşekli (COLT 2022), LIT-236, read in NOTE-780. Theorem 4: |E gen| ≤ √(2σ²·R_E(ε)/n) + ε, where R_E(ε) is the least I(S;Ŵ) over surrogates within ε in expected generalisation error. For a Lipschitz loss R_E is bounded by the rate-distortion function of P_W (Corollary 6). Theorem 10 is a tail version over a KL ball. The block-coding meaning is proved for finite alphabets only. No bound is evaluated for a learned model, and every bound depends on the unknown data distribution. | Active |
| THEORY-253 v4 | In Liu, Liu, Michaud, Gore and Tegmark’s Geometry model of independent skills with orthogonal task vectors, simulation shows skills progressing together under gradient descent, the weaker taking about 1/p of the time, and in sequence under sign descent, a weaker skill progressing fast only once a stronger is done and the ratio of their times approaching 2; on a rotated quadratic the sequence appears under sign descent only when the loss’s eigendirections are misaligned with the coordinates | Liu, Liu, Michaud, Gore and Tegmark (2025), LIT-951, read in NOTE-794. The Geometry model has loss Σ p_i L((θ − θ₀)·t_i) with orthogonal t_i. In it, SGD learns task 2 in time ∝ 1/p₂, while SignGD holds task 2 back until task 1 is done and then takes a constant extra delay (t₂/t₁ → 2). A rotated quadratic shows the split only when its eigendirections are misaligned with the coordinates (Fig. 12). The record reproduced both. The order is frequency order either way. The evidence is simulation and one fitted ODE; a derivation and a trained-network test are QUESTION-193. Sequential onset caused by dependency between skills is a separate experiment, not part of this account. | Active |
| THEORY-255 v3 | In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice | Aerts (2009), LIT-934, Theorem 9 and §§2.2–3.3, read in full in NOTE-808. The impossibility is a proof, and it needs only that membership is monotone in the subspace. The fits that place the k-type deviations in a tilted subspace lattice and the ∆-type ones in an emergent sector are models with free parameters, not tests. Negation appears only as the orthocomplement used to derive the conjunction model from the disjunction model; no negated concept is measured. | Active |
| THEORY-256 v3 | When a sequence generator’s latent dynamics factor as a tensor product, its predictive state embeds losslessly and linearly in orthogonal per-factor subspaces of total dimension Σ(d_n − 1); 4-layer GPT-2-style transformers trained by next-token prediction on five Mess3 and Bloch Walk factors settle near that dimension, in per-factor subspaces that are linearly independent and approximately orthogonal in their leading components, and under the token noise tested reach it first even when noise makes it lossy | Shai et al. (2026), LIT-986, read in NOTE-758. The lossless embedding is an elementary theorem (Prop. 2.2, Thm 2.3, App. A). The rest is experiment on five synthetic factors: about 10 effective dimensions against 242 for the joint state, approximate orthogonality in the first two components, and under token noise a collapse to the factored form before expansion. It does not show that the preference is architectural rather than a feature of training, a question the paper leaves open, and it varies neither the objective nor the data domain. | Active |
| THEORY-259 v2 | For a recursive family of likelihoods and the universal prior, the Bayesian two-part code length −log Pr(D|H) − log m(H) equals the ideal-MDL length K(D|H) + K(H) plus the randomness deficiency of D for H, up to a constant fixed by the family; so ideal MDL and MAP selection agree to within a constant among hypotheses for which the data are random, and can part where they are not | Vitányi & Li (1999/2000), LIT-225, read in NOTE-801. With the universal prior m and a recursive family Pr(·|·), −log Pr(D|H) − log m(H) = K(D|H) + K(H) + d(D|H) + O(1), where d(D|H) = log(m(D|H)/Pr(D|H)) is the universal randomness deficiency of D for H; d ≥ −K(Pr(·|·)) − O(1), and d is bounded exactly when D is Martin-Löf random for H. So ideal MDL and MAP agree to within a constant on admissible hypotheses (Theorems 3–5, re-derived by the record) and part where the data are not random for the hypothesis (0ⁿ against the fair coin). The paper’s slack α = K(Pr(·|H)) + K(P) is an upper bound, not the gap. | Active |
| THEORY-260 v3 | An individual string’s information splits non-trivially into structure and noise only in a model class without a universal member: with partial recursive models every string’s minimal sufficient statistic is trivial, while with total recursive functions it is not, and the finite-set, computable-distribution and total-function classes agree to within logarithmic terms | Vitányi (2001/2006), LIT-229, read in NOTE-790. Sophistication, the shortest model that completes an optimal two-part code, is =⁺ 0 for every string if partial recursive models are allowed (Lemma 6.2). With total recursive models it ranges from about 0 for random strings to above n − log n − 2 log log n for some string of every length (Theorem 6.5), and it agrees with the finite-set and probabilistic versions to within O(log n) (Lemmas 7.1–7.2). It is not computable, and a sufficient-statistic oracle would compute K (Theorems 8.1, 8.3). It says nothing about any computable or learned model. | Active |
| THEORY-261 v2 | In sparse distributed memory, recall of uniform random data from exact cues is a signal-to-noise problem whose optimum activates about (2MT)^-1/3 of the hard locations, and capacity then grows in proportion to the number of hard locations, rising toward about a tenth of them at 0.999 bit fidelity, not with the word length | Kanerva (1992), LIT-921, §§3.5–3.7, read in full in NOTE-779, after Jaeckel (1989a). An approximate derivation for uniform random data and exact retrieval cues. It does not cover noisy cues, which lower capacity, or clustered data, which the uniform design handles badly, and it does not contain the critical-distance analysis that is sometimes attributed to it. | Active |
| THEORY-263 v2 | A coding rate that varies with the realised sample and output makes rate-distortion generalisation bounds data-dependent, and with the rate set to the pointwise information density log dP_{W|S}/dP(W) against a fixed prior the bound is the disintegrated PAC-Bayes bound | Sefidgaran & Zaidi (2023; IEEE Trans. Inf. Theory 2024), LIT-233, read in NOTE-806. Theorem 1: with a sample-dependent rate R_{S,W}, gen ≤ √(4σ²(R_{S,W} + log(√(2n)/δ))/(2n−1) + ε). With R_{S,W} = log dP_{W|S}/dP(W) the coding condition holds at ε = 0 and the bound is the disintegrated PAC-Bayes bound. The general Theorems 3–5 recover Rivasplata et al.’s PAC-Bayes bound and Xu–Raginsky’s. The coding construction is proved for finite alphabets only. The general proofs are change of measure. The rate is a Shannon code-length difference against a prior, not Kolmogorov complexity. | Active |
| THEORY-265 v2 | In Tissot et al.’s minimal model of bistable deciders in a modular coordination game (thirty modules of ten, couplings 1 within modules and 0.01 across), phase synchronisation within modules, driven only by each unit’s local payoff, makes each module respond to the others as a single unit, and synchrony matched to the modules is needed: global synchrony does not | Tissot, Levin, Buckley & Watson (2024), LIT-989, read in full (v1 and v2) in NOTE-773. In a simulated game of 30 modules of 10 units, each unit shifts the phase of its decision cycle toward partners it currently agrees with. Phases synchronise within modules, each module then switches as a block, and 50 of 50 runs with the learned phases reach a global optimum against 0 of 50 without; imposed global synchrony does not help. It is one clean game, one parameter setting, and a postulated phase rule; the modules are built in, and timing finds them rather than drawing them. Re-implemented by the sixteenth status review: learned phases, and module-matched phases imposed by hand, reach the optimum; random, shuffled and global phases do not make modules respond as units. The effect matters only where modules are isolated enough to lock in. | Active |
| THEORY-266 v2 | For a loss in [0, 1] that is L-Lipschitz in a distortion, with probability at least 1 − δ the average generalisation error along an optimisation trajectory is bounded, under a square root, by the rate-distortion function of the trajectory’s law given the sample, a coupling term that carries the trajectory’s dependence on the sample, and a distortion term 4Lε | Sefidgaran & Zaidi (2023), LIT-233, Theorem 7, read in NOTE-806. For an L-Lipschitz loss in [0, 1], with probability ≥ 1 − δ, the mean of gen(S, W^t) over iterations is at most √((RD(S, ε; π_S) + log(√(2n)M/δ))/(2n − 1) + 4Lε). RD is computed for the sample at hand. The coupling coefficient M depends on the unknown distribution and absorbs I(W^T; S). The CIFAR-10 experiment drops M and 4Lε and shows only that the estimated rate and the test gap fall together over learning rates. For FCN4 the plotted rate term is about 3, which is vacuous for 0–1 loss. | Active |
| THEORY-267 v3 | In Rasooly and Rozzi’s pre-registered randomised experiment on 817 Manifold play-money markets, about 59% of a ±5-point price shock that no trader’s information supported was still present after 60 days in unresolved markets, with about 25% undone in the first week and reversion slowing after it; their model of finitely many risk-averse traders gives such partial, slowing reversion even with no learning, and the data cannot tell that from learning | Rasooly and Rozzi (2025), LIT-958, read in full in NOTE-778. A pre-registered field experiment on 817 Manifold markets found about 59% of a random ±5-point shock still present after 60 days in unresolved markets. Reversion was 25% in week one and 6% more over the next three weeks. The paper’s model gets partial persistence from finite, risk-averse traders even with no learning. The data cannot say whether traders came to believe the shocked price. Its evidence that more traders or an outside estimate reduce persistence is correlational and weak. | Active |
| THEORY-269 v2 | In OLMo-1B, OLMo-7B and GPT-J, how well an affine map from subject to object representation reproduces a factual relation (its LRE causality) tracks how often the relation’s subjects and objects co-occur in pretraining, with mean causality above 0.9 beyond about 4.4k, 2.0k and 1.1k co-occurrences respectively, and, across OLMo-7B’s eight checkpoints, at whatever step that count is reached | Merullo, Smith, Wiegreffe and Elazar (2025), LIT-972, read in full in NOTE-786. 24 factual relations, three models, eight OLMo-7B checkpoints; r = 0.82 between LRE causality and mean subject–object co-occurrence, above either term’s frequency alone. Correlational by the authors’ own statement. The linear objects are per-relation affine maps between hidden states, not attribute directions. | Active |
| THEORY-270 v3 | In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples | Goldwasser, Gruber, Kalai and Paradise (NeurIPS 2023), LIT-941, read in NOTE-804. Proved for maximum likelihood in stylised random models with uniform languages and one-to-one translators; in one model the bound O(log|Θ| / (α·min(m,|T|))) is tight up to a constant. Information- theoretic only. The prior substitutes for parallel data in learning the decoder, not for information in a message. | Active |
| THEORY-271 v2 | In a toy subject–verb–object corpus whose eight subjects carry every combination of three binary aspects, the subjects’ raw co-occurrence rows form a parallelepiped if and only if, for each object, its four sentences are equally probable; any other sentence distribution breaks it | Torii, Maeda and Hidaka (2024), LIT-977, Theorem 1 (§5.5), read in full in NOTE-781. Necessity is sound and short. The case analysis over permutations, and sufficiency, are loosely written. Exact shapes, raw counts, one design, and independent aspects that make the concept lattice Boolean. It says nothing about the logarithm or PMI, about noisy or approximate shapes, or about attributes that imply one another. | Active |
| THEORY-272 v3 | In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable | Hopfield (1982), LIT-918, read in full in NOTE-819. The energy descent is derived for symmetric couplings; everything quantitative, including the 0.15N figure, the basin width and the merging distance, is simulation at N = 30 and 100. It does not say that spurious states come only from overload: they appear at a low load too. Nor does it say anything about addressed memory. | Active |
| THEORY-275 v5 | In two optically trapped beads with a symmetric hydrodynamic coupling, one trap moved by a recorded noise drive, time-delayed mutual information and transfer entropy follow the drive’s path rather than the coupling’s symmetry: the flow between the beads reads two-way under moderate drive, in experiment and numerics, and one-way under strong drive, in numerics of the fitted model only | Das, Manikandan and Banerjee (2025), LIT-947, read in NOTE-788. Two colloids are coupled by the fluid through a symmetric tensor, and recorded noise moves one trap. Time-delayed mutual information and transfer entropy give drive → driven → other, with backflow at moderate drive. In numerics only, flow is one-way at strong drive. With no drive the coupled particles are independent at steady state. The paper does not test an unrecorded drive shared by both units, and it does not claim that the measures reveal coupling architecture. Whether the same holds in other reciprocally coupled systems is QUESTION-186. | Active |
| THEORY-276 v2 | In a network of identical units with identical couplings, the orbits of the graph’s symmetry group are synchronization clusters, and a cluster moved only by its own factor of the group can lose synchrony while the others keep it | Pecora et al. (2014), LIT-960, read in NOTE-775: eq. (1), the IRR block-diagonalization (eqs. 2–4, 7), the derivation of isolated desynchronization (eq. 6) and an 11-node electro-optic experiment. The pattern of clusters that can synchronize is read from the wiring; whether each does is set by the dynamics through its own transverse block. Symmetry is sufficient, not necessary, for clusters. The paper says nothing about external drive, so the use of these clusters as a sign of coupling rather than a shared cue is the record’s inference. | Active |
| THEORY-278 v3 | Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse | Han, Papyan & Donoho (2021), LIT-974, read in NOTE-783. MSE loss splits exactly into a least-squares term and a non-negative deviation term, and the deviation becomes negligible early in training in 15 dataset–network cells. On that central path, with λ = 0, the loss is ½ Σ 1/(ω_j² + C) in the singular values of Σ_W^(−1/2)M̃. Under a flow that re-whitens the features after every step, each ω_j grows as (4t/N)^(1/4), and their ratios tend to 1. That is NC1 and NC2, and NC3–NC4 follow. Not shown: that ordinary training is this flow, or anything for cross-entropy or with weight decay. | Active |
| THEORY-280 v2 | The bisimulation metric between two finite Markov chains with observed states and a known cost bounded by 1 can be estimated without a transition model from independent samples of each chain’s discounted state-pair occupancy, to accuracy ε with probability 1 − δ from a number of samples polynomial in the state-space sizes, 1/(1−γ), log(1/δ) and 1/ε² | Calo, Jonsson, Neu, Schwartz & Segovia-Aguas (2025), LIT-948, read in NOTE-763. Bicausal occupancy couplings are characterised by constraints that use only each chain’s sampled occupancy measure over consecutive state pairs. A stochastic primal-dual method on them gets an ε-accurate estimate with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) independent samples per chain. The states must be observed and enumerable, the cost known, and the samples drawn with restarts. A single running stream and latent states are outside it. | Active |
| THEORY-281 v2 | Perfect concept erasure caps what a representation can keep at the information not about the concept: for X of finite support, if Z is independent of A then I(Z;X) is at most H(X|A), and a deterministic erasure reaches that cap only when the concept groups’ distributions are permutations of one another | Chowdhury et al. (2025), LIT-963, read in full in NOTE-820. The cap (Lemma 3.3) is a two-line chain-rule argument that needs only the Markov condition A → X → Z. The permutation condition is proved necessary and sufficient for deterministic maps (Appendices A.3 and A.4), with finite supports and disjoint groups. The paper’s further statement, that the cap cannot be reached when the groups are not permutations, is false for stochastic maps, which reach it for any finite distributions with disjoint supports. | Active |
| THEORY-282 v2 | A subspace intervention can reproduce the output effect of changing a variable without localizing it, by pairing a direction that varies with the variable but cannot reach the output with one that can reach the output but does not vary on the data | Makelov, Lange and Nanda (2023), LIT-968, read in NOTE-782. Proved for the construction (§3.2, Lemma A.1); shown in GPT-2 Small’s MLP8 on IOI (46.7% FLDD, falling to 13.5% without the null-space component, 4.2% interchange accuracy) and in GPT2-XL MLPs on factual recall, where a residual-stream direction passes the same test. It does not say that subspace patching is unreliable in general, nor that a found direction with a null-space component is spurious: in a one-dimensional readout the test passes both kinds. The reply (LIT-961) contests the label, not the measurements. | Active |
| THEORY-283 v2 | For a hierarchical multi-index target with isotropic Gaussian inputs and even, separable links, direction k becomes weakly recoverable by the Bayes-optimal estimator at a sample count proportional to its squared weight to the power minus one, so features emerge one at a time through sharp spectral transitions in the number of samples; a target-agnostic spectral estimator attains the thresholds and the optimal weighted error, and a two-layer network whose first layer is that estimator, with a ridge readout, attains them in excess risk | Defilippis, Krzakala, Loureiro and Maillard (2026), LIT-940, read in NOTE-805. Take x ~ N(0, I/d), n = αd, and f⋆ = Σ a_k g_k(⟨w_k, x⟩) with even links of generative exponent two and a_k ~ k^(−γ). Direction k is weakly recoverable at α_k = Θ(a_k^(−2)) (Thm. 3.2). The optimal weighted error is Θ(α^(−1+1/(2γ))) until α ≍ m⋆^(2γ). The spectral estimator on Σ T(y_i)x_ix_iᵀ attains both (Thm. 3.6). The cascade is in the number of samples, not in training time. The network result is for a spectral first layer plus a ridge readout, not gradient descent. | Active |
| THEORY-284 v2 | If a feature’s values form a compact metric space carried continuously and injectively onto the unit sphere, and cosine similarity is locally a C² decreasing function of squared feature distance, then every path on the representation manifold is √(−2g′(0)) times as long as the feature path it images, so on-manifold shortest paths recover the feature’s intrinsic length metric up to one scale; a homeomorphism alone does not fix it | Modell, Rubin-Delanchy & Whiteley (2025), LIT-935, Theorem 1, read in NOTE-802. Under Hypothesis 1 (continuous injective ϕ_f from a compact feature space to the sphere) and Hypothesis 2 (CosSim = g(d²) locally, g ∈ C², g′(0) < 0), path lengths scale by √(−2g′(0)). So geodesics on the representation measure the feature, without knowing g. Both hypotheses are assumed, not derived from training. GPT-2’s years are homeomorphic to both a linear and a logarithmic scale, and only the metric test told them apart. | Active |
| THEORY-285 v2 | In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored | Linson, Anderson, Markham, Friston & Moutoussis (2025), LIT-983, read in version 2 in NOTE-766. A mechanism shown in illustrative simulation: policy selection by expected information gain, plus a confident prior, withholds the only disambiguating observations, so the prior keeps itself up. Exposure that restores sampling rebalances it; a precision-lowering drug only suppresses symptoms. No data are fitted, parameters are unreported, and the two intervention arms differ in environment as well as mechanism. | Active |
| THEORY-289 v2 | For ordinary least squares on T ≥ d + 2 isotropic Gaussian covariates reused every round, with each round’s labels generated by the previous fit, keeping every earlier round’s data keeps the expected excess test error below π²/6 times the first fit’s, though it rises every round, where replacing the data makes it grow linearly in the number of rounds | Gerstgrasser, Schaeffer, Dey et al. (2024), LIT-971, read in NOTE-815, Theorems 1–2 extending Dohmatob et al.’s replace result. Proved for ridgeless OLS with isotropic Gaussian features, the real covariates reused every round and only labels regenerated. The error is bounded, not constant: it rises every round towards its limit, and every model remains a noisier copy of the first. Small language and diffusion models behave alike over five to eight rounds; a VAE’s error still rises. | Active |
| THEORY-290 v2 | In a two-debater, three-proposition model of deductive debate whose payoff trades true beliefs against changes of mind, the most truth-conducive equilibrium can fall short of the truth-optimal profile because one debater free-rides on the other’s belief-disrupting arguments, and in a narrow band of mixed equilibria a higher weight on truth gives lower equilibrium accuracy | Handfield, Garcia, Hilbe and Yeo (2025), LIT-966, read in full in NOTE-795. With payoff w·v − d (true beliefs against changes of mind), debate between two deductive reasoners is a cooperative dilemma at low and moderate truth weights. Typically one argues boldly and the other free-rides (a snowdrift), and at low weights both exit. A small dip in accuracy as w rises occurs in a narrow band of mixed equilibria. It is shown numerically for two players and three propositions. It concerns collective accuracy, not a belief the pair settles on, and at high truth weight in short debates there is no dilemma. The title claims the model only; groups and richer strategies are QUESTION-173. | Active |
| THEORY-293 v3 | For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts’ total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio | Gu, Lyu, Li and Zhang (2025), LIT-984, read in NOTE-796. Proved for an idealised learner whose capacity is mutual information with the data (Theorem 4.3), on an orthogonal mixture of uniform facts and a second domain; seen in Pythia models from 14M to 6.9B on synthetic biographies mixed into web text. The theory is static. It does not say when in training the facts arrive, and it does not show that SGD reaches the optimal allocation. | Active |
| THEORY-294 v3 | For a wide two-layer ReLU network with zero bias on one-dimensional data whose two classes lie on opposite sides of the origin, the tangent kernel is a constant multiple of the initialization kernel, so lazy training leaves the within-class variability of its features, NC1, where initialization put it | Kothapalli & Tirer (2024), LIT-985, read in NOTE-774. NC1 = tr Σ_W / tr Σ_B of kernel-defined features is a ratio of linear functionals of the kernel matrix (Theorem 4.1). For one-dimensional data, two classes on opposite sides of the origin and zero bias, the ReLU tangent kernel of a wide two-layer net is a constant multiple of the NNGP kernel, so NC1 is the same (Theorem 5.1, re-derived). In higher dimensions it is measured no lower, and for ReLU higher; whether lazy training leaves NC1 in place there is QUESTION-192. The inference that collapse requires feature learning is the reading’s, not proved. | Active |
| THEORY-295 v3 | In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth | Devezer, Nardin, Baumgaertner and Buzbas (2019), LIT-988, read in full in NOTE-792. In a Markov-chain and agent-based model where each step tests one proposed linear model against the consensus by AIC or SC, replication cannot change the transition matrix. A boundary-testing population reproduces almost perfectly while rarely holding the true model, and correlations between reproducibility and truth depend on the strategy mix. The consensus is memoryless, scientists hold no beliefs, and an equal strategy mix is shown only never to be the worst, not best. | Active |
| THEORY-296 v2 | In a preregistered multimodal test with clearly seen images, category-selective posterior cortex showed no sustained content-specific synchronization with early visual cortex, which iEEG sampled with 12 electrodes: the preregistered phase coupling was brief, low-frequency and mostly stimulus-evoked, and exploratory amplitude coupling, gamma included, was also brief | Cogitate Consortium, Ferrante et al. (2025), LIT-973, read in NOTE-812. IIT’s proponents predicted sustained gamma-band synchronization between category-selective cortex and V1/V2 throughout a conscious percept. iEEG and MEG phase consistency found only early, brief, 2–25 Hz coupling, mostly evoked, with Bayesian support for the null; exploratory amplitude coupling was also brief. V1/V2 was sparsely sampled in iEEG. It tests IIT’s proposed implementation, not integrated information. | Active |
| THEORY-297 v2 | A bijection between shapes is equivalent, up to null sets, to the linear operator it induces on functions; in orthonormal bases a volume-preserving bijection’s matrix is orthogonal, and an isometry’s commutes with the Laplace–Beltrami operator, so it maps each eigenspace to the matching eigenspace | Ovsjanikov, Ben-Chen, Solomon, Butscher & Guibas (2012), LIT-993, Remarks 4.1–4.3, Theorem 5.1 with App. A, and §5.1, read in NOTE-822. Orthogonality was checked; commutation with the Laplacian is cited as well known. All of it assumes a bijection. The operator runs between two different spaces, but the paper gives no structure for a non-invertible or one-way map, and the stability of truncated eigenspaces is cited to Kato, not shown. | Active |
| THEORY-299 v2 | Structured and translational knowledge-graph embeddings are 0-cochains of a cellular sheaf of vector spaces pulled back from the schema, each triple scored by a norm of the coboundary on its edge and, in the squared-Euclidean form, the whole graph by the sheaf Laplacian; since the zero cochain is always a global section, what they learn is a contrastive margin, not the existence of a section | Gebhart, Hansen and Schrater (2021; AISTATS 2023), LIT-232, §4 and Appendices A and C, read in NOTE-843. A direct identification, covering Structured Embedding, the Unstructured Model, TransE, TransR and RotatE and not bilinear models. That existence of a section is trivial here, and that the only gluing question is whether fixed data extend, is the paper’s own remark taken further by the record; it is what separates this use of sheaves from the contextuality line. | Active |
| THEORY-301 v2 | The Procrustes distance between regularization-normalized representations and their average linear-decoding distance bound each other only through the participation ratio of the kernel difference: a small Procrustes distance forces a small decoding distance, but not conversely | Harvey, Lipshutz & Williams (2024), LIT-257, Proposition 3 and App. A.1, read in NOTE-834. With K_X = XG(X)⁻¹Xᵀ/M, α = Tr K_X, β = Tr K_Y and R_Δ the participation ratio of K_X − K_Y, P⁴/R_Δ ≤ E‖Xw* − Yv*‖² ≤ (2(α + β)P² − P⁴)/R_Δ for targets with E[zzᵀ] = I. The Procrustes–Bures equivalence is imported from LIT-250. Kernels with orthogonal supports show the converse fails: Procrustes distance maximal, decoding distance as small as 2/r. Finite samples, linear readouts only. | Active |
| THEORY-304 v3 | In plane Couette and Poiseuille flow, measured in the energy norm, the linearized operator is so far from normal that perturbations can grow transiently by O(R) and resonate by O(R²), and operator perturbations of size O(R⁻²) make it eigenvalue-unstable, at Reynolds numbers where eigenvalue analysis predicts stability | Trefethen, Trefethen, Reddy & Driscoll (1993), LIT-932, read in NOTE-839: computed scalings for the linearized shear flows, mostly credited to earlier work and unified through pseudospectra. That transition to turbulence happens by this linear growth plus nonlinear mixing is the paper’s conjecture from a 2 × 2 model, and is not part of this account. | Active |
| THEORY-305 v2 | SimCLR’s InfoNCE loss, over all n objects with a Gaussian kernel on the unit sphere, equals a graph-Laplacian quadratic form on the augmentation matrix plus the log of InfoNCE’s normalizers, which is spectral clustering only in a sense that admits any regularizer and does not make the minimizer the Laplacian’s eigenvectors | Tan, Zhang, Yang & Yuan (2023, ICLR 2024), LIT-227, read in NOTE-846: Theorem 3.1 and its proof (App. A), Lemma 2.5 and Definition 2.7. The identity between objectives is exact on a finite object set with symmetric π. The paper’s “spectral clustering” is any Laplacian form plus any regularizer; it names no minimizer and no population optimum, so it supports neither an eigenvector reading of InfoNCE nor the density-ratio optimum of THEORY-001. | Active |
| THEORY-306 v3 | MSN’s objective penalizes unequal cluster sizes explicitly and SwAV’s enforces equal sizes, and in one study with mini-batches of two to eight ImageNet classes, SimCLR, MSN and VICReg lost 11 to 18 points of ImageNet linear accuracy while MAE and data2vec lost under one | Assran et al. (2022), LIT-231, §§3–5 and Appendix C, read in NOTE-850. The prior is explicit in MSN’s loss (a KL to uniform over mean cluster assignments) and in SwAV’s equipartition constraint. For VICReg it rests on an informal reduction of a simplified loss to K-means and on K-means’s cited tendency to equal clusters, and for SimCLR on a cited equivalence, so the title claims the prior for MSN and SwAV only. The effect is shown with batches of two to eight ImageNet classes, one run per cell, not with natural long tails; the open half is QUESTION-171. | Active |
| THEORY-308 v2 | The neural Donsker–Varadhan estimator of mutual information is strongly consistent as an extremum estimator over compact families of critics, and, for critics bounded by M and L-Lipschitz in their d parameters, its uniform deviation from the population bound needs a number of samples set by d, M, L and the parameter radius, not by the data dimension | Belghazi et al. (2018), LIT-256, §3.3 and App. 8.2.2–8.2.3, read in NOTE-828. On a compact domain with Lebesgue densities, the supremum of the Donsker–Varadhan bound over some compact network family is within any η of the mutual information, and the empirical supremum converges to the population one almost surely. Uniform deviation ε needs Õ(d log d/ε²) samples for d parameters, under bounded, Lipschitz critics. The guarantee is for the exact empirical maximizer, not for gradient training. The finite-sample estimate is not a bound (Poole et al., LIT-246), and the proof has two gaps a reading found. | Active |
| THEORY-309 v2 | If contexts are defined by joint measurability alone, every nondisturbing correlation, maximally contextual ones included, is realized by trivial measurements on a one-dimensional quantum system, so Kochen–Specker contextuality needs a stronger operational notion of context | Budroni, Cabello, Gühne, Kleinmann & Larsson (2022), LIT-263, §IV.B.2, Eqs. 27–28, adapting Fritz (2012), read in NOTE-844. For a nonsignalling p(ab|xy), the POVMs G^{xy}_{ab} = p(ab|xy)𝟙 on a one-dimensional space are jointly measurable for each pair and reproduce p. So contextuality relative to joint measurability alone is trivial, and the review requires contexts of repeatable, nondisturbing measurements. Liang, Spekkens & Wiseman’s sharpness parameter is the other known escape. | Active |
| THEORY-312 v2 | The large-batch InfoNCE loss is the negative HSIC between representations and image identity plus a nonnegative log-mean-exp of the centred critic, so it bounds a kernel dependence loss from above | Li, Pogodin, Sutherland & Gretton (2021), LIT-248, eq. 28 and Theorem B.1, read in NOTE-830. On a finite dataset of N images with one-hot identity labels, the large-batch InfoNCE loss equals −HSIC(Z,Y) + E_Z log E_Z′ exp(k − μ₁) exactly. The second term is nonnegative by Jensen (the record’s step), and for a bounded kernel it is at least γHSIC(Z,Z) less a second-order remainder (Theorem B.1, printed with the remainder’s sign reversed). It does not say InfoNCE is HSIC, and it says nothing of the finite-batch estimator. | Active |
| THEORY-313 v3 | With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks | Bach (2022), LIT-255, read in NOTE-837. Σ_p = ∫ϕϕ* dp is uncentred and unit-trace, so p ↦ Σ_p maps distributions to quantum states, and D(p̃‖q̃) ≤ D(Σ_p‖Σ_q) ≤ D(p‖q) (Proposition 4(d), Eq. 8). The upper bound is joint convexity, with equality for every pair only if k(x,y) = 0 for x ≠ y. The lower bound is monotonicity under a measurement whose outcome law is p smoothed by h(x,y) = ⟨ϕ(x), Σ^{−1/2}ϕ(y)⟩². The O(σ²) gap is shown only for kernels on the torus with Fourier coefficients proportional to e^(−σ‖ω‖₁), with Lipschitz densities. It does not cover centred kernels, or similarity measures that compare Gram matrices of representations. | Active |
| THEORY-314 v2 | Every minimizer of an arbitrary cost on the training outputs plus a strictly increasing function of the RKHS norm lies in the span of the kernel functions at the training points, by the reproducing property and orthogonal decomposition alone | Schölkopf, Herbrich & Smola (2001), LIT-261, Theorem 1 with its proof (eqs. 18–24), read in NOTE-836 and checked. No convexity of the cost and no Riesz theorem is used; the RKHS is built from the kernel, so the reproducing property holds by construction. The theorem is stated over countable kernel expansions of finite norm, assumes a minimizer exists, and does not cover costs that see f other than through its values at the training points. | Active |
| THEORY-315 v2 | For a state whose four-state reflection orbit is realizable, generalized noncontextuality bounds two binary predictabilities by |⟨X⟩| + |⟨Z⟩| ≤ 1, so the form of a predictability trade-off, not its existence, can witness contextuality | Catani, Leifer, Scala, Schmid and Spekkens (2022), LIT-980, read in NOTE-825. Proved analytically, with a saturating model, for preparation noncontextuality and exact operational equivalence. The bound constrains a whole uncertainty relation only in a theory where every state has the orbit; the qubit’s circle violates it, and a classical theory without the symmetry escapes it. It needs no perfectly distinguishable states, and it is not a sheaf or Kochen–Specker criterion. | Active |
| THEORY-318 v2 | A bound k on the max-information between a dataset and an analyst’s chosen statistic multiplies by at most 2^k the probability that the statistic overfits that dataset, and an ε-differentially private mechanism on datasets of n points keeps k at most log₂e·εn, whatever the distribution on datasets | Dwork, Feldman, Hardt, Pitassi, Reingold & Roth (2015), LIT-964, Supplementary Material only, read in NOTE-848 (Skimmed: the main text was not reached). Theorem 2 and Lemma 5 are proved there and were checked: P[overfit] ≤ 2^{I∞}·max P[overfit], and ε-DP gives I∞ ≤ log₂e·εn. Thresholdout’s guarantee (Theorem 11), n = O(ln(m/β)/τ²·min{B, √(B ln(m/β))/τ}), is stated with its proof elsewhere. Statistical queries on a discrete universe only, and a procedure that prevents inflation, not a correction for a search already run. | Active |
| THEORY-319 v2 | When two views are redundant for a label, the predictor that infers one view from the other is within εX + 2√(εXεZ) + εZ of the Bayes predictor in mean squared error, and it is linear in the contrastive density ratio, so for a bounded label and a square-integrable ratio a linear readout of m random evaluations of the exact ratio approaches it at rate 1/m, with no symmetry between the views required | Tosh, Krishnamurthy & Hsu (2021), LIT-251, read in NOTE-841: Lemma 1, §2.3–2.4, Lemma 2 and Theorem 3. The label enters only through how redundant the views are about it; the representation is the density ratio p(x,z)/(p(x)p(z)), learned without labels. Population level. The error-propagation results need a uniform bound on the ratio, and the low-dimensional direct embeddings are shown to exist, not to be reached. | Active |
| THEORY-321 v4 | Under repeated DeGroot averaging in a strongly connected society, the consensus converges to the truth as the society grows if and only if the largest influence weight vanishes, so network structure alone, not the members’ common updating rule, decides whether the group belief is right | Golub & Jackson (2010), LIT-999, Proposition 2 with Lemma 1, Corollary 1, Proposition 3 and Theorem 1, read in NOTE-840 with the main proofs checked. The members are naïve, not sound: they ignore repeated information, and only the first round is Bayesian. The result is in a model with fixed weights, independent signals and n → ∞. Speed of consensus and its correctness are independent. | Active |
| THEORY-324 v2 | A system adapts by trial and error to an environment whose effect on its essential variables it does not know only if a second feedback, through those variables, changes the parameters of its primary feedback; and if that feedback changes them whenever, and for as long as, an essential variable is outside its limits, every equilibrium the whole can reach is an adapted one | Ashby (1952; 2nd ed. 1960), LIT-787, read in NOTE-851. Necessity is deduced from the trial-and-error rule against a Black Box environment; sufficiency from the theorem that a whole is at equilibrium only where every part is. The Homeostat shows it in a four-unit linear machine. The parameter changes are blind, and the account says nothing about how long the search takes, which is the subject of THEORY-307. | Active |
| THEORY-327 v2 | Fricker, in her 2003 paper and her 2017 retrospective, holds that testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, and that deliberate manipulation of credibility is a related but distinct wrong; and she accepts Anderson’s structural form, which is pre-emptive: a speaker’s word is never sought because of a structural prejudice, such as an un-refreshed list, that is no one’s fault | Fricker, in her own words, in two papers read in full: the paper that introduced testimonial injustice (2003, LIT-1049, NOTE-860), and her retrospective chapter (2017, LIT-1048, NOTE-853). A speaker receives the wrong degree of credibility “owing to a certain sort of unintended prejudice on the hearer’s part”. Deliberate manipulation of credibility is “a closely related but distinct injustice”. With Anderson, she accepts a structural form: when an un-refreshed list means some people’s views are never sought, “(through no one’s fault, let us add)”, that is pre-emptive testimonial injustice. It does not say every credibility deficit, or every omission from a list, is an injustice: a prejudice must cause it. | Active |
| THEORY-329 v2 | In 32,479 Weibo posts of 2011–2016 scored with an emotion lexicon, posts tagged fake by the platform’s fact-checkers carry more anger and less joy than true posts from verified accounts, and a post’s share of anger words goes with more retweets after controls; whether the gap is due to falsity, and whether anger causes the spread, is not shown | Chuai & Zhao (2020), LIT-379, read in NOTE-852. In 32,479 Weibo posts from 2011–2016, lexicon-scored anger share is higher and joy share lower in fake than in true news, and in highly than in lowly retweeted news, under three scoring methods. With content, author and topic controls, anger share predicts more retweets within 48 hours and joy share fewer (R² 0.08–0.13). The paper calls this causal; the design is observational, and the fake set is selected for having spread. Active as a report of the association (ADR-065); the paper’s causal thesis is THEORY-1221, Proposed. | Active |
| THEORY-332 v2 | For one index with no space limit, Bruno and Chaudhuri’s online tuner, which builds the index once the savings it would have given have risen by its build cost above their last low and drops it once they have fallen by the build cost below their last high, is not three-competitive: when every statement’s saving or loss is at most a fraction f of the build cost, a repeated workload makes it cost nearly 3 + 2f times the offline optimum, and one statement whose saving exceeds the build cost makes the ratio unbounded; no upper bound is shown | Bruno and Chaudhuri (2007), LIT-1095, read in NOTE-896. An always-on index tuner keeps, per index, the running sum of what each statement would have saved (updates counting as losses) and creates or drops the index when that sum moves by the build cost from its last extreme. The paper calls the single-index rule three-competitive, but its proof is one workload on which the ratio approaches 3, a lower bound. The reading’s own check found ratios up to 4.7 when one statement’s saving is near the build cost, and an unbounded ratio when it exceeds it; with every saving at most a tenth of the build cost no random workload passed 2.4, but a constructed one with steps at most fB approaches 3 + 2f (note of 2026-10-20). The multi-index tuner with a space limit is heuristic, and all costs are optimizer estimates. | Active |
| THEORY-333 v2 | Database cracking, which partitions a copy of a column in place as a side effect of each range query, measured in memory on 10^7 integers under uniformly random ranges, costs about 40% more than a scan on the first query and far less than sorting, gets cheaper per query with use, and stays cumulatively cheaper than sort-then-search until about 10^5 queries; that only the queried ranges get organized is not tested | Idreos, Kersten and Manegold (2007), LIT-1090, read in NOTE-892. Each range select splits, in place, the pieces of a column copy that hold its bounds and records the boundaries in an AVL tree, so later queries touch only edge pieces. In MonetDB on 10^7 values the first such select took 0.38 s against 0.27 s for a scan and about 3.3 s for sort-then-search, later selects 10 to 100 microseconds, and sorting overtook cracking in total only after about 10^5 queries. Every workload is uniformly random, so the paper’s claim that only ranges of interest get organized is asserted from the mechanism, not measured; there is no cost model and no rule for when a reorganization pays. | Active |
| THEORY-334 v2 | In copying a pattern of coloured blocks that stays in view, seven subjects told to work quickly looked back at the model about 1.3 times per block, and their commonest cycle, which looks at the model both before picking up a block and before dropping it, was the slowest per block in the authors’ observed tallies (about 3 s, against 1.5 s for a cycle that copies from memory); looks fell to 1.0 per block with model and copy 70° apart and to 0.7 with one-colour patterns that could be chunked, and in three subjects a colour change made during a saccade raised fixation time by about 129 ms after pickup against about 50 ms before; no statistics are reported | Ballard, Hayhoe, Pook and Rao (1997), LIT-1075, read in NOTE-883. Fixations are modelled as pointers that bind a program’s variable to a place in the world, so that only the task-relevant feature is fetched, just before use. In block copying, look-model, pick-up, look-model, drop was the commonest cycle (about 43% of cycles), at about 3 s a block against 1.5 s from memory; 70° separation cut looks from 1.3 to 1.0 per block, and one-colour patterns to 0.7. A colour changed mid-saccade mostly went unnoticed and cost time only once colour was held. The “carrying cost” of working memory is inferred from choice, not measured, and what is left in the world is raw input, not a computed result. Promoted 2026-10-20 on the observations alone (ADR-047 v3). | Active |
| THEORY-335 v2 | Selective accessibility, the account that comparing a target with a plausible anchor makes anchor-consistent knowledge more accessible and that this knowledge carries anchoring, fits the moderations of the effect by applicability, representativeness and plausibility that Strack and Mussweiler found, but its accessibility step is inferred, not measured: Harris et al.’s repetition of the recognition test of it, four times by lexical decision and once by continuous identification, found nothing in five preregistered experiments, with tasks that also missed direct picture priming | Strack & Mussweiler (1997), LIT-1064, read in NOTE-877, proposed the account and showed, in three small studies, that with the anchor number fixed the effect depends on whether the knowledge used to compare applies to the later estimate, and that plausible anchors slow the comparison and speed the estimate. Accessibility is inferred, never measured. Harris et al. (2019), LIT-1067, read in NOTE-876, repeated the recognition test (lexical decision, once continuous identification) of the accessibility step (Mussweiler & Strack 2000a, not held) five times at N = 128: anchoring was large and recognition of anchor-consistent words never moved (BF01 ≈ 21), but the same tasks missed direct picture priming. So the direct evidence is removed, not reversed; the account is neither refuted nor shown. Mussweiler & Strack 1999 (LIT-1078) is Unavailable. | Active |
| THEORY-336 v2 | An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it | Geisler (2011), LIT-1055, read in NOTE-870. A review that defines the ideal observer as optimal “given the available information and any specified constraints”, and writes it as an expected-utility argmax over the posterior given the stimulus S or a constrained representation Z = g(S;θ). Constraints are added one at a time (optics, receptor sampling and noise, ganglion-cell summation), and human performance that parallels the retinal-stage ideal while falling below it is read as loss in central factors. The empirical parts are the review’s account of studies the record has not read. The review gives no formula for efficiency and no notion of an optimum over a restricted class of computations. | Active |
| THEORY-337 | In the comparative-question paradigm, judging whether a quantity is higher or lower than a high or low anchor pulls the later estimate of it toward the anchor, with large effects in five preregistered experiments of 128 people each, lasting through dozens of intervening word-recognition trials | Harris et al. (2019), LIT-1067, read in NOTE-876, and Strack & Mussweiler (1997), LIT-1064, read in NOTE-877. In five preregistered experiments (N = 128 each) on temperatures and car prices, estimates after a high anchor exceeded those after a low one every time (d ≈ 0.85–1.39), with 40 to 70 word-recognition trials between the comparison and the estimate. Strack and Mussweiler’s three small studies show the same effect, and that it can shrink or reverse when the knowledge used to compare does not apply to the estimate. This is the effect, not its mechanism: why the estimate moves is THEORY-335’s question and is not settled. | Active |
| THEORY-338 | For a list searched from the front, where reaching the i-th item costs i, moving each found item to the front costs at most twice what any rule costs on the same sequence, including rules that know the future (four times counting its own moves), a fixed order by access frequency is within no constant factor, and paying to rearrange ahead of need never helps; for paging, no online rule beats the offline optimum by less than n/(n − n_MIN + 1), and LRU meets that bound | Sleator and Tarjan (1985), LIT-1093, read in NOTE-903. With no probabilistic assumption, move-to-front on a sequentially scanned list is within 2 of the best algorithm in its class on every sequence, the rival being free to rearrange as it goes and to know the whole future; a list sorted once by frequency, transpose and frequency count are not within any constant; and any algorithm can be replaced by one no more expensive that makes no paid rearrangement (Theorem 3). For paging the online price of not knowing the future is n/(n − n_MIN + 1), the size of fast memory when the memories are equal. It is one list, or one two-level memory, with uniform items: it says nothing about bins, partitions or search that is not a scan. | Active |
| THEORY-340 | For answering membership in a static set of n keys with the fewest worst-case cell reads, sorted order with binary search is optimal only when the key universe exceeds a Ramsey-number threshold and table cells may hold nothing but stored keys or pointer symbols; it is beaten for small universes, and by two probes once one cell may hold an encoded value | Yao (1981), LIT-1097, read in NOTE-891. In the setting later called the cell-probe model, where only cells read are counted and both the arrangement and the adaptive search are arbitrary, ⌈lg(n+1)⌉ probes are needed by any scheme whose cells hold only keys of the set or a fixed stock of pointer symbols, once the universe is astronomically large (Theorems 1 and 1′). One probe suffices for m ≤ 2n − 2 (Theorem 2), and with one extra cell holding any value two probes suffice for m ≥ 2^{16n²} (Theorem 3). Building the table is free, so the paper prices lookup only and says nothing about whether sorting pays for itself. | Active |
| THEORY-341 | Inferring not-P from failure to prove P is a deduction only under two separate conditions: Clark shows that when every branch of the search for a proof of P ends in failure, not-P is a theorem of the completed database, and a search that does not finish licenses nothing; Reiter’s closed-world assumption, adding the negation of every ground atom the database does not entail, keeps a consistent Horn database consistent and can make a disjunctive one inconsistent | Clark (1978), LIT-1087, read in NOTE-894, and Reiter (1978), LIT-1088, read in NOTE-899 from his 1977 workshop report. Clark proves negation as failure sound with respect to the completed database (the clauses read as if-and-only-if definitions, plus axioms that distinct names denote distinct things), given a finite failure tree for some selection rule; it is incomplete in general. Reiter defines the closed-world assumption by entailment, reduces closed-world query evaluation to open-world atomic queries when the assumption is consistent, and shows it consistent for Horn and definite databases but not for Pa ∨ Pb. Soundness here is relative to the completion, not to the world, and neither paper says anything about a search cut short by time. | Active |
| THEORY-342 v2 | In AlphaZero on 9×9 Hex, at fixed playing strength, training compute and tree-search compute substitute along a straight line in log–log space, about 15× less search for each 10× more training over about two decades of training compute, down to a one-node search; the trade is measured for one search and counts no games played | Jones (2021), LIT-1069, read in NOTE-884. Across about 200 AlphaZero models on Hex boards 3×3 to 9×9, the best Elo for a given training compute follows one change-point curve, shifted by a fixed factor of compute per board row. On 9×9 only, iso-Elo contours of training against test-time search compute are straight in log–log space (fit: log test = −1.2·log train + 0.004·Elo + 29) over roughly 1e14–1e16 FLOPS-seconds of training, ending at a one-node search. One game, one nonstandard implementation, no uncertainty on the fitted coefficients, and no count of how many searches the trained agent will run, so it gives an exchange rate, not an amortization. | Active |
| THEORY-344 v2 | An access method’s read, update and memory overheads, each a ratio to the base data alone with a floor of 1.0, are conjectured to trade so that bounding any two forces a lower bound on the third; the conjecture is unproved, its formal wording says no more than the floor already does, and the paper’s extreme designs show conflicts between pairs, read against memory and update against read, not a symmetric three-way trade | Athanassoulis et al. (2016), LIT-1085, read in NOTE-888. A six-page vision paper defines read, update and memory amplification and conjectures that an access method optimal on one cannot be optimal on both others. Its evidence is three one-paragraph propositions about extreme designs on an integer array, a qualitative triangle and a table of asymptotic I/O costs. The formal statement asserts only some lower bound on the third overhead, which every design has. The extremes are not symmetric: the memory-minimal design is also update-minimal. Build cost, workload and time are outside the triangle. | Active |
| THEORY-345 | Choosing indexes for a known mix of queries and updates, to minimize query cost plus a per-index upkeep, is NP-hard as the number of attributes and query types grows, and the tools built for it choose one stored order and a set of secondary indexes per table by searching configurations costed with the query optimizer’s own estimates, optimal at best relative to those estimates, with no proved guarantee for their heuristics and no term for the one-time cost of building | Piatetsky-Shapiro (1983), LIT-1089, read in NOTE-902; Finkelstein, Schkolnick and Tiberio (1988), LIT-1084, read in NOTE-898; Chaudhuri and Narasayya (1997), LIT-1083, read in NOTE-900. Minimum set cover reduces to choosing single-attribute secondary indexes under a known workload with a fixed upkeep per index. DBDSGN chooses the one clustered order and the secondary indexes per table against a frequency-weighted sum of access, update and upkeep costs plus storage, read from the System R optimizer; AutoAdmin picks at most k indexes by greedy search over what-if indexes. The model is steady state over a fixed workload: it prices upkeep against per-query savings, not a one-off build or sort against later lookups, and its optimality is relative to the optimizer’s estimates, never to measured execution. | Active |
| THEORY-346 | When information stays in view at a known place, people consult it less and rely on memory more as the time to reach it rises, in a graded trend across added delays from 0 to 3.2 seconds, even where the memory is weak and the display perfect; in one copying task a model that prices memory and display access in time alone, with no preference for either source, fits how many items people encode per look | Gray, Sims, Fu and Schoelles (2006), LIT-1073, read in NOTE-868, and Gray and Fu (2004), LIT-1058, read in NOTE-864. Adding a click to reach on-screen show information cut how often it was checked, and checks of finished settings came from memory. Lockouts of 0 to 3,200 ms on a Blocks World pattern cut looks and raised blocks placed per look, with linear trends. An ideal performer model with ACT-R memory equations and Q-learning on time alone fits four measures. That the control system has no bias toward the world is argued, not tested against a rival model. Errors in the 2004 study differed only against a group that memorized under several other changes, and the 2006 model has no wrong retrievals, no precomputation, no reuse across trials and no staleness. | Active |
| THEORY-347 | Belief bases with the same logical closure can revise to opposite conclusions on the same input, so a closed belief set does not fix its own revision; partial meet contraction of a base need not satisfy Recovery, and bases keep distinct inconsistent states apart, while a belief-set model reproduces the base difference only by attaching a different selection mechanism to the same set | Hansson’s Stanford Encyclopedia survey (2006, revised 2026), LIT-1072, read in NOTE-866, with Alchourrón, Gärdenfors and Makinson (1985), LIT-1050, read in NOTE-874. Bases {p, q} and {p, p ↔ q} have one closure, yet revised by ¬p one yields q and the other ¬q: statically but not dynamically equivalent. Base contraction escapes the Recovery that AGM proves for closed theories, and many inconsistent bases exist where there is one inconsistent closed set. A base marks which beliefs are held in their own right and recomputes the rest; it is not a record of reasons, sources or defeated alternatives, and the survey gives no result that bases handle iterated revision better. | Active |
| THEORY-348 | When staying in one state costs a fixed amount per request and leaving it costs a fixed one-time price, switching once the extra cost paid for staying equals that price is within 2 of the offline optimum on every request sequence, and no deterministic online rule does better; with a third option the best guaranteed ratio rises to 3 | Karlin, Manasse, Rudolph and Sleator (1988), LIT-1096, read in NOTE-893. A snoopy cache holding a shared block pays 1 per write to keep it current, or drops it and pays p to fetch it again on a later read. Dropping once the writes charged since the last read have cost p is strongly 2-competitive (Theorems 3.1 and 4.1); with a read- or write-through option the deterministic lower bound is 3 (Theorem 3.3). It is the first competitive analysis of a rent-or-buy decision, though the paper never uses the name, and its instance is when to stop maintaining a copy. Prices are fixed and known, the guarantee is worst case only, stale reads are forbidden rather than priced, and the authors call the practical significance “unclear”. | Active |
| THEORY-349 | For binary outcomes and a fixed representation of individuals, a predictor can pass every test in a bounded class without being close to nature: outcome indistinguishability to tests that see the sampled prediction is equivalent, with stated losses, to multicalibration over a closely related class of sets, a predictor meeting it exists at size polynomial in the tests’ size and the tolerance whatever nature’s complexity, and learning one is as hard as weak agnostic learning of the class | Dwork, Kim, Reingold, Rothblum and Yona (2021), LIT-1068, read in NOTE-885, and Hébert-Johnson, Kim, Reingold and Rothblum (2018), LIT-1052, read in NOTE-879. A predictor is outcome indistinguishable relative to a class of distinguishers when none can tell nature’s outcomes from outcomes drawn from the predictor, up to ε. With no access or sample access to the predictor this is equivalent to multi-accuracy or multicalibration over closely related classes, and a predictor meeting it exists at size O(s/ε²) (O(s/α⁴γ) for multicalibration), independent of nature. Learning a multicalibrated predictor is equivalent to weak agnostic learning of the class. Adequacy here is relative to the class, the outcome and the features, and is not unique; it is not closeness to the truth, and the papers say nothing about signal detection. | Active |
| THEORY-350 | Under a budget on maintenance time, a stored view’s upkeep depends on which other views are stored, so upkeep is not additive and adding a view can lower it, and greedily adding the single view of best saving per unit of upkeep can be arbitrarily bad; on OR view graphs a greedy over inverted tree sets gets 1 − 1/e of the saving of an optimum given as much upkeep, using up to twice the budget, if no tree-piece of the optimum alone exceeds the budget | Gupta and Mumick (1999), LIT-1092, read in NOTE-890, with the proofs of the journal version (2005), LIT-1080, read in NOTE-901, and the superadditivity first noted by Gupta (1997), LIT-1091, read in NOTE-889. Minimizing query cost subject to a cap on total maintenance time, one stored view can make another cheaper to keep current, so saving per unit of upkeep is not monotone and single-view greedy fails without bound (Example 2). Grouping views into inverted tree sets restores a 1 − 1/e guarantee, proved in 2005 (Theorem 8), for OR view graphs only, bicriteria, under an assumption on the optimum, with greedy stages exponential in general. The proof does not treat the negative effective costs the paper allows. Upkeep is a budget, not a cost traded against savings, and staleness and build cost are not modelled. | Active |
| THEORY-352 | A pseudorandom generator exists if and only if a one-way function exists, where both are secure against uniform adversaries and every reduction is an oracle Turing machine, as statements about ensembles indexed by n; the construction from an arbitrary one-way function is only weak-preserving, so security at one length is bought at a much greater one | Håstad, Impagliazzo, Levin and Luby (1999), LIT-1051, read in NOTE-880, with Goldreich’s Foundations of Cryptography, LIT-1061, skimmed in NOTE-863 from the 1995 draft. Theorem 6.2.2: “There are one-way functions iff there are pseudorandom generators.” The hard direction runs one-way function → false entropy generator → pseudoentropy generator → generator, by uniform reductions, with seeds of length n^10 in the direct construction. Adversaries are algorithms; the paper has no circuit version, and that the proof carries over to non-uniform adversaries is the reading’s inference, not the paper’s statement. Everything is asymptotic. Goldreich states the theorem and does not prove it. | Active |
| THEORY-353 | For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory | Alchourrón, Gärdenfors and Makinson (1985), LIT-1050, read in NOTE-874, with Hansson’s survey, LIT-1072, read in NOTE-866. Contracting a closed theory A by x keeps what a selected set of maximal x-free subsets of A share; the six basic postulates characterize exactly these partial meet contractions (Observation 2.5), and with the two supplementary postulates exactly those selected by a transitive relation (Corollary 4.5). Revision follows by the Levi identity. A selection function or ordering must be added to A, and in the finite case distinct ones give distinct contractions. Nothing here concerns iterated change, where the selection comes from, or why a belief is held. | Active |
| THEORY-354 | An observer’s power to refute a predictor depends on its access to the predictor as well as on its budget: against tests that see only the sampled prediction a predictor of size O(s/ε²) always passes, but if fine-grained clique counting is hard, or BPP ≠ PSPACE, some cheap tests that may query the predictor adaptively pass only predictors about as costly as nature’s own | Dwork, Kim, Reingold, Rothblum and Yona (2021), LIT-1068, read in NOTE-885. Against sample-access distinguishers of size s an outcome-indistinguishable predictor of size O(s/ε²) always exists. Against oracle-access distinguishers, under fine-grained hardness of clique counting or BPP ≠ PSPACE, there are natures and polynomially many cheap distinguishers such that no predictor much cheaper than nature is indistinguishable. Non-adaptive queries add little; the power comes from adaptive queries. The results are conditional and asymptotic, with upper and lower bounds in different parameterisations; they say nothing about human observers or about which access a real auditor has. | Active |
| THEORY-356 | When stored views are never updated, greedily storing the view of largest marginal saving in query cost gets at least 1 − 1/e of the optimal saving on a data-cube lattice or any OR or AND view graph, against the optimum for the same number of views or for the space the greedy ends up using, which may exceed the budget by one view | Harinarayan, Rajaraman and Ullman (1996), LIT-1094, read in NOTE-895; Gupta (1997), LIT-1091, read in NOTE-889; Gupta and Mumick (2005), LIT-1080, read in NOTE-901. A stored answer is worth the saving on every query derivable from it, less what answers already stored save, and greedy on that marginal saving gets 1 − ((k−1)/k)^k ≥ 1 − 1/e of the best saving for k views; under a space budget the same holds against the optimum in the space the greedy actually uses (up to one view over), or 0.63 − f when no view takes more than a fraction f of the budget. The bound is on saving against storing the base view alone, not on query cost, and it holds only with no updates: once upkeep enters the objective it is proved only for AND graphs in which every view is queried more often than it is updated. | Active |
| THEORY-357 | Computational indistinguishability is strictly coarser than statistical closeness: without any assumption there are ensembles on at most 2^(n/2) strings that no circuit of size 2^(n/8) tells from uniform, though none of them is efficiently samplable, and an efficiently samplable pair of this kind exists if a pseudorandom generator does, since its output covers at most 2^n of the 2^l(n) strings; all of it is about ensembles indexed by n, not single pairs of distributions | Goldreich, Foundations of Cryptography vol. 1, LIT-1061, read in NOTE-863 (Skimmed, 1995 draft, Chapters 1–3 in full), with Håstad, Impagliazzo, Levin and Luby (1999), LIT-1051, read in NOTE-880. Proposition 3.3 proves by counting that some ensemble far from uniform fools every small circuit; its proof is non-constructive and the ensemble cannot be sampled efficiently. A generator’s output is efficiently samplable, indistinguishable from uniform by definition, and statistically far by an image count (§3.3.1). That a samplable such pair exists only if one-way functions do is cited to Goldreich 1990, not read. The book’s statistical difference has no factor ½, so it is twice total variation. | Active |
| THEORY-358 | In visual search among up to about 36 well-spaced items, a target that alone carries one colour or one distinctive shape part is found in time almost independent of the number of items, while a target identified only by a combination of features its neighbours carry separately takes time that grows linearly with the number of items, about twice as fast when the target is absent, the pattern of a serial self-terminating scan, and practice did not remove it; that conjunctions rather than similarity cause it is the paper’s reading, not its result | Treisman and Gelade (1980), LIT-1082, read in NOTE-897. With coloured letters on cards of 1 to 30 or 36 items, a blue letter or an S among brown Ts and green Xs was found at 3.8 and 2.5 ms per item, and a green T among the same distractors at 28.7 ms per item when present and 67.1 when absent, linear and in the ratio of a scan that stops at the first find; 13 sessions of practice did not change it. The paper reads the contrast as feature integration: conjunction structure, not similarity, makes search serial. That reading is contested by the similarity account of Duncan and Humphreys (LIT-1081), which the record could not obtain, and it does not follow that any item unique in one attribute pops out in a real pile. | Active |
| THEORY-360 v2 | In Kirsh and Maglio’s Tetris keystroke data, rotation began in earnest by 400 to 600 ms after a piece appeared, sometimes before it was fully in view, and at each of three skill levels drops preceded by a move to the wall and back were longer than ordinary drops, about 19 to 20 squares against 13 to 16 | Kirsh and Maglio (1994), LIT-1054, read in NOTE-878, with Kirsh (1995), LIT-1076, read in NOTE-871. In Tetris keystroke data, pieces are rotated more than plan-then-act would need, rotation starts by 400–600 ms, and a translate-to-wall check precedes longer drops. The authors define an epistemic action as one whose primary function is to cut the space, time or unreliability of the agent’s own computation, and argue these moves are of that kind. The saving is argued from a three-player pilot comparing physical with mental rotation, not measured in performance. The trade is offloading at the moment of use, once, with nothing stored; the paper sets itself against caching. Promoted 2026-10-20 on the timing and Table 1 alone (ADR-047 v3); the epistemic-action reading is held as THEORY-823. | Active |
| THEORY-361 | When every object but one has full grade in exactly one of two attribute-sorted lists and the one has full grade in both, finding it under min takes at least about half the objects’ worth of accesses for any algorithm that only reads the lists from the top and looks up objects already seen, and, with the tied objects in random order, about as many in expectation even for algorithms that may look up any object; a query on one attribute alone costs k accesses whatever the number of objects | Fagin, Lotem and Naor (2001; full version read as arXiv cs/0204046v1), LIT-1115, read in NOTE-924: Example 6.3 and Theorem 6.4, the paper’s proof that no algorithm is instance optimal once lucky lookups are allowed. With 2n + 1 objects, n + 1 high in each list and only one high in both, an algorithm that cannot guess must read at least n + 1 entries before it sees the target, and on a randomized family any deterministic algorithm needs at least n + 1 accesses in expectation. The paper uses the case to bound instance optimality. That it explains linear conjunction-search slopes in human vision is the record’s analogy, not the paper’s claim. | Active |
| THEORY-362 v2 | For nonrecursive Datalog or SQL views with safe stratified negation and stratified aggregation, under set or duplicate semantics, storing with each derived tuple only the number of its derivations suffices to maintain the view under insertions and deletions of base tuples: signed counts propagated through delta rules stratum by stratum give each tuple’s change as its new count minus its old, a tuple leaves when its count reaches zero, and a MIN or MAX group whose extreme is deleted is recomputed from the stored lower stratum | Gupta, Mumick and Subrahmanian (1993), LIT-1114, read in NOTE-915. Algorithm 4.1 evaluates delta rules in order of rule stratum and combines signed counts; Theorem 4.1 states it derives count(t^ν) − count(t) for every tuple, and Theorems 6.1 and 6.2 extend this to negated and aggregate subgoals. Its cost is argued to be close to evaluating the view and is not measured. All three theorems are stated without proof in this extended abstract. It does not say that a count suffices when support is cyclic: the paper keeps counting to nonrecursive views. | Active |
| THEORY-363 | For finding the top k objects under a monotone combining function from m lists each sorted by one attribute, with sorted and random accesses the only costs, the threshold algorithm is correct and on every database costs at most a constant times what any correct algorithm that makes no wild guesses costs; the constant is m + m(m−1)c_R/c_S, no deterministic algorithm does better when the function is strict, and once wild guesses are allowed no algorithm has such a guarantee on databases with ties | Fagin, Lotem and Naor (2001; full version read as arXiv cs/0204046v1), LIT-1115, read in NOTE-924. The threshold algorithm reads m attribute-sorted lists in parallel, looks up each new object’s other grades, and stops when k seen objects score at least the combined score of the last grades read. For every monotone combining function it is instance optimal among correct algorithms that make no wild guesses (Theorem 6.1), with ratio m + m(m−1)c_R/c_S, tight for strict functions (Corollary 6.2, Theorem 9.1). It is a guarantee relative to a class of algorithms under an access-cost model. It does not say that top-k queries are cheap: the absolute cost depends on the data, and is linear in N for some conjunctive queries. | Active |
| THEORY-364 | When each cached value is an interval its source guarantees to contain the current value, and a query must return an answer interval no wider than R whatever the true values, choosing the cheapest set of exact reads for a SUM is a 0/1 knapsack problem, NP-complete in general and solved by taking the narrowest intervals first when read costs are equal, and for a COUNT with a selection predicate the cheapest choice is the ⌈|T?| − R⌉ cheapest undecided tuples | Olston and Widom (2000), LIT-1117, read in NOTE-922. In TRAPP a cache holds intervals that sources keep valid by pushing a refresh whenever a value escapes, and each query names a width R its answer may have. The SUM answer is the sum of the intervals, so meeting R at least read cost is knapsack (profit = read cost, weight = width, capacity = R); COUNT under a predicate has a greedy optimum. Staleness becomes bounded imprecision, never error, and only the query-time cost of buying it down is optimized: the sources’ upkeep, the choice of interval widths and any cost of imprecision are outside the model. The MIN rule’s optimality proof is in an unread technical report. | Active |
| THEORY-365 v3 | In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it | Malkov and Yashunin (2020), LIT-1098, read in NOTE-916 from arXiv v4. They build an index like a skip list, with proximity graphs in place of linked lists, and argue that search costs a constant per layer over O(log N) layers. That holds if each layer is an exact Delaunay graph of bounded mean degree. HNSW does not build such graphs, and Delaunay degree grows exponentially with dimension. For the graphs it does build, cost no worse than logarithmic is measured on uniform random data at d = 8, a saturating beam width at d = 4, and at d = 128 (200M SIFT) the growth departs from a logarithm. A proof for the graphs HNSW builds is QUESTION-187. The paper gives no worst-case bound and no recall guarantee. Construction is argued to cost about N searches. | Active |
| THEORY-366 | With a fixed number of data points and a query drawn independently of them, if the variance of the p-th power of the query-to-point distance divided by its mean tends to zero, the farthest point comes within any factor 1 + ε of the nearest with probability tending to one; independent coordinates meet this condition, while data on a low-dimensional subspace, or queries that fall inside well-separated clusters, escape it | Beyer, Goldstein, Ramakrishnan and Shaft (1999), LIT-1099, read in NOTE-917 from the authors’ 1998 technical report. Theorem 1: for n fixed data points and an independent query, var(d^p / E[d^p]) → 0 implies P[DMAX ≤ (1+ε)DMIN] → 1 for every ε > 0. The index m is arbitrary, so dimension enters only through the workload’s distance distributions. IID coordinates and several correlated or non-identical workloads satisfy the condition; a diagonal line, two latent degrees of freedom and queries inside clusters do not. The condition is sufficient, not necessary. The result concerns the ratio of distances at fixed n, and does not show that real high-dimensional data loses contrast, or that indexes fail. | Active |
| THEORY-367 v2 | For SPJU relational queries, which Buneman, Khanna and Tan’s tree model encodes, the minimal witnesses of an output tuple, the inclusion-minimal sets of source facts from which the query still derives it, are the minimal sets matched by single derivations and so are the same for every equivalent query, while where-provenance, the source location a value was copied from, can differ between equivalent queries | Buneman, Khanna and Tan (2001), LIT-1104, read in NOTE-920. Two meanings of the provenance of output data: why it is there (the witness basis, the source structures matched by each valuation that produces it) and where a value came from (the source paths bound to the output variable). The first is preserved by the paper’s rewrite system and, cut down to minimal witnesses, agrees across equivalent queries with equality conditions; the second is shown by example to depend on how the query is written. Proofs are omitted. It is not a theory of revision: additions to the source, negation, aggregation and strengths of support are all outside it. | Active |
| THEORY-368 v2 | On two-dimensional point sets constructed for the purpose, the authors’ code for fast-preprocessing DiskANN, HNSW and NSG, at parameters from the original papers, in most cases found none or few of a query’s five nearest neighbours until the search queue held about 10% of the points, for n from 10⁵ to 2·10⁶, and at that queue length ten graph indexes averaged recall@5 between 0.0 and 0.42; this is shown by experiment on one query per instance, and no theorem bounds these algorithms | Indyk and Xu (2023), LIT-1112, read in NOTE-910. The instances are grids of points in the plane in which a large cluster, reached first from the start vertex, lies closer to the query than the only points linked to its nearest neighbour. At queue length 0.1n, average recall@5 ranged from 0.0 (DiskANN) to 0.42 (KGraph). The paper calls the query time linear and says so “empirically”; the only argument beyond the experiment is an informal Ω(n/R) for DiskANN’s fast build. It does not show that these indexes fail on natural data or natural queries. | Active |
| THEORY-369 v2 | Under uniformly distributed, independent data and queries in the unit cube with Euclidean distance and a fixed number of points, an optimal nearest-neighbour search over any index that partitions points into blocks or convex clusters of at least two points must read a fraction of blocks that grows toward all of them as dimension rises; with a five-fold sequential advantage for scanning, estimates put the crossover at about 15–26 dimensions for rectangular and spherical bounding regions, and two tree indexes measured on 50,000 points lost to a scan beyond about 10 | Weber, Schek and Blott (1998), LIT-1107, read in NOTE-919. Under uniformity and independence, with the expected nearest-neighbour distance used as every query’s radius and integrals evaluated by Monte Carlo, the fraction of blocks read passes the 20% line at about d = 15–20 for rectangular regions and 26 for spherical ones. A crude bound for any convex clustering puts it below 610. The often-quoted “around 10” is the measured crossover for an R*-tree and an X-tree. The VA-file, a scan of compact per-vector approximations, is the paper’s alternative. Nothing is shown for data of low intrinsic dimension, and the finite-dimensional bounds are not rigorous as stated. The asymptotic statement, at fixed N and for clusters of two or more points, is proved by the record by a concentration argument. | Active |
| THEORY-370 | On an n × n lattice whose nodes also get long-range links drawn with probability proportional to lattice distance to the power −r, a searcher that knows only the target’s position and the links of nodes already reached delivers in expected time polylogarithmic in n only at r = 2: greedy routing then takes O((log n)²) steps, while for 0 ≤ r < 2 every such searcher needs Ω(n^((2−r)/3)) steps and for r > 2 Ω(n^((r−2)/(r−1))), even at r = 0 where short paths exist | Kleinberg (2000), LIT-1103, read in NOTE-923. Proved for the two-dimensional lattice: with nodes, local links and the number of long-range links fixed, changing only the exponent r of the long-range link distribution moves the expected delivery time of the best decentralized searcher from O((log n)²) at r = 2 to a power of n at every other r, including r = 0, where paths of polylogarithmic length exist. The upper bound is proved only for one local and one long-range link per node; the k-dimensional statement (r = k) is asserted, not proved; the family is inverse power laws of a distance every node can compute, so it does not show that no other link structure is navigable. | Active |
| THEORY-371 | For conjunctive queries without equality or comparisons, two queries are equivalent exactly when homomorphisms run both ways between the queries read as small databases, every such query has a minimal equivalent query unique up to isomorphism obtained by deleting atoms, and evaluating one against a database is NP-complete when the query is part of the input | Chandra and Merlin (1977), LIT-1111, read in NOTE-907. Reading a conjunctive query as its natural model, equivalence becomes a pair of homomorphisms (Lemma 13), the minimal equivalent query is unique up to isomorphism (Theorem 12), and evaluation with the query as input is NP-complete (Theorem 7), as is deciding whether one query is a folding of another (Theorem 11). The paper states equivalence, not containment, and does not state that containment or equivalence is NP-complete; those are corollaries later work attributes to it. Nothing here reaches queries with equality, comparisons, negation or union. | Active |
| THEORY-372 | For c-approximate near-neighbour search over n points in Euclidean space, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} are achievable on any data set whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming space), and no list-of-points structure, the model of known data-independent hashing, does better on random instances; without that model the curve is matched only at query time n^{o(1)} by one or two cell probes | Andoni, Laarhoven, Razenshteyn and Waingarten (2017), LIT-1120, read in NOTE-918 from the arXiv full version. A data-dependent tree of spherical caps, with different thresholds for filing points and for routing queries, realizes every pair of exponents on the curve, for worst-case data, with constant success probability; the equal-threshold point is LSH. On random instances with d = ω(log n), list-of-points structures cannot beat the curve (a lower bound the paper calls conditional), and one- or two-probe cell-probe structures need n^{(c/(c−1))²−o(1)} words, the curve’s space at ρq = 0. Optimality for data-dependent schemes, for more probes, and for d = Θ(log n) is not shown. The curve prices space against per-query time for one query at a time; it has no workload and no build cost beyond space. | Active |
| THEORY-373 | For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation | Green, Karvounarakis and Tannen (2007), LIT-1110, read in NOTE-913. Union and projection add tags, join multiplies them; the usual RA identities hold exactly when the tags form a commutative semiring, and a map of tags commutes with every query exactly when it is a semiring homomorphism. The polynomials N[X] are the free commutative semiring, so the annotated answer in any semiring is the polynomial answer evaluated at the input annotations (Theorem 4.3). It is a sufficiency result for revisions that re-annotate existing inputs, not a minimality result, and it excludes added facts and every nonmonotone operation. | Active |
| THEORY-374 v2 | Given views with conjunctive definitions, deciding that they answer a conjunctive query exactly is NP-complete in the size of the query and definitions, while a maximally-contained rewriting can be built in time polynomial in that size without saying whether it is exact; and finding all certain answers from the stored contents is polynomial in their size under the open-world assumption when the views are conjunctive and the query conjunctive or datalog, but co-NP-hard in that size under the closed-world assumption even for conjunctive views and queries | Halevy (2001), LIT-1116, read in NOTE-921, reporting Levy et al. 1995 (LIT-1100, read), Duschka, Genesereth and Levy, and Abiteboul and Duschka 1998 (not read). Reuse of stored views is graded: certifying an exact rewriting is NP-complete in the size of the definitions, building everything the views give (maximally contained) is polynomial, and the certain answers can cost a co-NP-hard search in the size of the stored data. Promoted 2026-10-21 on a re-derivation of the last two parts (ADR-047 v2); the primary papers they come from are unread. None of it prices recognition for definitions outside conjunctive queries and datalog, or says how costly recognition is on typical queries. | Active |
| THEORY-375 v2 | In one simulator evicting whole query-shaped cache regions, no replacement value function was best on every workload: a value based on distance from the current query beat LRU where queries clustered around a hot region or wandered with loops, while MRU was best on regular cyclic paths, by up to 2.3 times over the distance value and 9.6 times over LRU; none of the functions weighs what a region would cost to fetch again, and nothing is proved | Dar, Franklin, Jónsson, Srivastava and Tan (1996), LIT-1113, read in NOTE-904. With semantic regions as the unit of eviction, Manhattan distance from the most recent query beat LRU at every query size on a skewed selection workload, and a directional variant was best or tied on random map paths (LRU within 1.06) and on lollipop paths circling a square five or more times; MRU won on a single circuit and on plain squares (Manhattan 2.29 and LRU 9.57 on the 32 × 32 square). The study is simulation only, with synthetic paths and no refetch cost in any value; it shows that the best forecast of reuse depends on the workload, not that semantic distance is a robust rule. | Active |
| THEORY-376 v2 | In a simulated single-client, read-only database answering selection queries on one relation, describing cached data by the predicates of past queries let each query be answered partly from cache, with only the remainder Q ∧ ¬V sent to the server; this matched page caching on a clustered index, beat it on an unclustered one, and, on an unindexed attribute with a disk cache, answered over 60% of 1% queries without contacting the server; the cost of computing probe and remainder queries was not charged | Dar, Franklin, Jónsson, Srivastava and Tan (1996), LIT-1113, read in NOTE-904. In one simulator (one client, one server, a 10,000-tuple relation, 90% of queries in a hot 10%), semantic caching performed about like page caching on a clustered index, far better on an unclustered one, and on an unindexed attribute answered over 60% of 1% queries with no server scan. Space overhead depended on coalescing: 55 regions and about 275 factors for 1% queries without it. The result is measured, not proved; updates, joins and the CPU cost of reasoning about the cache’s contents were left out, so it does not show that predicate-described storage pays when finding the reusable part is costly. | Active |
| THEORY-377 v2 | In Olston, Loo and Widom’s simulations, a rule that multiplies a cached interval’s width by 1 + α on a pushed refresh with probability min{ρ, 1}, and divides it by 1 + α on a query’s exact fetch with probability min{1/ρ, 1}, settled within 1% of the best fixed width on one random walk and within 5% over eight walk settings; on a 50-host network-traffic trace, restricted to exact copies, it almost matched a tuned adaptive cache-or-not rule, and with a 50-value cache, where queries tolerated imprecision, it cost less; that it converges to the best width is argued, not proved | Olston, Loo and Widom (2001), LIT-1109, read in NOTE-912. The rule treats a pushed refresh as “too narrow” and a fetch as “too wide”, weighted by ρ = 2C_vr/C_qr, and needs no measurement of update rates, query rates or precision demands. Within 1% of the best fixed width on one random walk and within 5% over eight settings; with α = 1 and thresholds it tracked a 50-host traffic trace. With every copy forced exact it decided cache-or-not as well as an adaptive exact-caching rule, and did better where imprecision was allowed, except with a small cache. Convergence is argued from the balance condition only, all evidence is the authors’ simulation, and there is no worst-case guarantee. | Active |
| THEORY-378 | When items change as independent Poisson processes with known rates and a fixed total refresh rate is shared among them, refreshing every item equally keeps the copy at least as fresh and at most as old, averaged over time, as refreshing in proportion to change rate, for any spread of rates; the freshness-optimal share rises and then falls with change rate and gives nothing to items that change too fast for the budget | Cho and Garcia-Molina (2000), LIT-1119, read in NOTE-906. Freshness is the fraction of items equal to their source, age the mean time since an unseen change, both averaged over time and over equally weighted items. With fixed-interval refresh, uniform refresh is never worse than proportional on either measure, by convexity (proof in the authors’ technical report). Under a fixed budget the freshness-optimal allocation starves the fastest-changing items; the age-optimal one refreshes them somewhat more. It prices staleness as an objective under a fixed budget: there are no reads, no query savings, no build cost and no cost of a stale use, so it is a staleness term and not a model trading staleness against upkeep. | Active |
| THEORY-379 v2 | Fagin’s Algorithm finds the top k of N objects held in m attribute-sorted lists under any monotone combining function, and when the lists’ orderings are independent with at least m − 1 of them uniformly random it does so with O(N^((m−1)/m) k^(1/m)) sorted and random accesses with arbitrarily high probability; with dependent orderings the bound can fail | Fagin (1996/1999), LIT-1108, as reported by Fagin, Lotem and Naor (2001), LIT-1115, read in NOTE-924. Fagin’s own paper could not be obtained, and the reporting paper declines to describe the probability model (“off track”). It does correct one misreading: the only probabilistic assumption is independence of the lists’ orderings, not a uniform distribution of grades. For two lists the bound is √(Nk). It does not hold for anti-correlated attributes, where any algorithm can need about half the objects. | Active |
| THEORY-380 v2 | For datalog over a finite input instance annotated from a commutative ω-continuous semiring, with an answer’s tag defined as the sum over its derivation trees of the product of its leaves’ tags, the tags are the least fixed point of a finite polynomial system and factor through formal power series N∞⟦X⟧ in the input tuple ids; an answer tuple’s series has an infinite coefficient exactly when one of its derivation trees contains a tuple lying on a cycle of unit rules | Green, Karvounarakis and Tannen (2007), LIT-1110, read in NOTE-913. A recursive query can derive a tuple in infinitely many ways, so polynomials do not suffice and the semiring needs infinite sums. With ω-continuous semirings and the provenance semiring N∞⟦X⟧ the paper states the analogues of the relational results: a fixed-point characterization (Theorem 5.6), factorization (Theorem 6.4), and a test for infinite coefficients (Theorem 6.5), with algorithms that decide finiteness and compute coefficients. Proofs are not given. Plain N and N[X] are excluded, the provenance can be an infinite object held only as equations, and negation remains outside. | Active |
| THEORY-381 | In the cell-probe model, a near-neighbour structure on n random points that answers in t probes with m cells of w bits must satisfy m^t·w/n ≥ the robust expansion of the metric’s neighbourhood graph at measure 1/m^t (and (mwt/n)^t ≥ its vertex expansion if deterministic), so in expanding spaces such as the hypercube fast queries need space polynomially larger than the data; the bounds are average case, approximate, and fall off exponentially in t | Panigrahy, Talwar and Wieder (2010), LIT-1102, read in NOTE-914 from the arXiv preprint. For graphical neighbour search, to which decision approximate near neighbour reduces, on data drawn so that neighbourhoods rarely overlap: deterministic t-probe structures satisfy (mwt/n)^t ≥ Φv and m^t·t·w/n ≥ Φv(1/m^t); randomized ones satisfy m^t·w/n ≥ Φr(1/m^t, γ/t) and (mwt⁴/n)^{2t} ≥ Φr(1/m, γ/t). On the hypercube this gives space n^{1+Ω(1/(ct))} at approximation c and n^{Ω(1/(tε²))} at 1 + ε; a one-probe structure matches on Cayley graphs; low-contention dynamic structures need update time at least Φr(τ, 1/(4t²)) over poly(t). Tight only for constant t; nothing about worst-case or structured data, workloads or build time. | Active |
| THEORY-382 v2 | When a web server keeps each page fragment either cached and invalidated by updates or stored and refreshed after them, materializing trades staleness for response time, and in one synthetic simulation with fixed costs a periodic greedy that moves a capped number of fragments between the two, to keep the share of fresh fragments served above a set threshold, tracked that threshold by the authors’ reading of its curves, answered faster than caching alone and recovered from overload by materializing everything; no bound on its distance from the best plan is shown | Labrinidis and Roussopoulos (2003), LIT-1105, read in NOTE-911. Quality of Data is the share of fresh fragments in served pages; OVIS(θ) estimates per fragment the change in response time (hits and misses against refresh load) and in QoD from switching policy, and each period greedily switches up to MAX_CHANGE fragments to spend a QoD surplus on speed or buy back a deficit. The paper asserts the greedy step is best under the cap and proves nothing; the evidence is one simulator with uniform costs and no comparison with an optimum. The reading the evidence does not support: that this is a joint cost model of build, upkeep, staleness and savings. Staleness is a user-set threshold, not a price, and switching and storage are free. | Active |
| THEORY-383 v2 | On one Apache and Informix server answering cheap indexed selections over 1000 pages, a page whose query result was stored in the DBMS and refreshed on every update answered more slowly than one recomputed on each request at five or more updates per second, which the authors explain by reading the stored result costing about as much as recomputing it and its refresh loading the same DBMS, while a finished page stored at the web server and rewritten on every update answered 10 to 230 times faster than either | Labrinidis and Roussopoulos (2000), LIT-1118, read in NOTE-908. Three ways to serve a database-generated page under online updates, with no stale reads allowed: compute on request, store the query result in the DBMS, or store the finished page at the web server. A cost model sums access and update costs by frequency; the per-page selection problem is stated and not solved. On their testbed the page stored at the web server won at every load, and the result stored in the DBMS lost to recomputation at five or more updates per second. The reading the evidence does not support: that storing an answer pays whenever it is reused enough. Where it is stored decided the outcome, and the generalization to other systems is the authors’ expectation, not a measurement. | Active |
| THEORY-384 | A nearest-neighbour graph built by pruning each point’s links against all other points with a factor α > 1, so that from every point some link is α times closer to any other point, lets greedy search from any start reach a ((α+1)/(α−1) + ε)-approximate nearest neighbour in O(log_α(∆/((α−1)ε))) steps of cost O((4α)^d log ∆), for doubling dimension d and aspect ratio ∆; the ratio is tight and the dependence on ∆ cannot be replaced by one on n, and the build takes O(n³) time done naively | Indyk and Xu (2023), LIT-1112, read in NOTE-910. Proved for DiskANN’s “slow preprocessing” build, not the build DiskANN implements. The guarantee rests on one property of the graph (α-shortcut reachability, α > 1), gives a constant approximation, not exact search, is logarithmic in the aspect ratio ∆, which can exceed n by far, and has a per-step cost exponential in the doubling dimension. HNSW and NSG prune with α = 1 and are not covered. | Active |
| THEORY-385 v2 | In HNSW over 10-dimensional data in 100 isolated clusters, linking each new item to its nearest neighbours leaves part of the bottom layer unreachable from the entry point and caps recall well short of 1 however wide the query search, while linking a candidate only when it is nearer the new item than to any neighbour already kept keeps that layer connected and reaches recall near 1 with about half the links; on uniform data at 10 and 64 dimensions the two rules differ little; shown in Malkov and Yashunin’s own tests on 10 million points and in an independent re-run on 30,000, which found the nearest-neighbour rule capped at a recall@10 of 0.72 to 0.83 against 0.999, at matched build cost too | Malkov and Yashunin (2020), LIT-1098, read in NOTE-916. In HNSW, each new element’s neighbours are chosen either as its M nearest candidates or by a pruning rule that keeps a candidate only if it is nearer the new element than to any neighbour already kept, a rule related to the relative neighbourhood graph. On 10M random d = 10 vectors in 100 isolated clusters, nearest-neighbour links failed to reach high recall and the pruning rule did not. In every test reported, the rule was at least as good. Its gain was largest at low dimension, at high recall, and on clustered data, and small on uniform or very high-dimensional data. These are the authors’ benchmarks: one clustered dataset, synthetic, with no independent replication in this reading. Re-run independently at smaller scale on promotion (2026-10-20), with the same result. | Active |
| THEORY-386 | When a cached interval of width W is kept valid by its source, which pushes a refresh at cost C_vr whenever the value leaves it, and queries fetch the exact value at cost C_qr whenever it is too wide, and the chances of the two per step go as 1/W² and as W, the expected refresh cost is least at W* = (ρK₁/K₂)^(1/3) with ρ = 2C_vr/C_qr, where ρ times the chance of a pushed refresh equals the chance of a fetch | Olston, Loo and Widom (2001), LIT-1109, read in NOTE-912. The cost rate Ω(W) = C_vr·K₁/W² + C_qr·K₂·W is least where ρ·P_vr = P_qr. The minimum is calculus on assumed forms. P_qr ∝ W follows exactly from uniform precision demands; P_vr ∝ 1/W² comes from a Chebyshev bound on a random walk used as an estimate, and the paper’s simulation of a random walk measured both proportionalities. Staleness here is priced as refresh traffic: the source guarantees validity, so a stale read is imprecise but never wrong, and there is no build cost and no cost of a wrong or imprecise answer. | Active |
| THEORY-387 v2 | For positive recursive Datalog views under set semantics, deleting every derived tuple that has any derivation through a deleted base fact, then restoring each deleted tuple that still has a derivation from what remains, then adding what insertions derive, leaves exactly the tuples derivable from the new base (E − Δ⁻) ∪ Δ⁺, with no derivation counts stored; the extension to stratified negation and aggregation is asserted, not shown, and the size of the overestimate is not measured | Gupta, Mumick and Subrahmanian (1993), LIT-1114, read in NOTE-915. DRed runs stratum by stratum: semi-naive Δ⁻-rules compute an overestimate of deletions, δ⁺-rules put back those with a derivation in the reduced database, and Δ⁺-rules add insertions. Theorem 7.1 states the result contains t if and only if t has a derivation in (E − Δ⁻) ∪ Δ⁺; the proof is in a 1992 AT&T report the record has not seen. The paper asserts, and does not show, that DRed handles stratified negation, aggregation and rule changes, that it beats the PF algorithm by up to an order of magnitude, and that it is slower than counting on nonrecursive views. | Active |
| THEORY-388 | Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy | Indyk and Motwani (1998), LIT-1106, read in NOTE-905 from the authors’ 1999 expanded version, compared with the STOC text. A hash family under which points within r1 collide with probability at least p1 and points beyond r2 with at most p2 gives n^ρ random tables, each storing every point: O(dn + n^{1+ρ}) space and O(n^ρ) hash evaluations per query, ρ = ln(1/p1)/ln(1/p2), for the decision version, which ring-cover trees connect to approximate nearest neighbour with polylog overhead. Bit sampling gives ρ ≤ 1/(1+ε) in Hamming space and, by embedding, ℓ_p for p in [1, 2]; the STOC text states 1/ε and only for ε > 1. Grid bucketing gives Õ(d) queries for n·O(1/ε)^d preprocessing. Preprocessing is counted as space. Nothing is proved about whether these exponents can be beaten. | Active |
| THEORY-389 | For conjunctive queries and views without comparison predicates, a stored view can be used in an equivalent rewriting of a query exactly when the view’s body maps homomorphically into the query’s, a minimal rewriting needs no more literals than the query, and deciding whether a rewriting shorter than the query or one using only the views exists is NP-complete in the size of the query and view definitions; with comparisons in both, deciding whether any rewriting uses the views is Π2p-complete | Levy, Mendelzon, Sagiv and Srivastava (1995), LIT-1100, read in NOTE-909. Every use of a view in an equivalent rewriting comes from a containment mapping from the view into the query (Proposition 3.1); without comparisons a locally minimal complete rewriting has at most as many literals as the query (Lemma 3.5), which puts the problems in NP, and they are NP-complete (Theorems 3.7, 3.9, 3.10), with two independent sources of hardness. The results are worst case, in the size of the definitions rather than the stored data, for equivalent rewritings under set semantics only; they do not say recognizing reuse is costly in practice, and they do not cover contained (partial) rewritings. | Active |
| THEORY-390 | Any distribution of n × N matrices whose norms concentrate as Johnson–Lindenstrauss proofs require gives the restricted isometry property of order k with probability at least 1 − 2e^{−c₂n} for k ≤ c₁n/log(N/k), by a union bound over (12/δ)^k net points on each of the (eN/k)^k sparse supports; this makes k log(N/k) measurements sufficient for Gaussian, ±1 and sparse three-valued matrices, and does not show they are necessary | Baraniuk, Davenport, DeVore and Wakin (2008), LIT-1156, read in NOTE-931 from the author-posted copy. The only assumption is the concentration inequality Pr(|‖Φx‖² − ‖x‖²| ≥ ε‖x‖²) ≤ 2e^{−nc₀(ε)}, met with Achlioptas’s c₀(ε) = ε²/4 − ε³/6. The proof is the JL argument run over a net on every sparse support, plus a step from the net to the whole subspace. Necessity of k log(N/k) is recalled from Gelfand widths, not derived from concentration, and constants are implicit. | Active |
| THEORY-391 v3 | Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble | Donoho and Tanner (2009), LIT-1160, read in NOTE-933 from arXiv 0906.2530v1. The Gaussian transition is a theorem of the authors’ earlier papers, restated here. The universality is an observation from 2,948,000 optimizations at N = 200, 400 and 1600: strict finite-N universality is rejected, and agreement as N → ∞ is not rejected. Sparse and Hadamard matrices sit about one transition width low at small N. The title states only what was measured. Whether the agreement is a limit theorem is QUESTION-168. | Active |
| THEORY-392 v2 | Every map from an n-point subset of a metric space into Hilbert space extends to the whole space with Lipschitz constant at most K√(log n) times its own, by dimension reduction to K log n coordinates followed by Kirszbraun and the non-linear Hahn–Banach theorem; for L_p, 1 ≤ p < 2, the constant must be at least δ(Log n/Log Log n)^(1/p−1/2), so √(log n) is right up to a log log factor in the worst case | Johnson and Lindenstrauss (1984), LIT-1153, read in NOTE-930. Theorem 1 proves Marcus and Pisier’s conjecture L(X, n) ≤ K(Log n)^(1/2) for every metric space X; Theorem 3 gives L(p, n) ≥ δ(Log n/Log Log n)^(1/p−1/2) for 1 ≤ p < 2, hence L(∞, n) ≥ δ(Log n/Log Log n)^(1/2). The √(log n) comes from the √k cost of passing between ℓ₂ᵏ and ℓ∞ᵏ, with k of order log n. Whether the upper bound is sharp the paper leaves open, and the constants are loose. | Active |
| THEORY-393 | Any two families of n vectors of norm at most 1 have images in ⌊C log(2 + ε²n)/ε²⌋ dimensions whose cross inner products ⟨x_i, y_j⟩ are each within an additive ε of the originals, found by a randomized polynomial-time algorithm, which for small ε is below the log n/ε² of the Johnson–Lindenstrauss lemma; only cross products are kept, the new vectors’ norms are not bounded, and whether one set can be embedded that small is open | Alon and Klartag (2017), LIT-1151, read in NOTE-929 from arXiv v4. Theorem 1.2 answers, in a bipartite form, Larsen and Nelson’s question whether log n can become log(2 + ε²n) in the JL lemma. The proof is convex geometry: Khatri–Šidák bounds the Gaussian measure of a body of slabs, and a weakened low M*-estimate puts a random t-dimensional subspace close to every scaled a_i. Conjecture 2.4, the same with norms O(1), is proved only for t of order n (Theorem 2.5). It is not an embedding of one point set and says nothing about distances. | Active |
| THEORY-394 | Every n-point metric space embeds in Euclidean space of O(log n) dimensions with distortion O(log n), by distances to random sets of every scale followed by Johnson–Lindenstrauss, and constant-degree expanders need distortion Ω(log n) in every ℓ_p with 1 ≤ p ≤ 2, of any dimension, so O(log n) is tight in the worst case; the constants are unspecified, the lower bound rests on Leighton and Rao’s flow–cut gap, and nothing is shown for trees or for p > 2 | Linial, London and Rabinovich (1995), LIT-1155, read in NOTE-928 from the authors’ February 1995 manuscript. Theorem 3.2(1) states Bourgain’s theorem (LIT-1161, unread): every n-point metric embeds in O(log n)-dimensional Euclidean space with distortion O(log n). The paper reproves the distortion with a randomized map into ℓ_p^{O(log² n)} built from distances to O(log n) random sets of each size 2ˡ, and gets the dimension from Johnson–Lindenstrauss. Proposition 4.2 shows that every embedding of a constant-degree expander into ℓ_p, 1 ≤ p ≤ 2, has distortion Ω(log n), from the expander’s Ω(log n) gap between flow and cut. Together these answer Johnson and Lindenstrauss’s Problem 4 in the negative, which the paper does not say. | Active |
| THEORY-395 v2 | For ε between about lg^0.5001 n/√min{n, d} and 1, some n-point set in ℝ^d needs Ω(ε⁻² lg(ε²n)) dimensions under any map, linear or not, that keeps its pairwise squared distances within 1 ± ε, so a random projection with a union bound is optimal up to a constant for the worst set; nearly orthogonal sets can need fewer, and the bound says nothing about typical data | Larsen and Nelson (2017), LIT-1154, read in NOTE-925 from arXiv v2. Theorem 2 removes the lg(1/ε) gap that Welch, Levenshtein and Alon left, for arbitrary maps. The proof is an encoding argument over a family of k-sparse vectors with k ≈ ε⁻², ending at nm = Ω(kn lg(d/k)). It is a worst-case bound by existence with an untracked constant. It is not a phase transition, and the paper notes that ε-incoherent sets fit in o(ε⁻² lg n) dimensions for small enough ε. | Active |
| THEORY-396 | Every Lipschitz map from an n-point metric space into any Banach space extends to any containing space with Lipschitz constant at most C log n/log log n times its own, by averaging over random partitions at every scale, and every n-point subset of L₂ extends into any Banach space at C√(log n) after Johnson–Lindenstrauss dimension reduction; for maps into Hilbert space this is no improvement on K√(log n), and whether √(log n) suffices for every target is left open | Lee and Naor (2005), LIT-1152, read in NOTE-927 from the authors’ January 2004 preprint. Theorem 1.10 (Theorem 5.1(4)): ae(X) ≤ C log n/log log n for every n-point metric space X, bounding the extension constant over every superspace and every Banach target at once, improving Johnson, Lindenstrauss and Schechtman’s C log n. Theorem 1.12: eₙ(L_p, Z) ≤ C(p)(log n)^{1/p} for 1 < p ≤ 2, C√(log n) at p = 2, through the Johnson–Lindenstrauss lemma and e(ℓ₂ᵈ, Z) ≤ C√d. For Hilbert targets Johnson and Lindenstrauss’s K√(log n) (THEORY-392) stays the better bound, and the paper states as open whether O(√(log n)) holds for every metric domain and normed target. | Active |
| THEORY-397 | A random linear map into k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n dimensions keeps every pairwise squared distance of a fixed set of n points in ℓ₂ within 1 ± ε, because each difference vector’s squared length concentrates with tails exp(−(k/2)(ε²/2 − ε³/3)) and a union bound over the pairs costs only ln n; the proofs cover a random subspace, Gaussian entries and independent zero-mean ±1 or √3·{+1, 0, −1} entries, and no matrix with nonnegative entries | Dasgupta and Gupta (2003), LIT-1157, read in NOTE-932, give the explicit bound for a projection onto a random subspace; Johnson and Lindenstrauss (1984), LIT-1153, read in NOTE-930, prove the lemma with k = [K(τ) log n] and K ≥ (10/τ)² by Lévy’s isoperimetric inequality; Achlioptas, LIT-1159, read in NOTE-926 from a 2002 draft, carries the same k₀ to independent ±1 and sparse three-valued entries. The guarantee is for a fixed finite set, on squared distances, and it rests on entries being zero-mean. It says nothing about maps that go up in dimension or are followed by a threshold. | Active |
| THEORY-398 v2 | A Lipschitz map from a subset of a metric space whose points are at least ε apart and whose diameter is D extends to the whole space into Hilbert space with Lipschitz constant at most 6D/ε times its own, and, by Benyamini’s argument in the same paper, into any Banach space with at most 4D/ε, with no dependence on the number of points | Johnson and Lindenstrauss (1984), LIT-1153, read in NOTE-930. Theorem 2 gives 6D/ε for maps into ℓ₂, through ℓ₁ and a weak form of Grothendieck’s inequality; Benyamini’s appendix gives 4D/ε for any Banach target, through a retraction of ℓ∞(Γ) with Lipschitz constant at most 2. The bound trades the count of points for the spread of the set. Whether either constant is best possible is not discussed. | Active |
| THEORY-399 | Storing every inner product of n points of the unit ball of R^k to additive error ε takes Θ(n log n/ε²) bits once k ≥ log n/ε² and Θ(nk log(2 + log n/(ε²k))) below, for 1/n^0.49 ≤ ε ≤ 0.1, and comparing these counts across dimensions shows that some n such points cannot be mapped into fewer than Ω(min{k, log(2 + ε²n)/ε²}) dimensions with additive error ε for every ε ≥ 2/√n; this is a worst-case lower bound with unstated constants, the small-ε bit count is open for k above that dimension, and no one-family embedding that small is proved | Alon and Klartag (2017), LIT-1151, read in NOTE-929 from arXiv v4. Theorem 1.1 gives the bits of an additive ε-distance sketch for every k when 1/n^0.49 ≤ ε ≤ 0.1, by a volume argument above and a packing of net points against random probes below. Since an embedding is a sketch, Corollary 2.1 re-proves Larsen and Nelson’s bound in that range, and Corollary 2.3, using the Gaussian correlation inequality in place of a union bound, extends it to every ε ≥ 2/√n with log(2 + ε²n) in place of log n. The dimension statements are for additive error on squared distances of points of norm at most 1; the relative form for small ε follows but is not stated. The matching upper bound for one set is open. | Active |
| THEORY-400 | A metric space that admits, at every scale, random partitions into small pieces with each point padded away from piece boundaries with constant probability is absolutely Lipschitz extendable: maps from it into any Banach space extend to any containing space at O(1/padding), which gives C log λ for doubling constant λ, C r² for K_r-minor-free graph metrics, C(g+1) on genus-g surfaces and C√d from ℓ₂ᵈ; the bounds are upper bounds only, and the method needs a decomposition the domain may not have | Lee and Naor (2005), LIT-1152, read in NOTE-927 from the authors’ January 2004 preprint. Theorem 4.1 turns (ε, δ)-padded random partitions of the domain at every scale into a C/(εδ)-gentle partition of unity, and Lemma 2.1 turns that into extensions of constant 3C/(εδ) into any Banach space; Lemma 3.8 carries the partitions from X to any superspace. Theorem 5.1 gives ae(X) ≤ C log λ(X), ≤ C r² for K_r-minor-free graph metrics, ≤ C(g+1) on surfaces of genus g, and e(ℓ₂ᵈ, Z) ≤ C√d, generalizing Johnson, Lindenstrauss and Schechtman’s Cd and Matoušek’s trees. No lower bound shows any of them tight. | Active |
| THEORY-401 | A fair lottery on the natural numbers has a uniform, regular probability defined on every set of tickets, by normalised numerosity with values in the hyperrationals; it is fixed only up to a choice of free ultrafilter, it is hyperfinitely and not countably additive, and its standard part is ordinary asymptotic density | Wenmackers (2011), LIT-163, chapter 2, read in NOTE-139. P_num(A) = num(A)/α is defined on all of 𝒫(ℕ), gives each ticket 1/α > 0, and satisfies FAIR, ALL and a hypercountable form of SUM (eqs. 2.20–2.27). Proved, given the axiom of choice. It is not a free lunch: one solution per free ultrafilter, no invariance under relabelling, no countable additivity, and by the author’s own admission it is empirically indistinguishable from the finitely additive real-valued solution. | Active |
| THEORY-403 | The single-photon Mach–Zehnder phenomenology with balanced beamsplitters and phases 0 or π, including which-way detection, the bomb tester, delayed choice and the quantum eraser, is reproduced exactly by a local, deterministic classical field model with an epistemic restriction, so that slice of interference forces neither complementarity, nor observer-dependent reality, nor nonlocality | Catani, Leifer, Schmid & Spekkens (2021), LIT-019, read in full in NOTE-018. Each mode carries a binary occupation and a binary phase; an agent may know at most one of N, Φ, N⊕Φ; the beamsplitter swaps N_L with Φ_L⊕Φ_R. The model gives the quantum statistics for the phase shifter, which-way detection, the bomb tester (¼), delayed choice and the eraser. The existence proof holds for that fragment. The paper’s further claim, that the model is generalized-noncontextual and so classical by its preferred criterion, is asserted, not proved, and unbalanced beamsplitters already admit a no-go. | Active |
| THEORY-404 v2 | With a common prior, a sender who commits to a signal the receiver observes can induce any distribution of posteriors whose mean is the prior, so an openly designed experiment moves a fully informed Bayesian receiver whenever the concave closure of the sender’s value lies above its value at the prior; at the optimum, under the paper’s tie-breaking, the receiver is certain when she takes the sender’s worst action and indifferent when she takes another | Kamenica & Gentzkow (2011), LIT-1021, read in NOTE-717 with the proofs. Proved results (Proposition 1, Corollary 2, Propositions 4 and 5) for a single ideal Bayesian receiver, a finite state space, commitment and sender-preferred tie-breaking. A judge who knows the investigation was designed to convict convicts 60 percent of defendants when 70 percent are innocent. The paper does not analyse manipulation, does not compute the receiver’s welfare and does not cite Blackwell. | Active |
| THEORY-405 v2 | Adding one classifier-free guidance term per concept to a diffusion model’s unconditional noise prediction samples images that satisfy several concepts at once without retraining, and on synthetic scenes it composes object positions far better than compositional energy-based models but relations no better; the reading of the summed score as the score of a product of experts holds only for concepts conditionally independent given the image, unit weights, a conservative field and the noise-free distribution | Liu, Li, Du, Torralba & Tenenbaum (2022), LIT-770, read in NOTE-597. CLEVR object positions, 1/2/3 components: 86.42/59.20/31.36% against an EBM’s 70.54/28.22/7.34%. Relations: 60.40/21.84/2.80% against 78.14/24.16/4.26%. FFHQ attributes are comparable to LACE. Text-model composition (GLIDE, Stable Diffusion) is shown only in figures, with attribute confusion and hybrid objects among its listed failures. The product is derived at the data distribution and applied at every noise level, which NOTE-597 notes is in general not the score of the noised product. | Active |
| THEORY-407 | For a log-integrable linear cocycle over an ergodic probability-preserving system, vectors grow at finitely many exponential rates along an invariant flag, and for isometries of a proper metric space with integrable displacement the linear drift is detected by a horofunction, with no curvature assumed | Filip (2017), LIT-093, expository notes read in full in NOTE-078. Thm 2.2.6 is the Oseledets theorem, proved without Kingman; Thm 4.3.1 is the Karlsson–Ledrappier theorem, proved completely apart from cited measurable-selection results. Both are classical and not in doubt. The notes’ own renditions of two other steps are flawed: Prop. 3.4.8(iii) is mis-stated, and the §5.3 tracking proof misapplies its lemma, so the Karlsson–Margulis and Kaimanovich forms should be cited to their originals. The hypotheses are ergodic stationary sampling and N → ∞. | Active |
| THEORY-408 | In a one-locus model with non-overlapping generations, random mating and weak selection, gene frequencies change as if each individual maximized its inclusive fitness, so a trait that costs its bearer spreads only if its effect on relatives exceeds 1/r times the cost; the increase of mean inclusive fitness is proved only for traits whose total effect on others is not negative | Hamilton (1964), Part I, LIT-497, read in full in NOTE-401. Each effect an individual causes falls partly on genes identical by descent with its own (weight r) and partly on others, which only dilute. So selection tracks inclusive fitness. The threshold is written −k > 1/r (§5), not rb > c: “more than two brothers, or four half-brothers, or eight first cousins”. The maximization theorem needs non-negative total effects on others; for harming traits the change only points uphill. r is exact only for loci not under selection, and Hamilton could not bound the error (§4). | Active |
| THEORY-417 | ’Personal identity’ names several largely independent questions, among them who one is (characterization), what a person is, what it takes for one to persist, how many of us there are and what matters in survival; the persistence question concerns numerical identity, which ‘remaining the same person’ in the characterization sense presupposes, and posing it as ‘a person at one time and a person at another’ assumes that we are persons essentially | Olson (2002; rev. 2023), LIT-454, read in NOTE-374. Seven questions: characterization, personhood, persistence, evidence, population, personal ontology and what matters. The distinctions are conceptual and the record holds them as stated. They do not say which answer to any of the questions is right; the entry’s own answer, animalism, is argued in other sections and is not part of this account. The self is set aside by the entry, so the map does not cover the self literature. | Active |
| THEORY-419 v2 | In a structural causal model of autonomous equations, with intervention do(X = x) defined, as Pearl defines it, as a surgery that deletes X’s own equation, sets X to x and solves the rest, and with observing X = x violating no equation and amounting to ordinary conditioning, models that agree on every observational distribution can disagree on the effect of an intervention, and in a structural equation y = βx + ε the coefficient β is the rate of change of E[Y | do(x)], which equals the regression slope of Y on X only when X is uncorrelated with ε | Pearl (2009), LIT-894, skimmed in NOTE-696 from four excerpts the author posts: the contents, the second-edition preface, §§5.3.2–5.4.1 and the Epilogue, 46 of some 480 pages. The account is well established and the excerpts state it in full; Proposed because it rests on a skim. The surgery needs autonomous equations and a diagram saying which equation belongs to which variable. The excerpts state the three rules of the do-calculus without derivation, and contain no three-level hierarchy of association, intervention and counterfactual. | Active |
| THEORY-420 | In trained transformers, zeroing a decile of a weight matrix’s singular values does damage that falls monotonically with their size in square matrices, whose random-matrix null reaches zero, but not in rectangular ones, whose null has a positive lower edge: there the smallest decile is never the least damaging and can be among the most | Staats, Thamm & Rosenow (2024), LIT-856, read in NOTE-659. Freshly initialised matrices of BERT, Pythia and Llama follow the Marchenko–Pastur law; trained ones depart from it above the bulk, and, in rectangular matrices only, below it. Zeroing Llama-3 8B’s smallest Down-Projection decile leaves 2.0% on GSM8K from 43.2%, worse than every decile but the largest, while the same decile of the square Attention-Output leaves 40.0%. Whether the below-bulk outliers are what carries the damage is not tested: the decile mixes them with the lower bulk, and overlap with the input’s principal directions and ablation importance disagree for several matrix types. | Active |
| THEORY-421 v2 | In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary | Yang, Zhou, Ying, Chen & King (2023), LIT-871, read in NOTE-669. On two synthetic 3-ary trees of eight levels, HGCN leaves the root at hyperbolic distance 1.8–3.3 from the origin where the minimum is 1.2–2.4, the distances are roughly normal rather than leaf-heavy, and radius orders true levels for 69–75% of 5,000 random pairs. Recentring on the centroid plus an outward push raises that to 75–84%. The diagnosis rests on one architecture, two synthetic trees and one table without variance, and the paper does not separate a real failure to order levels from the origin’s arbitrariness under losses that see only pairwise distances. | Active |
| THEORY-422 | In the standard formal models of group learning, no communication network is best for every epistemic problem: costly transmission favours minimally connected or star networks, pooling of estimates by averaging favours regular networks, and learning by trying in bandit problems can favour sparse ones, so a group’s network cannot be judged until its learning problem is known | Zollman (2013), LIT-716, a survey read in NOTE-567. Each result it gathers is proved or simulated in its source (Bala and Goyal, Jackson and Wolinsky, DeGroot, Hegselmann and Krause, Zollman). Under their idealisations the optimal structure changes with the problem, so neither “too little communication” nor unequal centrality is by itself an epistemic pathology. It is a claim about models: utilitarian optimality only, with fairness, robustness and the route to the optimum left out. | Active |
| THEORY-424 v2 | Mutual-information measures of individuality that read a system’s next state off its present one measure association, not propagation: I(S_{t+1}; S_t) can be positive when the environment alone drives the system, and conditioning on the environment does not recover the interventional quantity, which can be zero where the conditional measure is one bit | Bourrat (2024), LIT-005, read in NOTE-027, against Krakauer et al.’s (2020) measures. A* = I(S_{t+1}; S_t) is confounded when the environment is a common cause (his Fig. 2), and his interventional  = I(S_{t+1}; Ŝ_t) is zero there. His claim that  equals Krakauer’s conditional A = I(S_{t+1}; S_t | E_t) is false in general: NOTE-027’s XOR system has A = 1 bit and  = 0.  itself depends on an intervention distribution the paper leaves open, and neither it nor the paper’s two ETI criteria is computed on any system. | Active |
| THEORY-425 | A connecting path trained only on clean data keeps only the behaviour the clean data ask for: between two backdoored or error-injected image classifiers, models near the ends of such a path keep most clean accuracy and lose the attack, and a path flat in ordinary loss can carry a barrier in adversarial loss that tracks the input Hessian’s top eigenvalue | Zhao, Chen, Das, Ramamurthy & Lin (2020), LIT-668, read in NOTE-530. A quadratic Bezier path trained on 50–2,500 clean samples between two backdoored VGG-16s on CIFAR-10 has, at t = 0.1, 88% clean accuracy and 1.1% backdoor success with 2,500 samples, and 63% and 2.5% with 50; injected weight errors fail at every point of such paths. Paths trained with the standard loss between regular and adversarially trained models are flat in standard loss with a PGD-loss barrier correlated with the input Hessian’s top eigenvalue (0.71–0.88). The proposition offered for that correlation rests on an assumption that would remove the barrier. | Active |
| THEORY-426 v2 | Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion; in single deterministic runs on a T-maze and an 8 × 8 maze it planned through short-term costs that agents with a shorter horizon did not; that the pruning keeps deep search tractable beyond toy problems is argued, not shown | Friston, Da Costa, Hafner, Hesp & Parr (2020), LIT-577, read in NOTE-454 as prose: the equations were not recoverable from the text layer. The recursion is a construction (strong). That it plans through costs rests on single deterministic runs of a T-maze and an 8 × 8 maze. The title keeps that scope. Active as a report of the construction and of those runs; the general form is THEORY-1245, Proposed. | Active |
| THEORY-427 | For gradient matrices of rank at most r, the Kronecker product of the square roots of the accumulated row and column gradient second moments dominates the full-matrix AdaGrad statistic up to the factor r, so preconditioning each side of a weight matrix by a fourth root, as Shampoo does, preconditions at least as strongly as full-matrix AdaGrad and has O(√T) regret for convex Lipschitz losses | Gupta, Koren and Singer (2018), LIT-621, read in NOTE-480. Lemma 8 is pure matrix analysis (operator monotonicity of the geometric mean); Theorems 7 and 10 give regret √(2r) D Tr(L^{1/4}) Tr(R^{1/4}), O(√T) when the spectral norm of the gradients is at most 1. Proved, in online convex optimisation, with D uncontrolled because the implementation omits the projection. Nothing here is a result about non-convex training; the deep network curves are four runs without seeds. | Active |
| THEORY-430 | A real hyperplane arrangement in ℝⁿ has (−1)ⁿχ(−1) regions and (−1)^rank χ(1) bounded regions, where χ is the characteristic polynomial of its intersection poset, so these counts depend only on which hyperplanes meet and in what dimension, not on the shapes of the regions | Stanley (2006), LIT-355, lecture notes read in full in NOTE-313, with Lectures 1–2 and 6 read closely. Zaslavsky’s theorem (Thm 2.5) is proved twice, by a deletion–restriction recurrence and by Euler characteristic; the bounded-region half has two unanswered “[why?]” steps in the first proof and an omitted fact in the second. Faces of every dimension are counted by Möbius values (Thm 2.6). The notes never use “topes” or “covectors”, and do not cite Zaslavsky’s own texts. A region is fixed by its side of each hyperplane only implicitly (p. 89). | Active |
| THEORY-431 | For classical systems with Markovian, microscopically reversible dynamics, the entropy production of a trajectory obeys an exact finite-time symmetry against the time-reversed process, P_F(+ω)/P_R(−ω) = e^ω, wherever ω is odd under reversal; Jarzynski’s work equality and the mean second law follow as corollaries, and the symmetry constrains fluctuations, not which steady state a system selects | Crooks (1999), LIT-522, read in NOTE-413, with Jarzynski (1997), LIT-517 (NOTE-417), and Seifert’s review (2012), LIT-523 (NOTE-420). The work equality ⟨e^{−βW}⟩ = e^{−βΔF} holds at any switching rate, and Crooks derives it from a trajectory-level entropy-production theorem; Seifert derives every known theorem of the kind from one master theorem. These are exact given their premises. They are not proofs of the second law, since the dynamics build irreversibility in, and they say nothing about extremum principles for steady states. Their practical reach is work fluctuations of order k_BT. | Active |
| THEORY-435 | In Moral Foundations Questionnaire responses, Care and Fairness are endorsed across the political spectrum, while Loyalty, Authority and Sanctity rise toward the conservative end, in every world region sampled | Graham, Haidt et al. (2013), LIT-499, read in NOTE-396, reporting large internet samples in twelve world regions; and Atari et al. (2023), LIT-500 (NOTE-398), with the MFQ-2 in 19 nations. The profile is robust as a description of self-report. It is a finding about what people say they weigh, mostly online, mostly English-speaking, and mostly WEIRD; and Purity, which carries the steepest gradient, fails cross-national invariance in the MFQ-2. | Active |
| THEORY-437 | When each agent’s feasible actions depend on the others’ actions, an equilibrium exists if every action set is a contractible polyhedron, every constraint correspondence has a closed graph with non-empty compact values, payoffs and the maximum-value functions are continuous, and every best-response set is contractible; convexity is not needed | Debreu (1952), LIT-067, read in full in NOTE-037. The theorem (p. 888) is proved by applying the Eilenberg–Montgomery/Begle fixed-point theorem to the product of best-response correspondences. A Remark reduces the continuity of the value function to a compact graph plus continuity (lower hemicontinuity) of the constraint. It is existence only, in finite dimensions with finitely many agents and real-valued payoffs. That it contains Nash’s theorem is asserted; the reduction is routine but not in the paper. | Active |
| THEORY-439 v2 | Given the same few demonstrations, temporarily updating a language model’s weights on them and then conditioning beats conditioning alone, most on tasks with structural rules or distribution shift and most when the update is trained in the in-context format: for models of 1–8B parameters, examples used as context and the same examples used as a training signal are not interchangeable | Akyürek et al. (2024; ICML 2025), LIT-796, read in NOTE-637. On BIG-Bench Hard with Llama 3.1 8B and 10 demonstrations over five seeds, zero-shot 40.9%, in-context 50.5%, per-task LoRA test-time training then in-context 57.8%; 12 of 27 tasks gain at least 2 points, and Boolean expressions falls. Training the update on leave-one-out in-context tasks beats training on bare pairs (57.8% against 51.5%). The update is discarded after each task, so it is as transient as a context. It is two benchmarks and one model family, and which task property predicts the gain is open. | Active |
| THEORY-441 | In spatially compact vacuum general relativity the Hamiltonian vanishes, so time is represented by no flow on a symplectic space and change by no functions on a space of states; this problem of time does not come from the 3+1 split, is absent from asymptotically flat general relativity, and does not show change to be illusory | Belot (2007), LIT-122, read in NOTE-118. A formal, not rigorous, derivation in the covariant phase-space formalism, with standard results cited. The vanishing Hamiltonian and the covariant derivation are strong; the canonical isomorphism of the reduced spaces is “presumed”; the claim that an artificially covariant theory escapes is hedged (“I think”). It is classical only. It diagnoses the problem and does not solve it. | Active |
| THEORY-445 | A network built from group correlations over a discrete split group of plane symmetries, pointwise nonlinearities and coset pooling is equivariant to that group by construction, through the regular representation alone: the group is fixed before training and built into the weight sharing, not found in the data | Cohen & Welling (2016), LIT-314, read in NOTE-288. G-correlation commutes with the left action of G by the change of variables h → uh (eq. 12), and so do pointwise nonlinearities and pooling over cosets, up to edge effects and for rotations that are symmetries of the grid. Matched-parameter p4 and p4m layers lowered rotated-MNIST error from 5.03% to 2.28% and plain-CIFAR10 error from 9.45% to 6.46%. The paper uses no irreducible representations, characters or Fourier analysis, learns nothing about which symmetry the data have, and does not separate the effect of equivariance from the changed channel structure. | Active |
| THEORY-450 | Across 1,176 premodern archaeological sites, the house-size Gini rises only weakly with the number of levels of political hierarchy and with polity population, and the spread of Gini widens with scale: scale raises the ceiling on domestic wealth inequality without setting its level | Feinman et al. (2025), LIT-087, read in full with its SI in NOTE-049. Bivariate fits on the GINI project’s site-level file (47,019 houses): levels of hierarchy give r² 0.16 (all sites), 0.22 (apex), 0.27 (sites with 50 or more houses); log polity population gives 0.11, 0.21, 0.08. The scatters fan out as scale grows, and the sites that reach Gini above 0.75 are autocratic apices of polities with three or more levels. The negative claim, that scale does not fix inequality, is what the record believes. The positive claim, that governance decides it, rests on an expert-coded index partly read from architecture and on a composite whose higher r² comes from summing two predictors; it is not part of this account. | Active |
| THEORY-455 | Every finite stochastic transition matrix on a configuration space equals the entrywise squared modulus of a complex matrix, unitary after dilation, so the Hilbert-space formalism can be written over any indivisible stochastic process as a non-unique representation; within that representation interference is the term by which the process fails to divide at an intermediate time, relative to the chosen configurations, and an ideal correlating environment restores divisibility | Barandes (2023), LIT-086, read in full in NOTE-080. Γ_ij = |Θ_ij|² = tr(Θ†P_iΘP_j) is an identity (Θ_ij = √Γ_ij works), and a unitary Θ on the original space exists only for doubly stochastic Γ. Interference is the cross-term of Eq. 43 against a would-be Markov factorization, and a perfectly correlating environment that then evolves separately makes the process divisible (Eqs. 45–56). The “emergent” Schrödinger equation is the definition of H, unitarity is a choice the claimed gauge freedom does not preserve, and the decomposition is relative to a configuration basis the paper never fixes. | Active |
| THEORY-456 | Iterating Ollivier–Ricci flow on edge weights lengthens edges between communities relative to edges within them, so a weight threshold can separate the communities; but the separation is proved only on symmetric clique graphs under the uniform neighbour measure, and there only by a constant factor for gateway edges, and on a two-block stochastic block model the method fails at an inter/intra ratio near 0.55, where other methods still recover the blocks and well short of the detectability threshold near 0.78 | Ni, Lin, Luo & Gao (2019), LIT-004, read in full with its SI in NOTE-006, which checked Lemma E.1 by linear programming and iterated the proof’s matrix. The flow w ← (1 − κ)d replaces each weight by the transport cost between the endpoints’ neighbourhoods; bridge edges, with few shared neighbours, are negatively curved and grow. Theorem 4.1 (α = p = 0, cliques G(a,b), a > b ≥ 2) shows only non-gateway intra-clique edges shrinking; gateway edges grow at the inter-community rate with a ratio k ∈ (0,1) from Maple. Experiments use α = ½, p = 2, which no theorem covers. The Kesten–Stigum figure is NOTE-006’s calculation. | Active |
| THEORY-462 v2 | If commonsense knowledge includes defaults that later evidence displaces, a monotonic logic, in which an added axiom can never remove a theorem, cannot represent that displacement as it stands; Minsky’s further claim, that no logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge even once repaired, is asserted, not shown | Minsky (1974), the frames memo, LIT-410, read in NOTE-337. Its appendix argues that “logistic” systems fail for commonsense on five counts, of which monotonicity is a correct observation about classical logic. That no repair is possible is asserted (“I think such attempts will continue to fail”), and consistency being undesirable is argued by analogy. The reading says the memo is a source of hypotheses, not evidence; this account takes from it only the half that holds by definition and marks the rest as the memo’s claim. Active as the definitional half and a report of how the memo supports the rest; the further claim is THEORY-1239, Proposed. | Active |
| THEORY-463 | In classical statistical mechanics a strictly irreversible approach to equilibrium cannot come from the isolated Hamiltonian dynamics alone: the fine-grained Gibbs entropy is constant and bounded systems recur, so each account surveyed either weakens the target to typical or long-run behaviour or adds something from outside the dynamics | Frigg & Werndl (2023), LIT-114, the SEP entry, read in NOTE-103. The impossibility half rests on standard consequences of Liouville’s theorem, time-reversal invariance and Poincaré recurrence. The classification half is the survey’s: it is stated account by account, not argued as a general result. The entry is classical only, excludes the direction of time, and reports rather than settles the disputes about each repair. | Active |
| THEORY-466 | Among contestants of equal prowess with dangerous weapons, individual selection can favour conventional fighting backed by retaliation: an escalating Hawk is not evolutionarily stable, Retaliator is at most neutrally stable because Mouse ties it, and in a contest settled by persistence alone no fixed persistence is stable while an exponential mixture is | Maynard Smith & Price (1973), LIT-505, read in full in NOTE-404. A simulation of five strategies, 2,000 contests per pairing, shows that “limited war” can be favoured by individual selection, without benefit to the species. The paper calls Retaliator an ESS “though Mouse does equally well”, which fails its own second condition. There is no Hawk–Dove game, no “dove”, no difference in prowess and no ownership, so it is not the source of display, deference or owner–intruder conventions. | Active |
| THEORY-469 | Because persistence (mean absolute fitness at least 1) and selection (fitness differences) are logically independent, niche construction theory’s evolutionary niche, read externally as the factors producing fitness differences in a population, differs in extension from the realized population ecological niche even after unit and modality are matched, so it is not a simple translation of Hutchinson’s niche | Trappes (2021), LIT-169, read in NOTE-155. The non-identity follows from a conceptual argument the reading grades strong. The three kinds of case that populate the gap are weaker: habitat degradation rests on a projected persistence failure, commensalism on one field study after the author’s first candidate failed, and non-limiting resources on a “possible example”. The result is scoped to Odling-Smee, Laland & Feldman’s 2003 niche concept and to the externalist reading of selection pressure. It does not say the two niches differ much; they “still coincide to a large extent”. | Active |
| THEORY-471 | Psychological states are not the machine-table states of an organism’s unique best description as a probabilistic automaton, because some are never occurrent, several hold at once, there are indefinitely many and they contain one another, none of which a machine-table state can be; this leaves open that they are computational states of such a system | Block & Fodor (1972), LIT-641, read in NOTE-491. Five independent arguments against the machine-state identity theory, each with a stated escape that keeps functionalism: dispositions with no occurrent state, simultaneous states, too-fine individuation, productivity and constituent structure. They refute the machine-table clause, not functionalism. The paper’s remaining argument (its third), inverted or absent qualia, is the only one aimed at functionalism broadly, and it is explicitly prima facie. | Active |
| THEORY-472 v3 | A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19) | Frank, Goodman, Lai and Tenenbaum (2009), LIT-824, read in full in NOTE-627. The derivation (utility −D_KL from intended to literal meaning gives P(w) ∝ |w|^(−α) for a point referent) is a proof under its assumptions. Learners’ bets track the parameter-free inversion at r = .93 on condition means (700 web participants); speakers’ tag-word use in free descriptions correlates at r = .19, .51 without colour terms and .02 for them. In these displays the informative-speaker learner and a plain size-principle learner coincide by construction, so the fit does not show that learners reason about the speaker. | Active |
| THEORY-473 | Scaled down to one organism, Hutchinson’s niche, defined as the conditions that give that individual an expected lifetime reproductive success of at least one, is observed only as a point; the rest of it is counterfactual, reached by translocation in only a few dimensions and otherwise inferred from other individuals, so in practice it is the niche of a phenotype class, and its authors call it mostly metaphorical | Takola & Schielzeth (2022), LIT-127, read in full in NOTE-170. The working definition (p. 5) gives the individualized niche an absolute-fitness boundary, expected LRS ≥ 1, and splits it into realized, potential, time-slice, prospective and trajectory-based niches. That the realized niche is a point and the potential niche unobservable follows from the definitions. The paper concedes there is “no empirical solution” at the individual level (p. 15), and its fix, marginalizing fitness functions across individuals, is schematic. It is a framework for structuring research, by its authors’ own account, not a measurement. | Active |
| THEORY-474 | Spekkens’ toy bit theory, restricted to maximal knowledge and post-selected measurements, has a universal, sound and complete ZX-style graphical calculus whose phase-free rules and symplectic check-matrix conditions are exactly those of pure qubit stabilizer theory; the two calculi differ in the phase group, the Klein four-group against Z₄ | Backens & Duman (2014), LIT-054, read in full in NOTE-066. A red–green spider calculus interpreted in the category of relations is universal and sound, and complete by Backens’ stabilizer route. A binary matrix is a valid toy state iff SᵀJS = 0, the stabilizer condition (Lemma 3). The phase-free rules coincide with stabilizer ZX. That the phase group is the only categorical difference is cited, not shown, and completeness leans on analogy for several steps. The paper does no ψ-epistemic analysis and says nothing about which operational difference the phase group carries. | Active |
| THEORY-475 | In simulations from the drift-diffusion model itself, with 12 subjects and 20 to 150 trials each, hierarchical Bayesian estimation recovers every DDM parameter with less error than per-subject maximum likelihood, χ²-quantile fitting or non-hierarchical Bayes, most of all at few trials; whether the gain survives misspecification, and how much of the power gain is the hierarchy’s, is not shown | Wiecki, Sofer & Frank (2013), LIT-563, read in NOTE-466. Experiment 1 gives the lowest trimmed recovery error for HB on every parameter, and HB beat ML on the same data sets in every condition. Experiments 2–3 report more power to detect drift differences and trial-wise covariate effects, but HB was judged by a posterior-interval test and the others by t-tests, and the clean comparison was omitted. All data were generated from DDMs, with sz = st = 0. It is the standard partial-pooling result shown for one model family, not a test of the DDM. | Active |
| THEORY-477 | None of the ten morphosyntactic criteria linguists use to identify words is necessary and sufficient for the orthographic word across languages, and the test batteries that combine them pick criteria opportunistically, so no cross-linguistically valid definition of the word is yet available | Haspelmath (2011), LIT-063, read in full in NOTE-073. Each of ten criteria (pauses, free occurrence, mobility, uninterruptibility, non-selectivity, non-coordinatability, anaphoric islandhood, nonextractability, morphophonological idiosyncrasy, biuniqueness) fails as necessary and sufficient against cross-linguistic counterexamples, and nine published batteries use no common set (Table 1). The target is orthographic practice by design. It does not say that words cannot be defined within one language, that sub-word units are privileged, or that morphology and syntax are shown to be one domain; whether units cluster on a boundness scale is open. | Active |
| THEORY-478 v2 | If the diffusion densities’ Hellinger distances obey a Hölder bound with exponent θ in d_T / t^(1/β) and are bounded below once d_T exceeds a multiple of t^(1/β), then for 0 < α < θ/β summing scale-weighted hyperbolic distances between diffusion densities over dyadic times gives a distance two-sided equivalent, up to constants, to min{1, d_T^(αβ)} for pairs above the finest scale: it keeps the tree’s neighbourhoods and not its global distances; whether a tree’s heat kernel, or sampled data, meets the bounds, and with which θ, is not shown | Lin, Coifman, Mishne & Talmon (2023), LIT-870, read in NOTE-675. The hyperbolic diffusion distance is equivalent to min{1, d_T^(2α)} for 0 < α < 1/2 (Proposition C.4), under heat-kernel bounds on the tree that are cited rather than proved; one Hölder step holds only for d_T ≥ 1, and repaired, the reader finds the admissible range halves. Pairs beyond a radius are only known to be far, which matches the method’s best mean average precision and its higher distortion than hyperbolic MDS on five graph benchmarks. A Euclidean distance on the same multiscale features does far worse. | Active |
| THEORY-479 | Shannon’s chain rule is the 1-cocycle equation of a cohomology of finite probability spaces whose action is averaged conditioning, and on connected, sufficiently rich structures of partitions every measurable 1-cocycle is a constant multiple of Shannon entropy, with no symmetry or continuity axiom assumed; in the unitary-invariant quantum version the first cohomology vanishes and von Neumann entropy is the 0-cochain whose coboundary is minus Shannon entropy | Baudot & Bennequin (2015), LIT-053, read in NOTE-088. Theorems 1 and 3 are proved for finite spaces, through the Tverberg–Kendall–Lee result on measurable solutions of the fundamental equation of information. The hypotheses are stated three different ways in the paper; Theorem 1’s is operative. The higher mutual informations appear as coboundaries, and Theorems B, D and F and the synergy reading are sketched or deferred to companion papers, so they are not part of this account. | Active |
| THEORY-480 v2 | Where a general-search monopoly was kept by exclusive distribution deals, the court in United States v. Google treated the user-interaction data that scale produced as a fruit of the violation and ordered the raw data and index metadata shared with qualified competitors, but not the engineered signals and models built from them, as a behavioural remedy while rejecting divestiture; that the sharing will let rivals compete is the court’s expectation, not a finding | Mehta (2025), LIT-006, the remedies opinion, read in full in NOTE-004. The court finds that exclusive distribution produced a scale advantage (9× the queries of all rivals; 93% of unique phrases seen only by Google), calls it a fruit of the violation, and remedies it with raw data and syndication rather than with divestiture, on a sliding scale of causal connection. Whether the shared data will let rivals compete is untested; the parameters that decide it were left to a Technical Committee. The opinion also finds no proof that search scale gives Google’s GenAI a quality edge (op. p. 156), which is an absence of proof, not a finding that there is none. | Active |
| THEORY-481 v4 | Under gradient flow averaged over tasks and data, a linear-attention layer trained on in-context regression learns as a two-layer linear network from an aligned start does: when tasks share the input covariance’s eigenbasis and the weights start and stay aligned with it, then in the long-context limit each input mode is learned on its own logistic curve whose timescale falls with the square of its variance, so learning is staged by the input spectrum, and the layer converges to the estimator preconditioned by the inverse input covariance; at finite context length the modes decouple exactly only when inputs are isotropic and the modes start equal, and otherwise couple at order 1/N | Mainali & Teixeira (2025), LIT-859, read in NOTE-658. The prediction of one linear-attention layer reduces to p₂ W Σ̂_x q₁ x_q, a product of two learned factors, so averaged gradient flow decouples into modes with τ_α dp_α/dt = q_α(1 − p_α q_α s^∞_α) and τ_α = (ηP s_α²)⁻¹, logistic from a balanced start. The fixed point is ŷ = W Σ̂_x Σ_x⁻¹ x_q as N → ∞. The paper’s finite-N form, the one Ahn et al. and Zhang, Frei & Bartlett found, holds at finite N only for isotropic inputs with equal modes. The trajectory is new. The decoupling into independent modes is exact as N → ∞. At finite N it holds only for isotropic inputs with equal modes: the paper commutes the non-diagonal empirical covariance at (A71) and (A73), and the correct Wishart moment couples the modes at order 1/N. Alignment of the weights with the data’s eigenbasis is assumed throughout, the null blocks vanish only in expectation over tasks, and the transfer to non-linear transformers is timing coincidences. | Active |
| THEORY-482 | For a closed quantum system that starts thermal and is measured projectively in energy before and after a unitary drive, the two-point-measurement work obeys the Jarzynski equality and the Tasaki–Crooks relation in exactly the classical form, because those energy measurements erase the coherences in which a quantum difference could appear | Vinjanampathy & Anders (2015/16), LIT-010, read in full in NOTE-011, which reproduces the derivation (Eqs. 24–30). With W_{m,n} = E_m^(τ) − E_n^(0), a thermal start and closed unitary evolution, ⟨e^{−βW}⟩ = e^{−βΔF} follows from Σ_n p_{m|n} = 1 (Eq. 29), identical in form to the classical relation. The identity is exact under its conditions. Its classicality is bought by the definition of work: two projective energy measurements destroy coherences, which is why the definition is contested. It says nothing about work defined without them. | Active |
| THEORY-488 v2 | Clark (2020), synthesising open writing by Russian military authors of 2015 to 2020, chiefly Chekinov and Bogdanov, Bartosh and Kiselev, with no stated rule for choosing sources, reports that they use “hybrid war” for a category of war marked off by its objective, reshaping a target state’s governance, geostrategic orientation and worldview, in which every action up to conventional military force is subordinate to an information campaign, and not, as he says Western usage does, for a set of means below the threshold of armed conflict | Clark (2020), LIT-023, an ISW report read in NOTE-012. A synthesis of open Russian military journals, official press and speeches, 2015–2020, with no stated method for choosing sources; the definition is the author’s synthesis of several writers, not a doctrinal text. The reading rates the definition moderate, the claim that it is institutional weak to moderate, and the US policy recommendations advocacy. Written before the 2022 invasion. | Active |
| THEORY-489 | Chains of human function learners, each trained on the previous learner’s responses, forget their starting function and in 28 of 32 families converge within a few generations to a positive linear function, the bias earlier studies report, whatever function trained the first learner; this is convergence to the learners’ favoured function, not shown to be convergence to their prior | Kalish, Griffiths & Lewandowsky (2007), LIT-820, read in NOTE-616. 288 participants in 32 nine-generation chains, four starting conditions. 28 families converged to a positive linear function; 3 converged to the negative linear one and 1 had not converged. The bias was taken from earlier studies, not measured on these learners, and stationarity was not established. It supports the claim that human transmission chains settle on what learners favour in this task, not that they reach the Bayesian prior, and it cannot tell sampling learners from MAP learners. | Active |
| THEORY-493 | From core mereology, classical extensional mereology (GEM) is reached by exactly three pairings of a decomposition principle with an unrestricted-sum principle: Supplementation with Sum2, Strong Supplementation with Sum3, and Complementation with Sum1; Supplementation with Sum3, the pairing given by Simons (1987), does not yield it, because which definition of ‘sum’ is used decides which decomposition axiom is needed | Varzi, SEP “Mereology” (2003, revised June 2026), LIT-124, read in full in NOTE-163. A result the entry reports with citations (Hovda 2009; Cotnoir and Varzi 2019) and a countermodel (its Fig. 4; Pietruszczak 2000a, Pontow 2004), and which corrects Simons (1987), Casati and Varzi (1999) and the entry’s own 2003 edition. The record has not checked the proofs. It is stated for first-order theories with a single primitive P and infinitary principles as axiom schemas, and it is a fact about axiom systems: it does not say classical mereology is true of anything. | Active |
| THEORY-496 | In few-shot classification and multiple choice by language models up to 175B parameters, whether each demonstration’s label is correct contributes little to the gain over zero-shot, while the distribution of the inputs, the label space and the paired format carry most of it | Min et al. (2022), LIT-814, read in NOTE-607. With k = 16 demonstrations, labels drawn at random from the correct label set cost 0–5 points against gold labels (2.6 on classification, 1.7 on multiple choice on average) across twelve model–inference combinations and 26 datasets. Out-of-distribution inputs, random English words as labels, or breaking the input–label pairing each cost far more. The random labels always come from the right label set, gaps reach about 14 points on single datasets, and the authors’ reading that the models learn no new correspondence at test time is untested on tasks pretraining cannot have taught. | Active |
| THEORY-498 | A mathematical structure whose domains are aspects of experience and whose axioms hold there is not thereby a structure of experience, because that test passes incompatible topologies on one set and every rescaling of a metric without consulting experience; requiring, for each relation, an aspect that changes exactly when a variation of experience breaks the relation removes those automatic passes, and has so far been met only by relative similarity and a visual topology, each on an assumed premise | Kleiner & Ludwig (2023; Synthese 2024), LIT-143, read in full in NOTE-135. The negative result is shown by construction: any set of aspects carries both the discrete topology and a coarser one, and a metric d passes whenever every C·d does. The replacement criterion (MSC) ties each relation to an aspect that co-varies with it under every variation. Its two positive cases rest on assumed transitivity and on the subsumptive unity thesis. The published version gives no structure that fails (MSC); the argument that metric spaces fail is in arXiv v1 only, and was withdrawn. | Active |
| THEORY-499 v2 | The breakup of the Eulalia parent body beside the Jupiter 3:1 resonance, 785–965 Myr ago, can plausibly have supplied the cluster of large lunar craters dated near 800 Ma, but shown only for the most favourable of five model solutions, the one carried through, with an average largest lunar impactor too small for Copernicus, and with the cluster’s absolute date inherited from Copernicus alone | Bottke, Vokrouhlický, Dykhuis & Zellner (2026), LIT-009, read in full in NOTE-002. A chain of multiplied factors (family fit, J3:1 influx, WISE incompleteness, collisional loss, impact probabilities) gives, for Solution 5 only, about 7,000 D ≥ 2 km fragments into the resonance and a largest expected lunar impactor of “>3 km”. A Monte Carlo puts the seven ~800 Ma craters “generally” within 1σ. It is an existence argument, not an estimate: one factor is tuned by eye, only the most favourable solution is propagated, and the seven craters are dated by N(1) against one sample-dated crater. The Earth, Mars and Venus links are the authors’ own speculation. Active as a reading of the model (ADR-065); whether the shower happened is THEORY-1195, Proposed. | Active |
| THEORY-501 | In weekly waste collection where each point’s visit days are free and its bin capacity must cover the longest gap between visits, a genetic algorithm that searches visit schedules and derives bins from them builds routes about 6% cheaper than eight-hour Gurobi incumbents but buys about 20% more bin capacity, ending level on overall cost at its best of 30 runs and 4–6% worse on a typical run, on five 12-point instances where the incumbents’ own optimality gaps are 26–32% | González et al. (2025), LIT-008, read in NOTE-015. The comparison is against incumbents whose distance from optimum is unknown, on five 12-point random subsets of one real instance, one of which was also used to tune the GA. Best of 30 against the MILP: −0.08% overall; against the MIQP: +1.70%; the GA’s mean or median is +3.75% and +5.61%. The abstract’s “able to match the results of exact solvers” says less than it appears to. On 40–163 points there is no bound or baseline. | Active |
| THEORY-502 | Forman–Ricci curvature of a graph edge, taken with no cells above the edges, depends only on the weights of the edge, its two endpoints and the edges adjacent to it, and with unit weights reduces to 4 − deg(v₁) − deg(v₂); it is a local, closed-form degree-and-weight statistic that sees no triangles or cycles, unlike Ollivier’s curvature, which needs a transport problem per edge | Weber, Saucan & Jost (2016; J. Complex Networks 2017), LIT-020, read in full in NOTE-026, which checked the formulas against page renders. The formula (Eq. 11) is Forman’s (2003) Bochner–Weitzenböck curvature specialised to a 1-dimensional complex by dropping the parent terms; the unit-weight form is NOTE-026’s substitution, not stated in the paper. The paper’s applications (an EMD distance between curvature distributions, a Ricci flow for change detection, a Laplacian flow for denoising) are one-table or one-figure demonstrations, and several of its formulas carry sign and consistency slips, so it is cited here for the definition only. | Active |
| THEORY-503 | The nearest rank-r matrix to a real matrix in the least-squares norm keeps its r largest singular values and zeroes the rest; it is unique when the r-th and (r+1)-th singular values differ, its squared error is the sum of the dropped squares, and the singular-vector factors that build it are not unique | Eckart & Young (1936), LIT-330, read in full in NOTE-301, which checked the error bound. The argument is first-order stationarity on the orthogonal group plus two cited theorems (the SVD and a simultaneous-SVD criterion), and assumes a minimiser exists. Only the Frobenius norm and real matrices are covered. Uniqueness is of the approximating matrix, under a spectral gap, never of the factors; the paper’s last section argues the canonic components cannot be read straightforwardly as psychological factors. | Active |
| THEORY-504 v2 | In the Gemma Scope sparse autoencoders on Gemma 2, the two fidelity measures disagree by site: residual-stream autoencoders cost more next-token loss than MLP or attention ones at comparable fraction of variance unexplained, and base-model autoencoders transfer to the instruction-tuned model by loss more than by variance explained | Lieberum et al. (2024), LIT-611, read in NOTE-482. Across every layer and sublayer of Gemma 2 2B and 9B, JumpReLU autoencoders on the residual stream raise the language model’s loss more, when spliced in, than MLP and attention autoencoders whose FVU is about the same; and autoencoders trained on the base model lose little loss on instruction-tuned rollouts while showing a larger FVU gap. The authors’ explanations (the residual stream is the bottleneck; fine-tuning re-weights old features) are conjectures, and the authors say no agreed metric of autoencoder quality exists. | Active |
| THEORY-505 | Questionnaire items written for morality-as-cooperation’s seven domains form seven factors when rated for relevance to moral decisions, in the UK, the US and Turkey, but not when rated as agreement with moral statements, and the two formats cannot be pooled into seven domains | Curry, Jones Chesters & Van Lissa (2019), LIT-496, read in NOTE-412, and the independent Turkish validation by Yilmaz, Harma & Doğruyol (2021), LIT-513 (NOTE-397). Relevance items fit seven factors in all three samples (CFI .98 UK, .95 Turkey). Judgement items do not (CFI .87, .83), and a seven-domain model of both fails (CFI .73, .75). The items were written from the theory and selected for fit within each domain, several subscales correlate above .8, and no item is non-cooperative. This is evidence about the instrument’s structure, not about what morality contains. | Active |
| THEORY-508 | The Belousov–Zhabotinsky oscillation is a relaxation cycle in which autocatalytic production of HBrO₂ oxidizes the catalyst until bromide, released as the oxidized catalyst is reduced by bromomalonic acid, switches the autocatalysis off; a closed batch oscillates only as a transient of up to several thousand cycles on its way to equilibrium, and only a continuous-flow reactor sustains it indefinitely | Zhabotinsky (2007), LIT-531, the Scholarpedia review by one of the reaction’s eponyms, read in full in NOTE-414. The mechanism is supported by the primary kinetics and by phase-resetting with injected Br⁻, Ag⁺ and Ce⁴⁺. The closed/open distinction matters for any use of BZ as a dissipative structure: a closed batch is a long transient, and only a fed reactor holds it away from equilibrium. The Oregonator models oscillation and excitability qualitatively, not quantitatively. The “several thousand cycles” figure is uncited, and the primary papers were not read. | Active |
| THEORY-509 | Under genericity conditions, two dense or top-k (k ≥ 2) mixture-of-experts layers compute the same function only if they differ by a permutation of the experts with their gates and a common translation of the gate weights, and the translation leaves every interpolation barrier unchanged, so expert order is the only gate symmetry alignment must undo | Tran, Trinh, Bui & Nguyen (2025), LIT-670, read in NOTE-513. Softmax is invariant to a common shift of scores and the sum to the order of terms, so G(n) = ℝ^d × ℝ × S_n preserves the layer; Theorems 4.1–4.2 show that generically it is all of the gate’s symmetry, for dense gating and for top-k with k ≥ 2 (top-1 has an extra scaling). Aligning expert order and then each expert’s hidden units gives barriers of 0.0044–0.0066 between GPT-2 MoE layers fine-tuned from different seeds, against 0.14–0.86 without the order step. Every experiment shares and freezes the backbone around the inserted layer, so it is not evidence about independently trained MoE models. | Active |
| THEORY-513 | Observational entropy taken relative to a maximum-entropy prior τ, S(τ) − D_M(ρ‖τ), lies between S(ρ) and S(τ) for states meeting the prior’s constraint, is finite and continuous whenever S(τ) is, and reduces to about two dozen textbook entropies by choice of measurement and prior; its second laws are bounds on the time-averaged state’s divergence from the prior, proved small only under sufficient conditions (nondegenerate energy gaps for equilibration, the eigenstate thermalization hypothesis for fit to the prior) | Schindler, Strasberg, Galke, Winter & Jabbour (2025), LIT-082, read in full in NOTE-043, with Proofs 1, 2, 7, 10 and 11 checked line by line. The bounds, the special cases and the decomposition are proved. The central target, a small divergence of the time-averaged state from the prior, is proved for no concrete system; in the paper’s own examples “in none of those cases was the smallness proved here analytically” (p. 12). Unpublished as of the reading. | Active |
| THEORY-515 | Sampling a posterior p(x|y) when only joint samples (X, Y) can be drawn is solved by a Schrödinger bridge on the joint space, with y held fixed along the path, between the joint distribution and any reference sharing its y-marginal; the bridge averages the per-observation bridges, its first iterative-proportional-fitting step is the conditional score-based model, and the error of later steps at the data end falls as 1/n times the reference’s distance from the bridge | Shi, De Bortoli, Deligiannidis & Doucet (2022), LIT-776, read in NOTE-589. Propositions 1–3 are proofs, read for structure and not checked line by line. The empirical gains over the conditional score model hold at equal, small step counts on most but not all image metrics, and in Lorenz-63 filtering at 20 steps. That a long noising horizon is not what makes conditional diffusion correct is NOTE-589’s interpretation, not something the paper tests. | Active |
| THEORY-516 | For any presheaf of propositions over a small category of contexts, a sieve-valued valuation satisfies the generalized functional-composition condition exactly when it is a subobject of that presheaf, and the construction runs unchanged in classical physics, where every microstate gives a global section yet truth values are still non-Boolean sieves; so the logic of the truth values is fixed by the base category, not by whether a presheaf has global sections | Butterfield & Isham (1998), LIT-381, read in full in NOTE-329: Theorem 4.1 proves that generalized FUNC is exactly naturality of ν : G → Ω, for any presheaf G of posets on any small category. In classical physics the value presheaf has a global section per microstate (3.6–3.7), yet the macrostate and microstate valuations are sieve-valued, and NOTE-329 shows excluded middle failing at a microstate. Isham & Butterfield (1999), LIT-389 (NOTE-334), adds that Ω is unique for a given topos and that a base with one arrow between any two objects gives two-valued logic. The choice of base, and so of the logic, remains free. | Active |
| THEORY-519 | Supervised learners compose only once each carries a request function that passes a corrected input upstream; with it they form a symmetric monoidal category, and for a fixed step size and an error whose derivative in its first argument is invertible, gradient descent with backpropagation is a faithful strong symmetric monoidal functor into it from differentiable parametrised functions | Fong, Spivak & Tuyéras (2017; LICS 2019), LIT-827, read in full in NOTE-625, which followed the chain-rule proof of Theorem A.1 line by line. Learners (P, I, U, r) form Learn (Prop. II.4); Theorem III.2 makes gradient descent with backpropagation a functor Para → Learn. Covered: fixed step size, one datum per update, deterministic updates, and errors with invertible ∂e/∂x, which excludes cross-entropy at 0 and 1. That learners cannot compose without a request is argued, not proved. Learn says how learners compose, not whether they learn. | Active |
| THEORY-522 | Open text-to-image diffusion models of 2022–2023 render a single named object almost always but fail most on relative position and next on attribute binding, by human rating and by detector checks alike; scaling the image model or continuing pretraining does not fix position, and interactions are rated near ceiling only by rubrics that ask whether an action is depicted | GenEval (Ghosh, Hajishirzi & Schmidt 2023, LIT-782, NOTE-593) and T2I-CompBench with its journal extension (Huang et al. 2023 and 2025, LIT-783 and LIT-798, NOTE-592 and NOTE-610), two independent benchmarks that do not cite each other. GenEval: single object 0.97–0.98 for every diffusion model, position at most 0.15, binding at most 0.35. Human ratings in T2I-CompBench: 2D spatial 0.31–0.46, shape binding 0.51–0.70, interactions 0.81–0.99. Human ratings cover only six 2023-era models; the 2024–25 generation is ranked by automatic judges validated on the older ones, so the ordering is not established for it. | Active |
| THEORY-524 | On cyclic measurement scenarios, where the contexts’ overlaps form one chordless N-cycle, the ℤ/2 Čech obstruction computed on the (N−1)-th iterated joint model is exact for logical and strong contextuality, the bound N − 1 being tight; beyond cycles no such result is established, the published extension is false on its own example, and the completeness conjecture fails on a Kochen–Specker cover that is never detected at any level | Carù (2018), LIT-280, read in NOTE-253, and the same result republished as Thm IV.32 of Carù’s 2019 thesis, LIT-281, read in NOTE-254. The cyclic theorem is proved and was confirmed on Hardy, on the (2,2,4) counterexample and on 659 random contextual cycle models. The failures beyond cycles (Prop 8.2 / IV.34, the §8 cover, Conjecture 9.1 / IV.41) are the readers’ computations and a lifting lemma, not the author’s. The joint models grow exponentially; nothing analyses the cost. | Active |
| THEORY-526 | The Čech cohomology obstruction of a support section is a sound but incomplete witness of contextuality: it vanishes exactly when the section lies in a compatible family of integer combinations of support sections, so it is non-zero on every section of every All-vs-Nothing model, over any coefficient ring, on a connected cover, yet it misses the Hardy model entirely, part of a strongly contextual Kochen–Specker cover, and every section of a strongly contextual model on the fully symmetric (2,2,4) Bell cover | Abramsky, Mansfield & Barbosa (2011), LIT-277 (NOTE-250); Abramsky, Barbosa, Kishida, Lal & Mansfield (2015), LIT-278 (NOTE-251); Carù (2017), LIT-279 (NOTE-252). The criterion (Prop 4.2 / Prop 16) and the All-vs-Nothing theorem (Thm 21) are proved; the three failures are explicit models, which the readers re-computed exactly over ℤ and ℤ/2. Two steps are the readers’: that the §8 cover and the (2,2,4) model admit no All-vs-Nothing argument over any ring, by Thm 21 read contrapositively. The witness is possibilistic, so it cannot grade anything below strong contextuality. | Active |
| THEORY-527 v5 | If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.’s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms | Kenton, Kumar, Farquhar, Richens, MacDermott & Everitt (2022), LIT-909, read in NOTE-707. The recovery result (Theorem 1) is proved under five assumptions, including the availability of interventions on mechanisms, and was re-derived on promotion (2026-10-20). The definition is a proposal; the verdicts on thermostats and RL are argued from it, and “reliably classifies humans as agents” is asserted. The labels depend on the chosen variables, which the paper states. | Active |
| THEORY-530 | The spectrum of a matrix bounds the norm of its rational functions up to a constant only if the matrix is diagonalisable, with the eigenvector condition number as the constant, while the closed numerical range of any Hilbert-space operator does so with a universal constant between 2 and 11.08, exactly 2 for 2×2 matrices | Badea & Beckermann (2013), LIT-011, a handbook survey read in full in NOTE-009. Example 107.2.5: σ(A) is K-spectral iff A is diagonalisable, with K ≤ cond(V). Fact 107.7.2: the closure of W(A) is completely K-spectral for every bounded A with a universal K ∈ [2, 11.08], and 2 suffices for 2×2. The chapter proves almost nothing itself; the numerical range result is cited to Crouzeix (2007). The bound 11.08 was the 2013 state; a later improvement to 1 + √2 is recalled in the reading but not verified. Whether the constant is 2 was open. | Active |
| THEORY-534 | The optimality of maximum likelihood was never one theorem: each historical proof holds under conditions its author left implicit, and of the classical counterexamples the practically important one is a parameter count growing with the sample, while Hodges-type superefficiency is confined to a set of measure zero | Stigler (2007), LIT-044, read in full in NOTE-074. Fisher’s 1922 proof rests on sufficiency, his 1925 proof on exact normality, and his private 1930 proof on multinomial data and smooth degree-0 estimators. Rigorous treatments (Wald 1943, Cramér 1946) came at the cost of unwieldy conditions. Neyman–Scott, where the MLE of σ² tends to σ²/2, is the failure Stigler weighs as practical; Hodges’s superefficiency is possible only on a Lebesgue-null set (Le Cam 1953). It is a history: the theorem statements are reconstructions, and it says nothing about modern overparameterized models beyond one hedged warning. | Active |
| THEORY-543 | Diffusion can destabilise a homogeneous steady state of reacting chemicals that is stable without it: a disturbance of one finite wave-length grows while uniform and very short ones decay, so a stationary pattern appears whose spacing is set by the reaction and diffusion constants and only its phase by the disturbance; with two substances this needs them to diffuse at different rates, and the pattern lasts only while free energy is supplied | Turing (1952), LIT-535, read in full in NOTE-427, with Cross & Hohenberg (1993), LIT-527, skimmed in NOTE-418. The linear analysis on a ring gives exact conditions (9.4a, b) for stationary waves of finite wave-length; that equal diffusibilities cannot satisfy them is NOTE-427’s substitution, not stated in the paper. Turing calls the linearity assumption “a serious one” and the biological examples “very limited”. Stationary chemical Turing patterns were first made in 1989–1991, and real morphogens implementing the mechanism had not been identified by 1993. | Active |
| THEORY-544 | In homotopy type theory extended with a strict directed interval and a layer of shapes, one contractibility condition, the Segal condition, gives a type identities, associativity and three-dimensional coherence, makes every function between Segal types a functor, and yields a dependent Yoneda lemma for covariant families; that such types are (∞,1)-categories rests on a semantics only sketched | Riehl & Shulman (2017), LIT-032, read in full in NOTE-069. The internal results of §§5–11 are proved: associativity from the Segal condition applied to A^2 (Prop. 5.9), inner 3-horns (Prop. 5.21), functoriality of every function (Prop. 6.1), the dependent Yoneda lemma (Thm 9.5), and uniqueness of left adjoints into a Segal type from a Rezk type (Thm 11.23). Segal and Rezk types match Segal and complete Segal spaces (Props A.21, A.25), but soundness of the type theory in bisimplicial sets is sketched and depends on an unproved initiality theorem. Coherence in all dimensions is conjectured, not proved. | Active |
| THEORY-545 | Aerts and Gabora’s founding quantum model of concepts is a classical probability model written in a Hilbert space: every context it uses is a projector diagonal in one basis, so the contexts commute and generate a Boolean algebra, its entangled ‘pet fish’ state gives the same statistics as a classically correlated mixture for every context in the model, and its guppy weights are copied from single-concept ratings rather than predicted for the conjunction | Aerts & Gabora (2005), Parts II and I, LIT-312 and LIT-340, read in NOTE-299 and NOTE-294. The structural facts are checks of the published model, not of the authors’ prose: contexts are sums of basis projectors, counts are chosen to match rounded frequencies, and the authors call the result a “quantization” of a set-theoretical model. Part I’s mark of non-classicality (eq. 19) is met by Boolean projection lattices too, and its lattice structure is supposed, not derived. The model as printed is also infeasible in two cells. It does not say that no quantum model of concepts does better. | Active |
| THEORY-547 v2 | A language model idealized as continuing any text with a plausible next word from the collection of all plausible texts follows narrative coherence and not truth, so its false statements are confabulations within the story the context sets up; whether trained and aligned chat models keep that nature is an open empirical question the idealization does not settle | Bottou & Schölkopf (2023), LIT-178, read in NOTE-102: an essay offering imagery, with no evidence beyond one Vicuna-13b transcript pair. For the “perfect language model” it defines, insensitivity to truth follows from the definition. That deployed chat models share it is assumed, and the essay’s own §2 leaves open whether alignment can turn a fiction machine into a verification machine. It credits the machine with knowledge of what makes sense in the story, denies it truth-directedness and intention, and never uses the word “sycophancy”. | Active |
| THEORY-548 v2 | In generalised Lotka–Volterra dynamics with every interaction delayed by τ, a feasible equilibrium that is stable without delay loses stability at τ_c = min over the community eigenvalues j of arctan(|Re j|/|Im j|)/|j|, where a pair of roots crosses the imaginary axis transversally (a Hopf point), so the critical delay is set by the community spectrum; in Saeedian et al.’s strongly self-regulated setting the oscillating regime’s community-level variability falls with diversity (S = 10–70) while species-level variability rises, and the fall matches what summing independently oscillating species gives | Saeedian et al. (2021), LIT-057, a preprint, read in NOTE-061. The critical delay is the linear delay result on the community matrix, re-derived on promotion (2026-10-20), and matched simulation at one strongly self-regulated setting. The variability result was re-run on promotion against an averaging null and matches it: the fall of community-level variability with diversity is what summing independent oscillators gives, not an ecological stabilisation. | Active |
| THEORY-550 | Unsupervised clustering is not made epistemically superior to supervised classification by the absence of labels: it is sensitive to sample bias, highly sensitive to how inputs are represented, and overfits, with the number of clusters acting as a capacity control just as parametric complexity does in supervised learning | Søgaard (2024), LIT-162, read in NOTE-108. A rebuttal of two arguments for clustering’s superiority, the No-Bias Argument and the Simplicity-Truth Argument, by toy counterexample and by the technical literature on overfitting in unsupervised learning (Hansen & Larsen 1996). The negative claims are firm. The paper’s positive step, that supervised classification is “at least as” justified, is under-argued and is not part of this account. | Active |
| THEORY-552 | Indeed’s US software-development job-postings index rose to 233.8 in February 2022 and fell to 65.2 by mid-February 2025, measured against 1 February 2020 = 100, while all postings stood near 110; it was the deepest fall of any Indeed sector, though not the only one to have more than doubled, and the postings data cannot apportion the fall among interest rates, over-hiring and AI | Orosz (2025), LIT-089, read in full in NOTE-039, which checked the figures against Indeed Hiring Lab’s own published series (current vintage). The headline numbers reproduce: 65.2 on 2025-02-13, a 3.59× fall from the peak, −8.3% year on year, all postings 109.5. Several side claims do not: the baseline is 1 February 2020, not January; the peak was February 2022, not mid-2022; six sectors peaked above 200. The index counts active postings, not vacancies or hires, and the post’s ranking of causes is informal. | Active |
| THEORY-553 | In the Döring–Isham topos representation of quantum theory a bounded physical quantity cannot take values in the topos’s real-number object; it is represented faithfully by an arrow from the spectral presheaf to a presheaf of monotone real functions over subcontexts, so its value at a context is a spread that widens as the context shrinks, and these values form only a commutative monoid, under which sums of daseinised quantities are not daseinisations of sums | Döring & Isham, paper III (2007/8), LIT-299 (NOTE-290), restated with additions in LIT-378 (NOTE-326). The Gel’fand transforms of daseinised operators grow as contexts shrink, so arrows into the constant presheaf ℝ are impossible (3.3–3.4). Each  gives a natural transformation δ̆^o(A) : Σ → ℝ^⪰ (Thm 3.1), and  ↦ δ̆^o(A) is injective, by de Groote’s results. ℝ^⪰ is a monoid, not a group; differences need a group completion, whose global elements are the functions of bounded variation on the context poset. No choice among the candidate value objects is made, and the state-dependent reading of the spread is cited, not proved. | Active |
| THEORY-555 | In London in 2018, Airbnb listings that were possibly breaking the 90-night short-let cap, defined as entire homes offered for more than 180 days a year by hosts with two or more listings, were about 0.16% of the housing stock (about 5,300 homes) but 3.5–7% in a few central small areas, and across small areas they correlate weakly with flats, private renting and low dwelling-type diversity | Shabrina, Arcaute & Batty (2021), LIT-094, read in full in NOTE-071. A descriptive estimate from Inside Airbnb and VOA counts under a proxy that rests on availability, not bookings, and on an untested 50% occupancy. The abstract’s “more than 2% … misused” is the share listed at all, not the share possibly misused. The paper’s borough rent result (n = 33, cross-sectional) and its “8% per doubling” translation do not belong to this account and should not be cited as an effect size. | Active |
| THEORY-559 | In Kolmogorovian probability, joint distributedness is transitive, so representing a measurement by one random variable in every context it occurs in already puts all measurements of a connected system on one probability space; within that representation a Kochen–Specker or Bell contradiction refutes the context-independent identification of measurements, not the existence of a joint distribution | Dzhafarov & Kujala (2017), LIT-836, read in full in NOTE-623. The transitivity step is an elementary proof; the reading of the reductio is an argument that rests on it. It binds the random-variable representation only: the sheaf account keeps one variable per measurement and asks whether marginals glue, and a reader who holds that view is not bound by it. The paper also proves that a coupling of a binary connection is multimaximal iff adjacent pairs, ordered by their means, are maximally coupled (Theorem II.3). | Active |
| THEORY-566 v2 | In Ganea et al.’s hyperbolic neural networks, linear maps and nonlinearities are Euclidean ones read through the chart at the origin, so a stack of them without biases is a Euclidean network conjugated by that chart, and the curvature acts only through the bias translations and the distance-based output layer | Ganea, Bécigneul & Hofmann (2018), LIT-879, read in NOTE-686, which checked the closed forms numerically. Möbius layers are defined as exp₀ ∘ f ∘ log₀, so any composition of them is exp₀ ∘ (the Euclidean composition) ∘ log₀, as the paper notes; only the Möbius bias, which is parallel transport from the origin followed by the exponential map, and the hyperbolic logistic regression’s distances to geodesic hyperplanes are not absorbed by the chart. The structural fact is proved; that the networks’ gains, confined to a synthetic task and to separating WordNet subtrees in low-dimensional Poincaré embeddings, come from those two operations is this record’s inference. NOTE-686 filed no THEORY. | Active |
| THEORY-567 | In the donation game, where a cooperator pays c for another to get b, defectors win in a well-mixed population, and each of five mechanisms makes cooperators evolutionarily stable above a benefit-to-cost threshold: kin selection r > c/b, direct reciprocity w > c/b, indirect reciprocity q > c/b, network reciprocity b/c > k and group selection b/c > 1 + n/m; the rules cover altruism only, not mutualism, coordination or conflict | Nowak (2006), LIT-501, read in NOTE-409, main text only. A four-page review that casts each known mechanism as a 2×2 game so that one set of criteria applies to all. The five ESS conditions are confirmed by the main text; the derivations are in the supporting material, which was not read, and in cited papers. Several rules hold only under weak selection, regular graphs or rare group splitting. Every mechanism is modelled with the donation game, so mutualism, coordination and conflict are outside it. | Active |
| THEORY-568 | Whatever redundancy measure a partial information decomposition uses, provided its atoms satisfy the consistency equation, the average number of single sources an atom can be read from is Σᵢ I(Xᵢ;Y)/I(X;Y) and the average number of sources it cannot do without is Σⱼ I(Xⱼ;Y|X₋ⱼ)/I(X;Y); so the redundancy–synergy index equals (r̄ − 1)·I(X;Y), and for three or more sources it counts a k-way redundancy k − 1 times but synergy once | Gutknecht et al. (2025), LIT-042, read in full in NOTE-045, with the combinatorial proofs of Props. 1–4 and 6 checked by hand. The averages need only the consistency equation and I(X;Y) > 0; reading them as average degrees, and the bounds of Appendix B, need non-negative atoms. Corollary 5 overstates Prop. 6 (⇔ for ⇐). The network results are on tiny quantised models evaluated on their own training sets, and are not part of this account. | Active |
| THEORY-569 | Measured as the contrast between a set of explanantia’s CohOG+ coherence with the explanandum and with its negation, explanatory power satisfies Roche and Sober’s Two Likelihood Inequalities for single hypotheses and fails the One, and in one causal chain and one collider it ranks the richer set above the sparser over a range of priors, where no two-place TLI measure can rank a conjunction so | Hartmann & Trpin (2026), LIT-165, read in NOTE-168, which recomputed all three numerical claims. One theorem (TLI for singleton explanans sets), one counterexample (OLI fails) and two parameterised existence examples (a distal cause for P(H1) < .7; a negatively interacting cause for P(H1) = P(H2) < .75). The escape from Roche & Sober is representational: a set is scored differently from the conjunction, and the two-place theorems still hold for conjunctions. No characterisation says when the richer set wins, and the paper’s claim that the measure “accommodates key scientific judgments” is more than the body shows. | Active |
| THEORY-574 | The structural width of Xin Li’s learning programme is a covering number of the margin-safe region by local pieces: its lower bounds come from metric confinement through a wedge point, not from topology, labels enter only through the margin, and the error floor below width binds only learners assembled from local experts, so it does not entail routing, nor training routing apart from the task loss | Li (2026), LIT-376 (Local Urysohn Width, v1), LIT-375 (Structural Learning Theory) and LIT-374 (Structural Decoupling), read jointly in NOTE-323, NOTE-324 and NOTE-322. In repaired form the formal core holds: width is incomparable with the per-cell VC dimension, fewer cells than width give an error floor (under an assumption that states it), and discovering all basins costs Ω(w log w) samples. But local correctness is automatic on any connected patch, the bouquet lower bound is metric (the paper’s Remark 8.11 concedes it), a global classifier has width 1 (Remark 3.4), and the additive Rademacher bound said to call for decoupled training holds for joint training too. | Active |
| THEORY-576 | In a bundled network, one fiber graph attached at a single node to every node of a base graph, the low Laplacian spectrum is the base spectrum divided by the fiber size to first order, so the gap exponent (Fiedler dimension) is set by the base while the density-of-states exponent (spectral dimension) is the fiber’s; the two come apart generically and coincide only in special cases, and relaxation then decays with the spectral dimension at intermediate times but equilibrates on a time set by the Fiedler dimension | Grimaldi, Villegas, Vezzani, Burioni, Cassi & Gabrielli (2025; PRE Letter 2026), LIT-028, read in full with its SM in NOTE-005, which re-derived the key steps and checked them numerically on the comb. λ_{n1} = ℓ^b_{n1}/N_f gives d_g = 2(d_f,f + d_f,b)/(d_f,f + 2d_f,b/d_g,b) in the base-controlled regime; d_s = d_s,f is earlier work by the same group. The comb has d_s = 1 and d_g = 4/3. Proved for one graph class only; the paper’s composite finite-size scaling prefactors (SM Eqs. 25, 29) are wrong, and its “generic property of composite architectures” outruns the proof. | Active |
| THEORY-580 v2 | On any cover of a space of input histories, the empirical models compatible with a causal structure are exactly the points of a polytope cut from a product of simplices by linear causality equations, so how much of a model a causal structure can explain is a linear program; causal separability is relative to an assumed ambient structure, and the causally separable fraction is never less than the separable local fraction | Gogioso and Pinzani (2023), LIT-788, skimmed in NOTE-624, with the proofs of the causaltope theorems (2.18, 2.19, 2.25) and of the lower bound (2.29) read and checked on promotion (2026-10-20), against the definitions they take from The Topology of Causality. The proofs are short because the work is in the definitions; the one step left implicit closes. The quantum switches witness indefinite order only relative to an ambient space with known no-signalling constraints, not absolutely. | Active |
| THEORY-581 v2 | Two exactly consistent descriptions of one system, at different scales and with equal past–future mutual information, can differ in how that information divides between redundancy and synergy; in small generated Boolean networks under the Williams–Beer redundancy measure the coarser scale was more synergy-biased in every random stochastic system, by at most about 0.07, and the direction mostly reversed for redundancy-biased deterministic ones | Varley & Hoel (2022), LIT-027, read in NOTE-020. Expanding a 3-node macroscale TPM into 4- and 5-node equivalence-class microscales holds mutual information fixed with zero inconsistency, so any change in the PID’s synergy bias B_syn is a change of type, not of amount. In 200 |Gaussian| TPMs the macroscale gained B_syn by about 0.005–0.07; in 185 near-deterministic ones most with macro B_syn < 0.5 lost it. One redundancy function, no robustness check, tiny constructed systems. The “unifying theory” of the title is a framing: the EI section re-describes a known decomposition and adds no result. | Active |
| THEORY-582 | Causal coupling between a process and something outside it does not by itself make that thing part of the process, so an argument from coupling to extended cognition needs a further premise saying what makes a process cognitive | Adams & Aizawa (2001), LIT-438, read in NOTE-368, and their reply to Clark (2010, read from a draft), LIT-455, read in NOTE-380. A thermostat’s bimetallic strip, the kidney’s filtration and a nerve at a neuromuscular junction are each coupled to more than themselves, and their processes do not extend. The 2010 text names the inference the “coupling-constitution fallacy”. The point is that coupling does not entail parthood. It is not shown that coupling never accompanies parthood, nor that the extended-mind authors relied on coupling alone. | Active |
| THEORY-584 | The quantum fidelity tr√(ρ1^{1/2}ρ0ρ1^{1/2}) is exactly the classical Bhattacharyya overlap of the two states’ outcome distributions minimized over all measurements, and the minimum is attained by a projective measurement whose eigenvalues are square roots of likelihood ratios | Fuchs (1995), LIT-018, read in full in NOTE-021. min over POVMs of Σ_b √(tr ρ0E_b)√(tr ρ1E_b) equals tr√(ρ1^{1/2}ρ0ρ1^{1/2}) (Eq. 3.28), attained in the eigenbasis of M = ρ1^{-1/2}√(ρ1^{1/2}ρ0ρ1^{1/2})ρ1^{-1/2} (Eq. 3.67), whose eigenvalues are √(tr ρ0E_b / tr ρ1E_b) (Eq. 3.92). Proved for invertible ρ1 with a separate support construction, in finite dimension. The fidelity is the unsquared one; Jozsa’s is its square. The thesis says nothing about trace distance, and the analogous problems for Rényi overlaps and the quantum Chernoff bound are left open. | Active |
| THEORY-585 v2 | In the Döring–Isham scheme for realising collections of systems in topoi, disjoint sums of quantum systems compose exactly; along a tensor product of finite-dimensional systems, the pull-back of a constituent’s quantity agrees with the composite’s own quantity at a context exactly when every projection in it with proper support on the first factor has the form P⊗1, so it fails for every non-scalar quantity and no operator on the composite realises the pull-back | Döring & Isham, paper IV (2007/8), LIT-310 (NOTE-285), restated in LIT-378 §§11–13 (NOTE-326). Classical physics satisfies the translation condition (3.18) exactly. For quantum disjoint sums the pull-back of δ̆(A₁⊕A₂) is δ̆(A₁) (Lemma 4.2, 4.24; checked). For H₁⊗H₂ the pull-back gives δ(Â₁)_{V_W}⊗1̂ at each context W, which equals δ(Â₁⊗1̂)_W on contexts V₁⊗ℂ1̂. That it differs elsewhere, and that no operator realises it, are asserted. The rules are “partly experimental”, and the trivial-system rule contradicts the series’ own definition of contexts. | Active |
| THEORY-587 | A U-turn stopping rule for Hamiltonian Monte Carlo leaves the target invariant once the trajectory is built as a random balanced doubling tree and the next state is drawn only from states that could have regenerated it; with a dual-averaged step size it matched the best hand-tuned identity-mass HMC on two of four test posteriors and beat it about threefold on the other two, and its ergodicity is not shown | Hoffman & Gelman (2011), LIT-085, read in full in NOTE-081. The invariance argument (conditions C.1–C.4 and the 2⁻ʲ tree probability) is complete, and so is the O(j)-memory kernel. Ergodicity is not addressed. The efficiency comparison is against HMC with an identity mass matrix on four smooth, near-unimodal targets, and the dual-averaging constants were tuned on one of the four. The account is of the 2011 algorithm; production NUTS has since changed its selection rule, U-turn criterion and adaptation. | Active |
| THEORY-592 | Every unital Banach *-algebra satisfying Gelfand and Naimark’s norm conditions is isometrically *-isomorphic to a norm-closed algebra of operators on a Hilbert space, built from positive functionals by what is now the GNS construction, and a commutative one is the algebra of all continuous functions on its compact maximal-ideal space | Gelfand & Naimark (1943), LIT-313, read in full in NOTE-303. Theorem 1 embeds every normed -ring in B(H) for a Hilbert space not assumed separable; Lemma 1 is the commutative case, resting on Gelfand’s 1941 paper and Šilov’s boundary. The paper never says “C-algebra” or “GNS”. It assumes two axioms it suspected redundant, and notes that a single norm identity may replace two others. It gives the object-level isomorphism R ≅ C(𝔐), not a categorical duality, and nothing on uniqueness or irreducibility of the representation. | Active |
| THEORY-595 | A filter over a set of worlds is a finitely consistent propositional theory and an ultrafilter a negation-complete one; read so, the ultrafilter extension theorem is Lindenbaum’s lemma for propositions, and Łoś’s theorem builds a model that makes true exactly what an ultrafilter says, which proves first-order compactness without proof theory, given the axiom of choice | Turner (preprint 2021, published 2025), LIT-145, read in NOTE-177. An expository paper: “None of the results are novel” (p. 2). Maximal, prime and ultra filters coincide (Thm 2); every filter extends to an ultrafilter (Thm 3); Łoś’s theorem (Thm 9); compactness (Thm 12). The results are standard and correct, though the write-up has slips. A filter is only finitely consistent, and that is the point. The pairings with Lindenbaum and Henkin are analogies, not reductions. | Active |
| THEORY-597 | Meditation-related experiences that Western Buddhist practitioners report as challenging span 59 kinds in seven domains, from fear and sleep change to altered self-other boundaries, and they are reported in daily practice, at under an hour a day and in the first days of practice, not only on intensive retreat; the frequencies come from a sample chosen for difficulty, and the causal link is the practitioners’ own attribution | Lindahl, Fisher, Cooper, Rosen & Britton (2017), LIT-066, read in full in NOTE-036. Interviews with 60 Western Buddhist practitioners, recruited for having had a challenging meditation-related experience, and 32 teachers and clinicians were coded into 59 experience categories in seven domains. Some practitioners reported onset within ten days of starting, at under an hour a day, or in daily practice. That rebuts the claim that difficulties arise only on retreat or in the predisposed. It gives no rates and no relative risk: the sample was chosen for difficulty, and causation is self-attributed, with a causality score of 2 or more built into inclusion. | Active |
| THEORY-598 | Group interactions that enter a dynamical model linearly, as in diffusion on simplices or random walks on hypergraphs with edge-independent vertex weights, reduce to pairwise dynamics on a reweighted graph; the abrupt transitions, bistability and stabilised coexistence reported for higher-order systems come from group terms that enter nonlinearly, and whether the representation is a hypergraph or a simplicial complex is itself a modelling claim that changes the outcome | Battiston et al. (2020), LIT-091, read in full in NOTE-057. A review: the reduction results are its own §V algebra (three-body linear diffusion becomes a weighted graph Laplacian, Eq. 35) and reported results it cites (Chitra and Raphael on hypergraph walks); the nonlinear effects are one-paper summaries, mostly mean-field or all-to-all. The review never states the condition as a result, and its introduction and conclusion claim more (“no doubts”, “profoundly change”), citing linear cases among the examples. The condition is the record’s reading of what its body shows; edge-dependent vertex weights also escape the reduction. | Active |
| THEORY-599 | In trained neural-network weights stored as BF16 or FP32, nearly all the redundancy a lossless coder can remove lies in the floating-point exponent, which occupies about 40 of its 256 values and compresses to about a third under single-symbol Huffman coding, while sign and mantissa are near-incompressible; the redundancy belongs to the number format, not to the learned function | Hershcovitch et al. (2024), ZipNN, LIT-268, read in full in NOTE-241. Fifteen models broken down by byte group (Table II), with the exponent skew shown on four (Fig. 2) from 1 GB middle slices. A parameter shuffle changes Zstd’s exponent compression by at most 0.05%, so the gain is order-0. The paper never computes an entropy. Its explanation of the skew is partly wrong as stated, and its remark that the redundancy shows overparametrization unused does not follow from what it measured. | Active |
| THEORY-601 | Among US undergraduates in the HERI surveys, students with higher verbal ability and stronger self-reported intellectual dispositions select into philosophy, and after adjusting for their entering SAT or freshman scores philosophy majors still score higher than other majors on GRE Verbal (SMD 0.48) and LSAT (0.35) but not on GRE Quantitative, and report more Habits of Mind and Pluralistic Orientation; the adjusted advantage is an association, not a shown effect of studying philosophy | Prinzing & Vazquez (2025), LIT-155, read in full in NOTE-167. HERI/CIRP freshman–senior data on 649,511 students at 804 institutions, 4,843 philosophy majors. Selection is real (SAT Verbal OR 1.57, SAT Math none). Baseline-adjusted mixed models and an IPTW check agree on the verbal and logical-test advantage and on its absence for quantitative tests. All test scores are self-reported, the SAT is an imperfect baseline for the LSAT, and nothing rules out faster verbal growth or harder test preparation among those who choose philosophy, so the paper’s causal title is not part of this account. | Active |
| THEORY-602 | An ergodic source’s entropy is the least average rate at which it can be encoded, a noisy channel has a capacity below which any rate can be sent with arbitrarily small error and above which the equivocation cannot fall below the excess, and a source can be reproduced at a given fidelity over a channel exactly when its rate for that fidelity does not exceed the capacity; Shannon rests the justification of entropy on these theorems, not on the axioms that single out −Σ p log p | Shannon (1948), LIT-764, read in NOTE-591: Parts I–II closely, Parts III–V and the later appendices skimmed for their theorem statements. The noisy-channel achievability proof is a random-coding sketch at the typical-set level, not a full ε–δ argument, and the strong converse is not proved; the theorems themselves were later made rigorous. Shannon says the uniqueness theorem for H is “in no way necessary” to the theory. The theorems assume ergodic, finite-state sources and channels and say nothing about meaning. | Active |
| THEORY-603 | In the Iowa gambling task, every one of ten controls and none of six patients with bilateral ventromedial prefrontal damage developed anticipatory skin-conductance responses, and three patients who could state the right strategy still did not switch to the good decks; the controls’ preference for the good decks became significant only once they reported a hunch, so the study does not show advantageous choice or deck-specific bodily signals before any awareness | Bechara, Damasio, Tranel & Damasio (1997), LIT-383, read in NOTE-331. The robust results are the group difference in anticipatory SCRs (Fisher’s exact, one-sided P < 0.001) and the insufficiency of stated knowledge in three patients. The abstract’s “before they realized which strategy worked best” and “before they knew” outrun notes 8 and 9: the pre-hunch choice preference was not significant, and the pre-hunch SCR effect reported is group × period, with no bad-versus-good deck test. Nothing here shows that the SCR, or what it indexes, causes the choice. | Active |
| THEORY-605 | In any ontological model that reproduces the quantum predictions and treats independently prepared systems as having product distributions over a joint state that is just the list of their states, distinct pure quantum states have distributions over the ontic state whose supports overlap in measure zero, so the pure state is fixed by the ontic state; approximate agreement within ε bounds the overlap by 2ε^{1/n} | Pusey, Barrett & Rudolph (2011), LIT-062, read in full in NOTE-030. For any two non-orthogonal pure states, n copies prepared independently and an entangled measurement in which each outcome is impossible for one product preparation force zero overlap; with ε error, D(µ0, µ1) ≥ 1 − 2ε^{1/n} (Eq. 7). Determinism and locality are not assumed. Preparation independence, in its strong product-space form, carries the weight, and dropping it permits ψ-epistemic models. The theorem excludes overlap, not every reading of ψ as knowledge, and it does not reach theories outside the ontological-models framework. | Active |
| THEORY-608 | On one physical piano-movers puzzle scaled to the solver, ant groups solve better as they grow, while human groups barred from speech and gesture solve worse than individuals and talking groups only about as well as one; large ant groups keep speed and heading through wall contact, and the human mechanisms offered are fitted or observed informally, not tested | Dreyer et al. (2024), LIT-075, read in NOTE-040. Large ant groups (n = 28) reach about 90–100% success by 400 corridor widths, small groups (n = 48) about 50%. Restricted human groups do 32% worse than individuals’ median, talking groups 7% better. There are no inferential statistics. The ant persistence mechanism is supported by collision data and a model whose carrier numbers fall short of calibration; the human majority vote (greediness P = 0.2) is tuned by eye, and “follow a random leader” rests on unrecorded observation. One ant species, one puzzle, one human population. | Active |
| THEORY-609 | When implementing an automaton requires reliable, counterfactual-supporting transitions for every formal state, Putnam’s construction fails, but any system with a clock and a dial still implements every inputless finite-state automaton, and any system with the right input–output dependencies, an input memory and a dial implements every finite-state automaton with those dependencies, so finite-state automata cannot carry computational sufficiency; combinatorial-state automata are argued, not proved, to escape | Chalmers (1996), LIT-404, read in NOTE-342 from the author’s HTML copy. The failure of Putnam’s construction and the clock-and-dial and input-memory theorems are constructive proofs given the definitions, so FSA-functionalism “still almost reduces to behaviorism”. That a combinatorial-state automaton can be implemented only by the right fine-grained causal structure is a sketch: the obvious blind construction grows by a factor of n per step, and that every construction does is asserted. Logically possible false implementations and the one-region-per-component condition are left open by the author. | Active |
| THEORY-611 v2 | Classifier-free guidance extrapolates between a conditional and an unconditional score learned by one network; with exact scores this equals classifier guidance by the classifier the generative model defines through Bayes’ rule, but learned scores need not be conservative, so the guided field is in general the gradient of no density and the tilted distribution that motivates guidance is not shown to be what is sampled | Ho & Salimans (2022), LIT-839, read in full in NOTE-603, with the derivations of Section 3 checked line by line. The non-conservativeness argument is credited to Salimans & Ho (2021). The FID–IS trade-off is shown on class-conditional ImageNet only. NOTE-603 declined a THEORY because the anthology holds the account of what the guidance weight does to metrics; this one states only the probabilistic point, which the record’s CLAIM-123 uses. | Active |
| THEORY-617 | In additive Gaussian noise independent of an input of finite power whose law does not depend on the signal-to-noise ratio, the derivative of mutual information with respect to that ratio is half the minimum mean-square error of the optimal noncausal estimator, so mutual information is accumulated estimation error over SNR; in continuous time the causal filtering error equals the noncausal error averaged over SNR, and the identity fails with feedback | Guo, Shamai & Verdú (2004), LIT-058, read in full in NOTE-086. Proved five ways for every finite-power input, in scalar, vector, discrete-time and continuous-time forms. It concerns the optimal, conditional-mean estimator; carrying it to a trained diffusion model’s learned denoiser needs the mismatched version, which is later work. Theorems 3, 4 and 11 are stated without proof. | Active |
| THEORY-619 | Over the poset of commutative von Neumann subalgebras, coarse-graining a projector to a smaller subalgebra is exactly the direct image of its clopen region of the Gelfand spectrum under restriction, so the coarse-graining presheaf is the presheaf of clopen spectral regions; and a quantum state’s sieve-valued truth values are fixed by one coherent family of regions, its supports: a proposition is true at a subcontext exactly when the support there lies under the coarse-grained proposition | Butterfield & Isham (2001), “Topos Perspective on the Kochen-Specker Theorem: IV”, LIT-382, read in full in NOTE-328. §2.4 proves G ≅ CloΣ: restricting the characters of V₁ that make P̂ true gives exactly the characters of V₂ that make its coarse-graining true, via a cited state-extension theorem. Theorem 3.1 shows that any assignment defined by “support ≤ coarse-grained proposition”, with supports forming a global element of G, is a sieve-valued valuation obeying FUNC; a state’s supports qualify. Both theorems are close to definitional. Nothing here builds Heyting operations on clopen sub-objects. | Active |
| THEORY-620 | Walsh’s 1923 functions are a closed orthonormal set on (0, 1) of ±1-valued step functions, each a parity of finitely many binary digits, whose expansions behave like Fourier series: grouped uniform convergence for continuous functions, convergence under bounded variation, localisation, Cesàro summability, and divergence at any chosen point for some continuous function | Walsh (1923), LIT-300, read in full in NOTE-281, with the explicit formula and the Lebesgue-constant recurrence checked for small n. Closure is inherited from Haar’s system, to which Walsh’s is related by an orthogonal change of basis within each dyadic level. Theorems II–X give the Fourier-like convergence theory. The paper never mentions groups or characters: that these functions are the characters of the Cantor group, and of the Boolean cube on n digits, is a later identification. | Active |
| THEORY-623 | In the two tests of contextuality in language data that the record has read, the contextuality is put there by the experimenter or not secured: meaning-selection models of ambiguous phrases estimated from corpora are all signalling and their two Contextuality-by-Default verdicts are plausibly noncontextual by their own bootstrap, while BERT’s 52 million anaphora models have a support fixed by the sentence schema, so whether each counts as contextual depends only on how uncertain the model’s three referent choices are | Lo, Sadrzadeh & Mansfield (2024), LIT-844, read in NOTE-648, and Wang, Sadrzadeh, Abramsky & Cervantes (2021), LIT-842, read in NOTE-645. Both readings recomputed the criteria. That 71.1% CbD-contextual measures BERT’s uncertainty is NOTE-648’s derivation from the stipulated support (CNT1 = 2 − Δ). The only possibilistically contextual phrase model has hand-set supports. Neither study uses human judgements, so neither bears on whether people’s interpretations are contextual. | Active |
| THEORY-624 v2 | Orseau, McGill and Legg cast whether an identified system is an agent or a device as Bayesian model comparison between descriptions of the function it optimises and descriptions of its input-output mapping, so that the verdict is relative to the observer’s prior; on seven toy gridworld trajectories goal-seeking comes out agent-like and wall-following device-like, and they argue that under a speed prior everything may look like an agent | Orseau, McGill & Legg (2018), LIT-911, read in NOTE-705. A formal version of Dennett’s choice of stance: the posterior probability that a system is an agent, from mixtures of goal-optimizing (inverse RL) and input-output descriptions. Illustrated on seven gridworld trajectories, with no human study. It presupposes an identified system, and its verdicts depend on the prior; under the speed prior everything may look like an agent. | Active |
| THEORY-629 | Every continuous linear functional on a Hilbert space is the inner product with a unique vector of the same norm, given in any orthonormal basis by the conjugated values of the functional on the basis; completeness is what makes it so, and Riesz’s 1907 statement on L² asserted only that a representing function exists | Adler (2021), LIT-243, read in full in NOTE-854: two complete proofs, by projection (Thm 3.5) and by an orthonormal basis (Thm 4.9), with completeness used once, in the projection theorem. Riesz (1907), LIT-326, read in NOTE-289, is the priority note: existence of k with U(f) = ∫ f k for continuous linear U on square-summable functions, stated without proof, uniqueness or norm. The theorem says nothing about kernels, factorisation or uniqueness of representations. Both readings advise citing a textbook for anything leaned on. | Active |
| THEORY-632 | Purity is moralized in self-report: in morality-as-cooperation’s own questionnaire studies, purity items are rated about as morally relevant as loyalty and authority items, and harmless purity violations are condemned, so that theory’s prediction that non-cooperative behaviour is morally neutral fails unless purity is itself cooperative | Curry, Jones Chesters & Van Lissa (2019), LIT-496, read in NOTE-412, against the prediction stated in Curry (2016), LIT-503 (NOTE-408), with the harmless-wrongs studies Haidt (2001) reports, LIT-511 (NOTE-403). In two samples, MFQ purity items are rated morally relevant at the level of ingroup and authority items, and the authors concede this “appear[s] to contradict” the theory. It is self-report from people asked about purity, not a blind count of what societies moralize, and it does not say whether purity is cooperative. | Active |
| THEORY-634 | In a ring of M chemotactic catalyst species, each making the next one’s substrate, the non-reciprocal nearest-neighbour coupling makes the linear stability matrix circulant: the uniform state goes unstable for any M exactly when the species attract themselves, and the parity of M then decides the leading mode, a real alternating mode for even M that sorts species by parity into two static clusters, and a complex pair for odd M that can give persistent oscillation | Ouazan-Reboul, Agudo-Canalejo & Golestanian (2023), LIT-036, read in NOTE-079. The spectrum is exact for equal parameters (Eq. 5): instability iff μ(p) − μ(s) < 0, real leading eigenvalue for M = 2K, complex pair for M = 2K + 1. Brownian dynamics at M = 5 and 6 bear out the linear predictions; the odd-cycle oscillation is one trajectory at one parameter set. Direct metabolic partners repel; same-parity second neighbours co-aggregate. The origin-of-life reading is untested, and the authors concede in the review file that real catalysts’ mobilities are unknown. | Active |
| THEORY-635 | If the quantum state of a Bohmian system is identified with the velocity dispositions it induces on the particles, inequivalent states can share one dispositional history: every energy eigenstate of a free particle in a box induces zero velocity everywhere, so the reading erases differences of energy and of typicality unless the map from states to velocity fields is injective for generic solutions, which is conjectured, not proved | Belot (2011), LIT-100, read in full in NOTE-110. Under the guidance law ψ(t) gives a vector field X_t on configuration space, read as a holistic disposition. For a particle in [0,1], each φn(x) = sin(πnx) gives zero velocity everywhere, so infinitely many energy eigenstates share one dispositional history, and the distinction between φ1 and φ2 in energy and typicality is lost. The rescue rests on Goldstein’s unproved genericity conjecture (fn. 41), and the problem arises only for closed systems (fn. 42). It is a counterexample to one reading, not a verdict on ψ’s category. | Active |
| THEORY-636 | In vignette studies of US online participants, judgements that an outcome happened because the agent wanted it move with judgements that it was done intentionally when luck is manipulated, and predict them better than know-how or probability-raising do; but they account for only about half of the luck effect, and in one morally neutral case the two come apart | Quillien (2026), LIT-132, read in NOTE-092. Three Prolific studies (N = 198, 673, 142; ratings 1–7). Causation is the strongest predictor of intentionality (standardized β .46, .56, .53, against know-how .33, .20, .27 and probability-raising .14, .16, .10), and a pre-registered Knowledge × Process interaction predicted by the causal account holds (F(1, 423) = 32). The evidence is correlational, the mediation exploratory and partial (64%, 45%, 49%), and in the neutral Kraemer vignette causation shows no effect where intentionality does. The “model” is verbal; robustness is never measured. | Active |
| THEORY-639 | Inductive risk arises not only when a hypothesis is accepted but in choosing a significance level, in characterising data and in interpreting results: in the dioxin cancer studies each of these choices traded false positives against false negatives, three panels scored the same rat-liver slides differently, and threshold and no-threshold readings of the same data underlie regulatory limits about 100 times apart | Douglas (2000), LIT-139, read in NOTE-145. Three worked cases from one domain, dioxin toxicology and its regulation, on secondary data (EPA 1994; Kociba et al. 1978; Goodman & Sauer 1992). The structural point and the variation in slide scoring are strongly supported. The 100-fold figure is cited from Finkel (1988) and Greenlee et al. (1991). This is the descriptive half of the paper; the claim that values are therefore required is THEORY-442. | Active |
| THEORY-640 | Measured as mutual 10-nearest-neighbour overlap, competent vision models agree more with one another and better language models agree more with vision models, but the cross-modal score reaches only 0.16 of a possible 1, is the maximum over all layer pairs, and was adopted after a global kernel metric showed only a very weak trend | Huh, Cheung, Wang & Isola (2024), LIT-302, read in NOTE-286. Across 78 vision models, those solving more VTAB tasks agree more in mutual k-nearest-neighbour structure; language models with lower bits-per-byte agree more with vision models on 1,024 image–caption pairs. The cross-modal score “only reaches 0.16” of 1, is maximised over layer pairs, and the metric was chosen after CKA “revealed a very weak trend”. It is local neighbourhood agreement that rises with competence, not kernel equality and not convergence to a model of reality. | Active |
| THEORY-641 v2 | Under a well-specified, identifiable parametric Markov model of the target given the joint past, with ergodic lagged states, transfer entropy is the limit of the normalised log-likelihood ratio for the null of no dependence on the source’s past, so it is Granger-sense predictive improvement; for finite-state chains the plug-in estimator is exactly that statistic, and under the null and the usual regularity it is asymptotically χ² with the difference in parameter counts as degrees of freedom | Barnett & Bossomaier (2012), LIT-048, read in full in NOTE-053. The consistency result is a one-line consequence of Props. 1–2, and Prop. 2 is argued informally. The χ² null is imported from Wald (1943), whose regularity conditions are not checked. The non-central χ² for T > 0 is at best a local-alternative approximation. The finite-state identity and the Gaussian equivalence (half the Granger statistic) are exact. | Active |
| THEORY-642 | Basic two-level type theory, an inner homotopy type theory with an outer level of strict equality, is conservative over homotopy type theory, so an inner type it proves inhabited is already inhabited in HoTT, but a fibrant-replacement type former would force uniqueness of identity proofs on the inner level | Annenkov, Capriotti, Kraus & Sattler (2017; MSCS 2023), LIT-095, whose §§1–3 were read in full in NOTE-047, the reading of the nLab page it supersedes (LIT-040). Props 2.18–2.19 prove that inhabitation of inner types is reflected, through a presheaf model over any model of HoTT. Thm 2.20 proves that a fibrant-replacement former implies UIP for inner types. The conservativity is the weak, inhabitation-reflecting kind. Results that use the axioms relating the two levels’ natural numbers, which untruncated semisimplicial types need, are not known to transfer. | Active |
| THEORY-645 | A countable discrete abelian group and a compact second-countable abelian group can each be recovered as the character group of the other, and a locally compact, connected, second-countable abelian group is a compact group plus ℝʳ; Pontryagin’s 1934 paper does not prove duality for general locally compact abelian groups | Pontryagin (1934), LIT-333, read in full in NOTE-284. Theorem 5 and the First Fundamental Theorem give the duality between countable discrete and compact second-countable abelian groups, with characters valued in ℝ/ℤ. The compact half rests on Peter–Weyl with Haar measure, used to extract one-dimensional characters. The Second and Third Fundamental Theorems are structure theorems, not duality. Commutative only; for non-abelian groups the relevant object is Peter–Weyl, and characters are traces, not homomorphisms. General locally compact duality is van Kampen’s (1935). | Active |
| THEORY-649 | When agents holding uniformly random consistent theories over two or three atomic sentences each average their theories once, bit by bit, with all peers within a Hamming bound of confidence, an individual agent ends up with the inconsistent theory at most 1.81% of the time, but the chance that some agent in the group does reaches 6.4% for two sentences and 16.9% for three | Wenmackers (2011), LIT-163, chapter 5, read in NOTE-139. An exact count, extended by sampling, of how often one round of bounded-confidence averaging over theories yields the inconsistent theory. The per-agent fraction F_AG peaks at 1.8112% (M = 2) and 0.3215% (M = 3); the per-profile fraction F_OP reaches 6.4298% and 16.867%. The thesis’s headline, that the risk “can be made arbitrarily small” by adding sentences, rests on two values of M and is contradicted for F_OP. | Active |
| THEORY-651 v2 | Disinformation incidents labelled with DISARM techniques can be carried in standard STIX 2.1 without extending it, as an intrusion set that uses attack patterns, is attributed to a threat actor and targets locations; the DISINFOX mapping that moved 118 incidents into OpenCTI this way keeps only a name, a free-text description, a first-seen date and labels beside those three relationships, with no object or property for an incident’s narratives, channels, evidence or countermeasures or for the confidence of its attribution | Sánchez González et al. (2025), LIT-083, an unreviewed preprint read in NOTE-028. The mapping is explicit (four STIX object types, three relationship types) and was exercised end to end on 118 incidents; the reading rates that demonstration strong. Nothing beyond ingestion was measured. Attribution is one ungraded edge to an actor, typically a nation-state, which is the layer the EEAS’s Exposure Matrix grades in four tiers. What the mapping is, and that the incidents were ingested, is held; whether the representation survives exchange and whether sharing helps is THEORY-1238’s question. | Active |
| THEORY-652 | Over any field, a finite-dimensional associative algebra with no nilpotent two-sided ideal is uniquely a direct sum of simple algebras, and each simple algebra is a full matrix algebra over a division algebra, which over the reals can be the complex numbers or the quaternions | Wedderburn (1908), LIT-309, read in full in NOTE-280. Thm 17 with Thm 10 gives the unique decomposition into simple algebras; Thms 20–22 build n² matrix units from primitive idempotents and give simple = primitive × full matrix algebra. Primitive here means one idempotent, which makes the factor a division algebra once semisimple (Thm 15). The field is arbitrary, which was the paper’s novelty. That the centre is spanned by the block identities follows from Thm 23’s corollary; the paper never says “centre”. The general splitting of an algebra into nilpotent plus semisimple parts is claimed in the summary but proved only in special cases. | Active |
| THEORY-653 | In the ethnographic record of the 60-society HRAF Probability Sample, seven cooperative behaviours are described as good in 961 of 962 valenced mentions, but each is recorded in between 90% and 15% of the societies, and the coding could barely have registered a negative | Curry, Mullins & Whitehouse (2019), LIT-047, read in NOTE-034. Coding eHRAF paragraphs on the 60 Probability Sample societies found 962 morally valenced mentions of helping kin, helping one’s group, reciprocating, bravery, deference, dividing disputed resources and respecting possession; 961 were positive. Presence ran from property (54/60) to fairness (9/60) and tracked ethnographic coverage (r = .43). The behaviours are defined in virtue terms and conflicts were set aside, so a negative was nearly impossible to code. It is a description of the coded record, not a test of whether morality is only about cooperation. | Active |
| THEORY-654 | For full-state linear-quadratic regulation of a single-input plant, the return-difference identity keeps the sensitivity at or below 1 at every frequency, so the loop tolerates any gain increase, a gain cut to one half and at least 60° of phase error for any Q ≥ 0 and r > 0; the guarantee does not carry over when the state is replaced by an observer’s estimate | Başar, Meyn & Perkins (2020), LIT-051, graduate course notes read in full in NOTE-062, with every worked example recomputed. §10.6 derives |1 + L(jω)|² = 1 + |P(jω)|²/r ≥ 1 for the full-state loop, hence |S| ≤ 1 and the classical margins. The derivation is for one input; the multi-input version needs more, which the notes do not discuss. The Pendubot thread shows a controllable, observable plant whose one-sensor observer-based design has gain margin below 10⁻². The result is Kalman’s and standard; the notes are an adequate free citation for it. | Active |
| THEORY-659 | In the EEAS’s 2024 sample of 505 foreign information manipulation incidents, the overt state voice is a small core: of the 2,055 channels that recur across incidents, 20% are attributed to Russia, 3.5% to China and 76.5% are not attributed, after some 28,000 disposable inauthentic accounts are set aside; the sample is not representative, and the four-tier Exposure Matrix used to grade channels’ ties to a state is unvalidated | European External Action Service (2025), LIT-084, the 3rd EEAS report on FIMI threats, read in NOTE-060. A description of what the EEAS monitored, which the report itself says is not a basis for general trends. The 76.5% is a share of a filtered co-involvement graph, not of the 38,000 channels, though the report’s conclusions restate it as such. The Exposure Matrix grades attribution in four tiers by evidence and treats it as revisable; no inter-rater test or worked classification is given, and the strongest evidence is proprietary or classified. | Active |
| THEORY-661 v4 | In a nearly decomposable system, where interactions within subsystems are much stronger than interactions between them, each subsystem’s short-run behaviour is approximately independent of the others’, and in the long run each depends on the others only through their aggregate behaviour; so a hierarchy defined by intensity of interaction separates fast dynamics inside its parts from slow, aggregate dynamics between them | Simon (1962), LIT-905, read in full in NOTE-704. The two propositions are a theorem of Simon & Ando (1961), cited and not reproduced, and established since. Simon widens hierarchy to any nesting of interrelated subsystems, with or without authority, identified by who interacts with whom. His claims that social systems are prominently nearly decomposable, and that higher organisational levels run slower, are assertion and “probably true”. The watchmaker argument is a model whose numbers he says not to take seriously. | Active |
| THEORY-664 | In a finite-dimensional probabilistic theory with unrestricted effects and local tomography, if every system can steer each of its own states from a composite with a copy of itself, every irreducible state space is homogeneous and weakly self-dual; steering does not yet give strong self-duality, so it stops short of the Jordan-algebraic and quantum state spaces | Barnum, Gaebler & Wilce (2009), LIT-081, read in full in NOTE-082. A bipartite state steers its marginal iff its map on effects is a strong quotient of order intervals (Thm 5.4). With matching dimensions a steering state is an order-isomorphism, and a cone is homogeneous iff every interior state purifies to an isomorphism state (Thm 4.1). Hence universal self-steering gives homogeneity and weak self-duality (Prop. 5.8). The weaker Prop. 5.7 is thinly argued. The C*-algebraic conclusion of §6 needs strong self-duality, which the paper assumes rather than derives. | Active |
| THEORY-668 | When the encoder is restricted to prunings of one fixed hierarchy of nested partitions, any non-negatively weighted sum of relevant information, irrelevant information and compression decomposes into per-node terms, so a bottom-up recursion finds the unique optimal tree with fewest leaves exactly in linear time, while greedy top-down splitting can miss information lying below a level that carries none; penalising irrelevant information only prunes the information-bottleneck tree | Larsson, Maity & Tsiotras (2022), LIT-868, read in full in NOTE-676, every proof followed step by step. The decomposition is the chain rule for refinements, and the recursion is the classic bottom-up pruning of a fixed tree. NOTE-676 declined a THEORY as exact but narrow; it is filed because the reading supports it as stated and its greedy-failure point bears on how hierarchy is measured. The hierarchy is given, not found, and the evaluation is one image. | Active |
| THEORY-672 | In the iterated Prisoner’s Dilemma with a probability w of meeting again, no strategy does better against an established TIT FOR TAT than TIT FOR TAT does against itself if and only if w ≥ (T−R)/(T−P) and w ≥ (T−R)/(R−S); always defecting is stable for every w, a cluster of reciprocators can invade it, and a nice stable strategy resists clusters, but the stability is only neutral, since always cooperating ties | Axelrod & Hamilton (1981), LIT-498, read in full in NOTE-411. Against TIT FOR TAT the best reply is always-defect or alternation, which gives the condition on w. The cluster result and the “ratchet” are explicit payoff comparisons. “Evolutionarily stable” here means no mutant does better, which is weaker than Maynard Smith and Price’s definition: always cooperate does exactly as well. The model is deterministic and error-free. Tournament robustness is a separate, empirical claim about the strategies people submitted. | Active |
| THEORY-674 v2 | In Garg and Fetzer’s GPT-4o-mini extraction from NBER and CEPR working papers, edges kept in at least 4 of 9 passes and never checked against human labels, the mean per-paper share of edges labelled as backed by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose from 7.7% in 1990 to 31.7% in 2020, at every overlap threshold; the paper’s per-decade slope fell from +0.0187 to +0.0011 once method mix and graph volume were controlled | Garg & Fetzer (2024; arXiv v2 2026), LIT-079, read in full in NOTE-083. GPT-4o-mini, nine passes per paper over the first 30 pages of 44,852 working papers, edges kept in at least 4 of 9 passes. The direction of the trend holds across overlap thresholds, its level does not (mean causal share 0.152 at EO = 1, 0.270 at EO = 9), and it nearly vanishes once method mix and graph volume are controlled. Validation covers paper-level method labels only, with RCT precision about 0.02. The association of causal structure with top-five publication rests on year-fixed-effect regressions over different subsamples and is not part of this claim. Promoted 2026-10-20 on the measurement alone (ADR-047 v3); the economics claim is held as THEORY-824. | Active |
| THEORY-676 | Dewar’s information-theoretic derivation of maximum entropy production does not reach the far-from-equilibrium regime where the principle is applied: the 2003 route needs a mean-field step and an unargued assumption about path counts, the 2005 route’s orthogonality condition, which yields both maximum and minimum dissipation, holds only with linear constitutive relations, and the 2003 self-organized-criticality corollary is an artefact of a quadratic expansion | Grinstein & Linsker (2007), LIT-525, read in NOTE-422, against Dewar (2003), LIT-521, read in NOTE-426. The self-organized-criticality point was checked against Dewar 2003 and is correct. The linearity point concerns Dewar 2005, which the record could not read; it is held on the Comment’s account of its equations and on an exactly solvable spin chain, re-derived. The Comment removes one derivation and one application. It does not say MEP is false, and it says the existence of far-from-equilibrium extremal principles “has not been settled”. | Active |
| THEORY-677 | Under uncertainty about which beings are sentient, an appeal to precaution does not by itself fix which error is worse to make: precaution on behalf of a possibly sentient being and precaution on behalf of human liberty point opposite ways, and judging which mistake costs more turns on the very science in doubt | Schwitzgebel & Sinnott-Armstrong (2026), LIT-166, read in NOTE-116 from the September 2025 draft. A conceptual argument with worked cases, which the reading rates strong: “By itself, an appeal to precaution does not specify the type of error most to be avoided” (p. 8). Birch’s framework puts the burden on whoever risks harming a sentience candidate; Mill’s harm principle, read as a precaution, puts it on whoever would restrict liberty. The record holds the point, not the reviewers’ reading of Birch’s books, which it has not read. | Active |
| THEORY-679 | For dynamics obeying detailed balance in a heat bath, the heat released in a transition between observer-defined macrostates is bounded below by the log-odds against its reversal less the internal entropy change; where a replicator’s likeliest way back is a copy decaying within one generation, the bound grows with the copy’s durability and the speed of copying, not with the information copied | England (2012; J. Chem. Phys. 2013), LIT-515, read in NOTE-419. The general bound, β⟨ΔQ⟩ + ln π(II→I) + ΔS_int ≥ 0, is a coarse-grained second law and was re-derived. Applied to replication it gives a heat cost of about 2n ln(n τ_decay/τ_div) per copy, which falls away as division slows. The numbers for E. coli (within a factor of three of the bound) and RNA rest on an upper bound on the reverse probability that is argued from biology, not proved, and on an internal-entropy term set “arbitrarily”. It prices finite-time irreversibility, not information, and draws no line between living and non-living replicators. | Active |
| THEORY-683 | Permuting a fixed set of few-shot examples moves a GPT-family model’s classification accuracy between near chance and near the supervised state of the art, at every model size tested; the good orders are specific to the model, the spread survives more examples and calibration, and bad orders mostly fail by predicting one label | Lu et al. (2021, ACL 2022), LIT-833, read in NOTE-626. All 24 orders of four balanced SST-2 examples give accuracies from about 50% to above 85%, in GPT-2 and GPT-3 models of eight sizes; Spearman correlations of order accuracy between sizes lie between −0.35 and 0.27 (175B against 2.7B: 0.05). The variance stays high up to 32 examples and after calibration. It is a measurement on causal GPT models and text classification, with the transfer analysis resting on one example set, and the paper finds no property of an order that predicts its effect. | Active |
| THEORY-685 | The Mizohata–Takeuchi conjecture fails for every compact C² hypersurface with a point of non-zero curvature: for each large R a function and a weight violate it by a factor of order log R, which refutes Stein’s conjecture as stated and closes the Mizohata–Takeuchi route to loss-free endpoint multilinear restriction | Cairo (2025), LIT-030, read in full in NOTE-051, which checked the counting and exponent bookkeeping. The paper states the theorem for any C² hypersurface not lying in a hyperplane; the proof needs a curved point, which a connected such surface has. The incidence lemma (§4) is the step a non-specialist could not verify. The loss is log R, not a power, and the counterexample is a family indexed by R. It does not refute endpoint multilinear restriction itself. Its proposed local replacement has since been refuted with power loss on a dense family of surfaces, per the abstract of a later paper. | Active |
| THEORY-686 v2 | An observation modelled as a monotone map between preorders can show of a joined system more than the join of its observations of the parts, and out of a preorder with all joins the observations that preserve all joins, the empty join and infinite joins included, are exactly the left adjoints | Fong & Spivak (2018), LIT-803, Chapter 1, read closely in a Skimmed NOTE (NOTE-601) whose other chapters were seen only at their openings. Def. 1.93, after Adam, defines a generative effect as Φ(a) ∨ Φ(b) < Φ(a ∨ b). Prop. 1.111 and Thm 1.115 give that left adjoints preserve all joins and, from a source with all joins, only they do. The results are standard order theory, re-derived by the record because NOTE-601 is a skim. The absence of generative effects is weaker than preserving all joins, and does not characterise the left adjoints. | Active |
| THEORY-688 | In an actor-critic learner that can choose temporally extended options as well as primitive actions, options that fit the task speed learning, while options with ill-placed subgoals slow it, and options that fit an earlier layout keep the agent from finding a newly opened shortcut, so hierarchy helps only with the right subroutines | Botvinick, Niv and Barto (2009), LIT-581, read in NOTE-451. Toy simulations in a rooms domain with pre-trained options. Doorway options sped learning; window options slowed it; with a shortcut opened, primitive-only agents found the shortest path on 75% of runs while doorway-option agents never used it (mean 11.79 against 9.73 steps). It is a result about the model. That people or brains work this way is the separate, proposed account THEORY-484. | Active |
| THEORY-689 | A mobile robot built from parallel layers of simple finite-state machines, each linking sensing to action by itself and combined only by one layer suppressing or inhibiting another’s signals, avoids obstacles, wanders and heads for distant places with no central world model, goal list or controller; shown at insect scale, on three layers of one physical robot | Brooks (1991, written 1987), LIT-436, read in NOTE-351. An existence proof by construction: the absence of central representation and control is true of the architecture by design, and the robot ran “for well over a year”. The evidence is described, not measured; three layers ran on a robot and six in simulation. The claims that coherence exists only “in the eye of an observer”, that representation is the wrong unit for most of intelligence, and that the method scales are assertion or labelled hypothesis. | Active |
| THEORY-692 | Near equilibrium, steady states minimise entropy production and every fluctuation decays; far from it, the excess entropy production has no fixed sign, so a fluctuation can grow past an instability of the thermodynamic branch into a structure held up only by flows across the boundary, and thermodynamics supplies only a sufficient condition for such a state’s stability, not a principle selecting which structure appears | Prigogine (1977), LIT-516, read in full in NOTE-424, with Cross & Hohenberg (1993), LIT-527, skimmed in NOTE-418. Minimum entropy production holds only for steady states in the strictly linear regime with constant coefficients. Beyond it, non-negative excess entropy production suffices for stability; its failure is not shown to give instability. Which structure appears is left to kinetics, boundaries and fluctuations. Cross and Hohenberg find no global minimisation principle beyond a perturbative one near threshold, and quote Glansdorff and Prigogine conceding that “the search for a universal kinetic potential has proved to be unsuccessful”. | Active |
| THEORY-695 | Two quantum states can be broadcast if and only if they commute, and cloned if and only if they are identical or orthogonal; the theorem is proved for invertible density operators and marks a line within quantum theory, not a witness of nonclassicality | Fuchs (1995), LIT-018, §4.3, read in full in NOTE-021, from the Barnum–Caves–Fuchs–Jozsa–Schumacher manuscript. Fidelity cannot decrease under partial trace or under the broadcasting map, so broadcasting forces equality conditions that make ρ0 commute with the optimal-fidelity operator. Proved for invertible ρ0, ρ1 only. Spekkens’ epistricted classical theories (LIT-007) reproduce no-broadcasting, so the theorem states where broadcastability ends inside quantum theory and does not by itself certify nonclassicality. | Active |
| THEORY-696 | Under multiplicative noise, members that pool their resources and split them equally every step each grow faster in the long run than they would alone, with no complementarity, relatedness or reciprocity: N independent geometric Brownian motions that share keep their drift μ and cut their volatility to σ/√N, raising each one’s time-average growth rate from μ − σ²/2 to μ − σ²/(2N); the gain vanishes as their noise becomes perfectly correlated, and a sufficiently superior member does better alone | Peters & Adamou (2022), LIT-035, read in NOTE-059. The mathematics is correct and self-contained, and the paper’s own citations show the pooled-variance result was known (McCloskey, Winterhalder, Uitdehaag, Kennedy et al.). Sharing leaves the ensemble rate μ unchanged, so an analysis built on expectations sees no reason to share, and the time average does. It holds with zero cost, perfect pooling, unbreakable pacts and unconstrained growth. It is a sufficient mechanism in that model, not a solution of the cooperation puzzle, and not a showing that cooperation is the default. | Active |
| THEORY-697 | Sparse distributed memory’s read weights each stored pattern by the number of hard locations within the read radius of both query and pattern, a ball-intersection count that falls roughly exponentially in Hamming distance over a local window, so softmax attention over cosine similarity of unit-norm queries and keys, with a fitted inverse temperature, approximates the SDM read there; the approximation cannot hold globally, since the binary count vanishes beyond twice the radius, and its inverse temperature comes from a regression, not from the paper’s analytic bound | Bricken & Pehlevan (2021), LIT-269, read in full in NOTE-242, which re-evaluated the intersection formula and the fit. The exact intersection (Eq. 2) is checked against Kanerva’s lune formula and simulation; eight SDM and fitted-softmax variants retrieve almost identically on random patterns, MNIST and CIFAR-10, with three named exceptions. The intersection is a staircase in distance, the fit holds to about dv = 20 for d = 15, the analytic β of App. B.2 is about 2–2.5 times too small, and every optimal radius assumes random patterns and addresses. The GPT-2 “confirmation” and the cerebellar implementation of attention are argument, not demonstration. | Active |
| THEORY-698 | Thresholding linear attribute probes on a language model’s embeddings yields a complete concept lattice for any incidence whatever, so the lattice is guaranteed by construction; what the Lattice Representation Hypothesis measured is per-attribute agreement with GPT-4o annotations at F1 70–83, and no lattice law | Xiong (2026), LIT-267, read in NOTE-240. Theorem 1 is the basic theorem of formal concept analysis applied to the crisp relation got by thresholding Fisher-LDA probes, and it holds for every binary relation; the paper’s own proof plan says so. The empirical content is that the thresholded probes on three 7–8B models agree with a GPT-4o-annotated incidence at macro F1 69.7–83.2 and predict WordNet subsumption at F1 57.1–77.1. No closure, absorption or recovered-lattice comparison is tested, and the paper’s join is not a lattice join as written. It is not evidence that language models encode a lattice beyond the incidence. | Active |
| THEORY-700 | For jointly Gaussian stationary processes, Granger causality in Geweke’s generalized-variance form is exactly twice the transfer entropy, conditional and multivariate, at the regressions’ lags, because least-squares residual covariance is the partial covariance and a Gaussian’s conditional entropy is half its log-determinant plus a constant | Barnett, Barrett & Seth (2009), LIT-1171, read in full in NOTE-934. F_{Y→X|Z} = 2T_{Y→X|Z}, exactly, for population quantities. The proof is two identities and was checked. Sample agreement depends on the entropy estimator. Both quantities are predictive; the authors keep them apart from physical causation. | Active |
| THEORY-701 | Transfer entropy from J to I is the Kullback entropy of assuming that I’s next state, given its last k states, does not depend on J’s last l states, so it is zero exactly when J’s past leaves I’s transition law unchanged; unlike time-delayed mutual information it is directional and zero for static correlation, and it removes a common input only when that input is conditioned on | Schreiber (2000), LIT-1177, read in full in NOTE-940. The definition and its zero condition are exact. The comparison with delayed mutual information rests on two coupled-map lattices built with a known direction. The continuum-limit claim is asserted. The Letter’s tent-map coefficient appears to be twice what Eq. 4 gives. | Active |
| THEORY-702 | Transfer entropy is the average of a local transfer entropy, the log ratio at each observation of the destination’s next-state probability with and without the source given the destination’s past, which can be negative where the source misleads; the destination history must be long for it to exclude self-influence routed through the source; and conditioning also on the destination’s other causal parents (complete) can differ sharply from conditioning on its past alone (apparent), as in elementary cellular automata, where long-history local transfer entropy picks out gliders and domain walls | Lizier, Prokopenko & Zomaya (2008), LIT-1175, read in full in NOTE-943. The local decomposition is an identity; the claims about k and about particles rest on elementary CAs judged by eye. Three worked local values and rule 18’s XOR structure were checked. | Active |
| THEORY-704 | In a stationary multivariate time series whose components include every relevant process, with interventions that are isolated exogenous changes and no instantaneous causation, a missing Granger-causal edge from a to b means intervening on a has no effect on b at the next step, non-ancestry means no effect at any horizon, and a set whose observed past blocks every back-door path into b’s ancestors identifies the effect from observational data, as a multi-step linear-predictor coefficient in the Gaussian case | Eichler & Didelez (2007), LIT-1164, read in full in NOTE-941. The back-door results are proved; the edge-to-effect statements in the conclusion are said to follow by the same reasoning; the front-door proof is deferred. All conditions are sufficient, and Assumption 2.3 is untestable from observational data. Two worked effects were checked. | Active |
| THEORY-705 v2 | For a target and two sources with finite alphabets, if unique information is to depend only on the two source–target pair marginals, then the minimum of the conditional mutual information over all joints with those marginals is an upper bound on every non-negative decomposition’s unique information, and the shared and synergistic information it implies are lower bounds; this unique information vanishes exactly when the other source is Blackwell-sufficient, and the measures are non-negative, satisfy the bivariate partial-information axioms and the identity property, and do not extend to three sources on the partial-information lattice | Bertschinger et al. (2014), LIT-1162, read in full in NOTE-937. The bounds, the Blackwell characterisation and the properties are proved. The motivating property is argued, not derived. Example 30 and the Table 1 rows reproduce, except two entries that conflict with the paper’s own Proposition 18. Under this measure, the record’s transfer-entropy examples split exactly as under I_min. | Active |
| THEORY-706 | Any redundancy function defined on the lattice of antichains of sources, ordered so that a collection precedes another when every source of the latter contains one of the former, has a Möbius inverse that decomposes the information a set of sources gives about a target into partial-information atoms summing to every subset’s mutual information; for the I_min redundancy, the expected minimum specific information over sources, every atom is nonnegative, and interaction information is a signed mixture of atoms, synergy minus redundancy for three variables | Williams & Beer (2010), LIT-1172, read in full in NOTE-936. The lattice, the inversion and the nonnegativity proofs are exact and were followed; the lattice sizes and the 3-parity and three-copies cases were checked. What is not shown is that I_min measures shared information: it gives two independent copied bits 1 bit of redundancy (Harder, Salge and Polani). | Active |
| THEORY-707 | Transfer entropy is not a measure of information flow, if flow means information in the target attributable solely to the source’s past: as a conditional mutual information it equals the source’s unique information plus its synergy with the target’s past, so it can credit a full bit to a source that carries none alone (Y_t = X_{t−1} ⊕ Y_{t−1}) and report zero from each of two sources that jointly fix the target (Z_t = X_{t−1} ⊕ Y_{t−1}); conditioning on further series, as causation entropy does, has the same fault | James, Barnett & Crutchfield (2016), LIT-1165, read in full in NOTE-945. Both examples are exact and were recomputed. The decomposition holds for any PID satisfying the consistency equations. The conclusion is relative to the authors’ definition of flow; transfer entropy as predictive improvement is untouched. The claims about directed information, interventional flow and Gaussian systems are cited or conjectured. | Active |
| THEORY-708 v2 | The I_min redundancy credits two independent sources with shared information whenever each reduces surprise about the same target outcomes, so it gives two independent bits copied into the target 1 bit of redundancy; a bivariate redundancy built from reverse information projections, each source’s information less what is lost when its posteriors are replaced by mixtures of the other’s, satisfies symmetry, self-redundancy, monotonicity and the identity property, gives a nonnegative decomposition, and agrees with I_min on XOR, AND and their compositions but can reverse the state-independent/state-dependent split of a transfer entropy | Harder, Salge & Polani (2013), LIT-1168, read in full in NOTE-944. The counterexample, the axioms and the nonnegativity proof are exact; every Table I value and both transfer-entropy examples were recomputed. The identity axiom and “mechanistic redundancy” are positions, not results, and the measure is bivariate only. | Active |
| THEORY-709 | Mutual information between continuous variables can be estimated from k-nearest-neighbour statistics, with the joint k-th-neighbour distance in the maximum norm fixing the scale at which marginal neighbours are counted, so that the biases of the three entropy estimates largely cancel; the one approximation is a locally constant density, the systematic error scales with k/N, and exactness for independent variables is observed in every tested case but not proved | Kraskov, Stögbauer & Grassberger (2004), LIT-1163, read in NOTE-938 (formal sections in full, numerics at lower resolution). The estimators are derived up to one approximation and one neglected correction; the rest is numerical. Checked by simulation on correlated Gaussians. | Active |
| THEORY-710 | In a finite causal Bayesian network, a set of nodes is causally independent of another, imposing a third, for every choice of mechanisms exactly when every directed path between them passes through the third; information flow, conditional mutual information computed from interventional instead of observational distributions, vanishes under causal independence, and it can differ from transfer entropy and conditional mutual information in either direction | Ay & Polani (2008), LIT-1174, read in full in NOTE-946 from the 2006 preprint. Theorem 2 and Prop. 3 are proved; Examples 3–6 were recomputed. Example 6’s one bit holds at k = 1 and fails for k ≥ 2 as stated. The measure needs the mechanisms, and estimation and identification are not addressed. | Active |
| THEORY-711 | Directed information from an input sequence to an output sequence, the sum over time of the information the inputs up to now carry about the current output given the past outputs, is at most their mutual information and equal to it when the channel is used without feedback; in a causal system it bounds what the output reveals about the source, so feedback cannot raise a memoryless channel’s capacity; and the usual definition of a memoryless channel holds only without feedback | Massey (1990), LIT-1166, read in full in NOTE-939 from the author’s preprint. Three short theorems, all checked; the feedback inequality also checked numerically. “Causal” is Massey’s factorization condition, not intervention. That directed information is full-history transfer entropy plus instantaneous terms is the record’s derivation. | Active |
| THEORY-712 v2 | Under any partial information decomposition satisfying the consistency equations, a transfer entropy T_{Y→X} = I(X_{t+1}; Y_t | X_t) is the unique information of the source’s past about the target’s next state plus the synergy of source and target pasts, which Williams and Beer call state-independent and state-dependent transfer; time-delayed mutual information is the unique part plus redundancy; with the I_min redundancy the state-independent part alone suffices for perfect open-loop control, and a two-source transfer entropy splits into unique, redundant and synergistic transfer | Williams & Beer (2011), LIT-1176, an arXiv preprint, read in full in NOTE-942. The two-term split is an identity from the consistency equations; the sizes of the terms, the control theorem and the multivariate split use I_min, which the record has not read argued for. The one-parameter family and the three corners of the two-source family were recomputed. The text mislabels the synergistic corner. | Active |
| THEORY-713 | No redundancy measure on the Williams–Beer partial-information lattice can satisfy symmetry, self-redundancy, monotonicity, the identity axiom and non-negative atoms once there are three sources: for two independent bits, their XOR and a target that copies all three, identity forces three one-bit atoms under a two-bit total, so one atom is at most −1 bit, and a multivariate decomposition that keeps identity must give up non-negativity or the lattice | Rauh, Bertschinger, Olbrich & Jost (2014), LIT-1202, read in full in NOTE-965. A proved impossibility, checked by hand and by script. It also uses invariance of redundancy under relabelling the target, which is not stated as an axiom. I_min escapes it by failing identity, Ince’s I_ccs by accepting the −1-bit atom. | Active |
| THEORY-714 v3 | Ince’s I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target’s surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.’s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality | Ince (2017), LIT-1201, read in NOTE-968. The definition is exact, and three of the four properties the paper claims hold; subset equality does not (re-derived 2026-10-20), and every discrete example reproduces. That the dropped properties ought to be dropped, and that unique information depends on the source–source marginal, are argued, not proved. Whether it is the right redundancy measure is open (QUESTION-223). It meets Rauh et al.’s −1-bit bound exactly on XORCOPY. | Active |
| THEORY-715 | The information about a target carried by the Gács–Körner common random variable of the sources is a redundancy defined for any number of sources that is nonnegative, target-monotone and no larger than I_min, and is a lower bound on any reasonable redundancy; but it counts only deterministic shared structure, so it is zero whenever the sources’ joint distribution has full support, fails the identity property and continuity, and gives negative synergy already for two sources | Griffith, Chong, James, Ellison & Crutchfield (2014), LIT-1203, read in full in NOTE-967. The lemmas are short and exact, and every example was recomputed, with Harder, Salge and Polani’s copy family and a three-source copy added. It is the record’s one redundancy defined for n > 2, and it does not have the identity property there or for n = 2. | Active |
| THEORY-716 | Synergy can be measured as the whole minus the union of its parts, the union information being the least information the sources could jointly carry about the target with every source–target marginal held fixed; this synergy lies between WholeMinusSum and Williams and Beer’s S_max, which respectively subtract redundancy and count unique information as synergy, and for two sources it is Bertschinger et al.’s synergy, so independent sources can share information about a target, as the authors’ last version concedes against their first | Griffith & Koch (2014), LIT-1204, read in full in NOTE-966, v6 with v1–v2 checked. The definitions, the criticisms of S_max, WMS and ΔI, and the properties proved in Appendix C are exact, and every Table 1 value for two predictors was recomputed. The identification with Bertschinger et al.’s CI for two sources is the record’s derivation. Monotonicity is claimed without proof, and the S_max bound’s proof has a step the reading could not reconstruct. | Active |
| THEORY-717 | For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix | Blackwell (1953), LIT-781, Theorems 4–6, read in NOTE-595: the Sherman– Stein theorem, with a new minimax proof. With n states and finite outcome sets, write experiments as n × N Markov matrices P and Q. Every risk vector Q attains in every decision problem is attainable with P exactly when Q = PM for some Markov matrix M. The arbitrary-outcome extension is THEORY-156, still Proposed. A qualitative order: most pairs are incomparable, and it says nothing about how far apart they are. | Active |
| THEORY-718 | In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n | MacKay (1992), LIT-623, Eqs. 5–6 with Eqs. 16 and 19–20, read in NOTE-471: for y = Φw + noise with Gaussian noise of precision β and the prior w ~ N(0, α⁻¹I) on k weights, the posterior is exactly Gaussian, so the evidence is exactly P(D|w_MP) times the Occam factor P(w_MP)(2π)^{k/2} det^{-1/2}A, with A = αI + βΦᵀΦ. If ΦᵀΦ/n converges to a positive-definite matrix and w_MP stays bounded, the log Occam factor is −(k/2) log n + O(1). Exact for this model class only. The general regular case and the singular case are THEORY-087, which is Proposed. | Active |
| THEORY-719 | On a finite augmentation space, over unconstrained functions of the two views, the spectral contrastive and NT-Logistic losses are both minimized at the positive-pair density ratio | Johnson, El Hanchi & Maddison (2023), LIT-258, App. B.2–B.3, read in NOTE-751: on a finite augmentation space, with every function of the two views allowed, the spectral contrastive loss is minimized by K⁺ = p⁺/(p⊗p) exactly, and the NT-Logistic loss with equal positive and negative classes by the log-odds log K⁺, so its exponentiated score is K⁺. Population optima over unconstrained functions only; not what an encoder reaches, and not InfoNCE, which is THEORY-001’s open part. | Active |
| THEORY-720 | Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned | Karvonen (2018), LIT-846, Theorems 4.1, 4.3 and 4.5, read in NOTE-651; Abramsky, Barbosa, Karvonen & Mansfield (2019), LIT-847, Theorems 21 and 22, read in NOTE-647. A classical simulation answers each measurement of the target by a jointly measurable set of the source’s, with shared classical randomness, so the cover may change. Non-contextual models are exactly those the empty model simulates, NCF cannot fall along a simulation, strong contextuality is reflected backwards, and a model clones itself only if it is non-contextual. No-signalling models on finite scenarios only. Which maps are simulations is THEORY-174’s open half. | Active |
| THEORY-721 | On a complex Hilbert space no orthonormal basis or tensor factorisation of given dimensions is distinguished by the space alone, and orthonormal bases correspond exactly to commutative special †-Frobenius algebras; over the reals that correspondence fails | Three results THEORY-017 rested on, held apart from its synthesis. (a) The unitary group acts transitively on the orthonormal bases of a finite-dimensional Hilbert space, and on its tensor factorisations of given dimensions, so neither is fixed by the space; a non-degenerate self-adjoint operator fixes a basis up to phases, and its invariant content is its spectrum. (b) Over ℂ, commutative special †-Frobenius algebras correspond exactly to orthonormal bases: Coecke, Pavlovic & Vicary, LIT-274, Thm 5.1 and §6, in finite dimension; Abramsky & Heunen, LIT-275, Thm 22, for the non-unital ones satisfying the H*-axiom, in any dimension. (c) Over ℝ it fails: ℂ as a real algebra, with multiplication scaled by 1/√2, is a commutative special †-Frobenius algebra with no nonzero copyable vector. | Active |
| THEORY-722 | In Contextuality-by-Default 2.0, whether a system of non-binary variables is contextual depends on which variables represent the measurements: coarse-graining can turn a noncontextual system into a contextual one, while for connections of two variables, and for consistently connected systems, CbD 1.0 and 2.0 agree | Dzhafarov & Kujala (2017), LIT-831, Examples 1–3 and Section 4, read in NOTE-643 (claims C4 and C6). Under multimaximal couplings, lumping values of non-binary variables can destroy every multimaximal coupling, so a noncontextual system becomes contextual. The verdict on a fixed set of measurements therefore depends on which variables represent them. For connections of two variables, hence cyclic systems, and for consistently connected systems, the 1.0 and 2.0 definitions agree. It does not say the coupling choice changes verdicts in larger connections, which is THEORY-168’s asserted part. | Active |
| THEORY-723 | For the final embedding and unembedding of a softmax model, the output distribution fixes the representations only up to an invertible linear map, and whether closeness in distribution bounds representational dissimilarity depends on the distance: not for KL in the saturation regime, but yes for the log-likelihood variance distance | Nielsen et al. (2025), LIT-253, read in full in NOTE-738, with Harvey, Larsen & Williams (2023), LIT-250, for the kernel side. What is observed fixes a representation only up to that observation’s symmetry group: a centred linear kernel up to a translation and an orthogonal map (THEORY-004), and a softmax output distribution up to an invertible linear map of the embedding with the inverse transpose on the unembedding (Thm 2.2). KL can go to zero while the embeddings stay non-linearly related (Thm 3.1, Cor. 3.2), but d_LLV ≤ ε gives d_f,g ≤ 2Mε (Thm 4.7). Final layer only. It does not say what the Platonic hypothesis means; that is THEORY-002. | Active |
| THEORY-724 | For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching | Farrell & Ioannou (1996), Part II, LIT-929, eq. 12 and §4, read in NOTE-842. Because d‖u‖²/dt = u†(A + A†)u, the first Lyapunov exponent lies between the long-run averages of the extreme eigenvalues of the Hermitian part. For a normal operator the largest is the spectral abscissa, so an operator that is normal and uniformly stable at every instant has a negative exponent. A switched undamped oscillator, neutral at every instant, has a positive exponent. The stochastic genericity results are THEORY-325’s, still Proposed. | Active |
| THEORY-725 | On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter | List & Pettit (2006), LIT-705, read in full in NOTE-550, for the court agenda only. Proposition 1: majority voting on each proposition is not robustly rational, by the discursive dilemma. Proposition 4: the premise-based procedure, majorities on P, Q and the biconditional with R derived, is robustly rational and makes the group’s set a function of the members’ sets. Under it the members’ judgments on R do not fix the group’s (weak discontinuity), and under the distributed procedure they need not enter (strong discontinuity, Proposition 5). The general impossibility for proposition-wise rules is THEORY-127, still Proposed. | Active |
| THEORY-726 | Weight decay on factors is a nuclear-norm penalty on their product: with tied weights λ‖W‖²_F = λ‖WWᵀ‖*, and for untied factors the least (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is ‖M‖*, so for any loss that sees the weights only through the product, square-penalized training is nuclear-norm-penalized training of a product of bounded rank | Boncoraglio et al. (2025), LIT-857, Appendix A, read in NOTE-661, and re-derived here. ‖W‖²_F = Tr(WWᵀ) = ‖WWᵀ‖*, since a PSD matrix’s singular values are its eigenvalues. For untied factors, Hölder’s inequality for Schatten norms and AM–GM bound ‖M‖* by (‖U‖²_F + ‖V‖²_F)/2 over every factorization UVᵀ = M, and the balanced SVD factorization attains it. The product’s rank is bounded by the factors’ inner dimension. It says nothing about what the trained product looks like, which is THEORY-180’s spectral law. | Active |
| THEORY-727 | For a fixed linear operator in an inner-product norm, growth over time t is the top singular value of the propagator, which the eigenvalues bound only from below; it starts at the numerical abscissa, which can be positive when every mode decays; and the forced response is bounded by the condition number of the eigenvectors over the distance to the spectrum | Farrell & Ioannou (1996), LIT-931, read in NOTE-831, with Trefethen et al. (1993), LIT-932, read in NOTE-839. For du/dt = Au with A fixed: a normal A grows as its spectral abscissa; in general the growth over t is σ_max(e^(At)), with the eigenvalues as a floor; the initial growth rate is the top eigenvalue of (A + A†)/2; and harmonic forcing is bounded by κ(E)/dist(iω, Λ(A)), with the ε-pseudospectrum marking where eigenvalues of A + Δ, ‖Δ‖ < ε, can go. All in a chosen norm. The white-noise variance result is THEORY-323’s, still Proposed. | Active |
| THEORY-728 v2 | On a finite data set, a symmetric positive affinity normalized to a Markov operator is conjugate to a symmetric one, and its eigenvectors scaled by powers of their eigenvalues embed the data so that Euclidean distance equals the diffusion distance weighted by the inverse stationary law | Coifman et al. (2005), LIT-981, read in full in NOTE-1441, with the derivation written out here; the 1/π-weighted distance is the companion’s (LIT-1246), not the paper’s own. For n points and a symmetric affinity K with positive entries, A = D⁻¹K is a Markov matrix with stationary law π ∝ d. It is conjugate to the symmetric D^(-1/2) K D^(-1/2), so its eigenvalues are real and in [−1, 1], and in [0, 1] when K is positive semi-definite. The map x ↦ (λ_j^m ψ_j(x))_{j≥1} turns the m-step diffusion distance, weighted by 1/π, into Euclidean distance exactly. Truncation gives a lower bound. It says nothing about the continuum limit, which is THEORY-565. | Active |
| THEORY-729 | An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms | Corfield (2017), LIT-115, §3, read in full in NOTE-141, with the proof the paper omits written out here. For F : C → D^op left adjoint to G : D^op → C, let C′ and D′ be the full subcategories on which the unit and the counit are isomorphisms. Then F and G restrict to inverse equivalences C′ ≃ D′, and D′ is a full subcategory of D^op, so C′ is dually equivalent to a full subcategory of D. Pontrjagin duality is the example. That most dualities are of this form is THEORY-604’s thesis, still Proposed. | Active |
| THEORY-730 | For an embedding whose Gram matrix is the absolute co-occurrence deviation matrix at full rank, words on a periodic lattice whose positive and negative co-occurrence parts each depend only on separation have Fourier modes of the lattice as principal components, in degenerate cos/sin pairs with amplitude the square root of the kernel’s Fourier transform, and the exponential kernel gives a closed loop with integer frequencies on a periodic lattice and an open curve with non-integer frequencies on an interval | Karkada, Korchinski, Nava, Wyart & Bahri (2026), LIT-860, Propositions 1, 3 and 4 and Corollary 2, with the proofs in Appendix E followed in NOTE-664. The result is conditional throughout. It concerns an idealised embedding with W Wᵀ = |M*| and d ≥ rank M*, not a trained one. It needs the positive and negative parts of M* to be translation invariant separately (Assumption B.1), not M* alone. It says nothing about whether any corpus or model satisfies these conditions, which is THEORY-183. | Active |
| THEORY-731 | For softmax cross-entropy with balanced classes, the Gauss–Newton matrix splits exactly into four positive semidefinite second-moment terms, the class-centre term has rank at most C with eigenvectors in the span of the class centres and the spectrum of a C × C Gram matrix, and so the k-th eigenvalue of the whole is never below the k-th eigenvalue of the class-centre term | Papyan (2019), LIT-619, read in NOTE-479. Exact algebra, re-derived here. For softmax cross-entropy, the Gauss–Newton matrix is G = (1/n)ΔΔᵀ, an uncentred second moment of logit derivatives indexed by class and logit. Grouping them gives G = G₀ + G₁ + G₂ + G₃, with every term positive semidefinite. G₁, the class-centre term, has rank at most C, and its spectrum is read from a C × C Gram matrix. So λ_k(G) ≥ λ_k(G₁ + G₂) ≥ λ_k(G₁) for every k, in any network, trained or not. That G₁ carries the outliers of trained networks is measured, not derived, and stays Proposed in THEORY-078. | Active |
| THEORY-732 | For ReLU networks interpolated linearly between two endpoints, weak additivity and commutativity at every layer make each layer’s features exactly the interpolation of the endpoints’ features, output-layer linearity bounds the path’s error by twice the endpoints’, and for one hidden layer with interpolating minima linear mode connectivity alone gives the exact case | Zhou et al. (2023), LIT-674, read in NOTE-512 with Appendix A’s proofs checked: for an L-layer ReLU MLP on a finite dataset, weak additivity of ReLU over the two networks’ pre-activations and the commutativity W_A H_B + W_B H_A = W_A H_A + W_B H_B at every layer make the interpolated network’s features at every layer exactly the interpolation of the endpoints’ features (Theorem 1). Output-layer linearity, at any positive scale, keeps the path’s error within twice the endpoints’ (Lemma 1). Lubana et al. (2022), LIT-669, Lemma 2: for one hidden layer and interpolating minima, linear mode connectivity forces shared activation patterns, and with shared patterns the hidden features are linear along the path. It says nothing about when the conditions hold. That trained pairs satisfy them is THEORY-111’s empirical claim. | Active |
| THEORY-733 | For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale | Barnett and Seth (2023), LIT-979, read in NOTE-791: the VAR form (3.26) of exact dynamical independence, with the genericity count of §3.3 made a proof for VAR order p ≥ 2 and corrected for p = 1. Linear coarse-grainings Y = LX of a stable VAR with non-degenerate Gaussian innovations only. It says nothing about near-independence, about the multimodality of the dynamical-dependence cost, or about nonlinear maps. | Active |
| THEORY-734 | In the hyperbolic plane, a root carrying deg_max ≥ 15 chains of length m, embedded with worst-case distortion 1 + ε for ε ≤ 1, forces some node to hyperbolic distance at least (ℓ/(4ε))·ln(deg_max/(π√2)) from the origin, so fixed-point Poincaré coordinates need Ω((ℓ/ε) log deg_max) bits | Sala, De Sa, Gu and Ré (2018), LIT-874, Lemma D.1, read in NOTE-679, with the proof redone here without the paper’s small-angle approximation. Only the root’s children are used to fix the embedding’s scale c, and then the chains turn the scale into depth. Plane only, one family of trees, fixed-point coordinates in the Poincaré disk, deg_max ≥ 15 and ε ≤ 1. It says nothing about H^r for r > 2, or about whether any tree needs fewer bits under another number system. | Active |
| THEORY-735 | Under a local quadratic model of the loss with the greedy optimal step, per-step progress at batch size B is ΔL_max/(1 + B_noise/B), and at fixed B the steps and examples needed satisfy (S/S_min − 1)(E/E_min − 1) = 1, with E_min/S_min the progress-weighted mean of B_noise | McCandlish, Kaplan, Amodei et al. (2018), LIT-348, §2 and App. D.1, as read and re-derived in NOTE-293. A derivation under stated assumptions: a second-order model of the loss along the step, an unbiased minibatch gradient of covariance Σ/B, and the step size that maximises the expected decrease. It says nothing about whether real training meets those assumptions, or how well the noise scale predicts the measured critical batch, which is THEORY-033. | Active |
| THEORY-736 | Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst | Kuditipudi et al. (2019), LIT-671, Theorem 1, read in full in NOTE-544, with Draxler et al.’s (2018) concatenation bound from LIT-653. Dropout stability gives every solution a half-width copy with the same loss up to ε. Half the units are then free, so working units can be swapped in and permuted at no cost. That joins any two such solutions. It is a statement about dropout-stable solutions only: whether training produces them is THEORY-109’s open question, and Theorem 4 of the same paper builds disconnected global minima. | Active |
| THEORY-737 | In a spectral contrastive graph that joins labelled known classes to unlabelled data, the least-squares residual of a novel labelling on the top-k eigenspace, with the labelled entries left free, equals exactly the part of the labelling outside that eigenspace that the labelled rows of the discarded eigenvectors cannot absorb | Sun, Shi, Liang & Li (2023), LIT-235, Theorem 5.5 with its proof in C.1, read in NOTE-833. The paper states the identity as ≤, and its proof is a chain of equalities. Re-derived here in three steps from orthogonality and least squares. With the paper’s Lemma 5.1 it bounds the novel labelling’s 0–1 linear-probe error by twice this residual. It is about the exact rank-k factorization of a population graph, not about trained networks. | Active |
| THEORY-738 | In the unconstrained-features max-margin problem with k classes under step imbalance and feature dimension at least k − 1, the unique optimal logits are the simplex-encoded labels, every margin equals one, and the classifier and embedding Gram matrices are the two square-root factors of that matrix’s singular value decomposition; the geometry is the simplex equiangular tight frame exactly when the classes are balanced or there are two of them | Thrampoulidis, Kini, Vakilian and Behnia (2022), LIT-994, Theorem 1 and Corollary 1.3, read in NOTE-764 and re-derived here. By THEORY-726 the problem is nuclear-norm minimisation of the logits under margin constraints. The polar factor of the simplex-encoded label matrix certifies it optimal, because under step imbalance its off-diagonal entries are negative. It does not say that gradient descent or trained networks reach this geometry, which is THEORY-286’s. | Active |
| THEORY-739 | For the quartic proxy of word2vec with tied weights, symmetric reweighting and constant weights, the rank-d minimisers are the top d eigenvectors of the relative deviation-from-independence matrix M*, scaled by the square roots of their eigenvalues, up to rotation; and from an initialisation aligned with those eigenvectors, gradient flow acquires each one in an independent logistic rise, in order of eigenvalue | Karkada, Simon, Bahri and DeWeese (2025), LIT-855, Theorem 1 and Proposition 2, read in NOTE-660 and re-derived here. The quartic proxy is a function of WWᵀ alone, so von Neumann’s trace inequality gives its rank-d minimisers. Gradient flow from an aligned start keeps the eigenbasis and reduces each mode to a logistic equation. It does not say that word2vec, or the proxy from random initialisation, behaves this way; that is THEORY-182’s. | Active |
| THEORY-740 | A two-layer linear network trained by gradient flow on white inputs from a decoupled, balanced start learns each singular mode of its input–output correlation in a logistic rise that waits a time of order its inverse strength times the log of the inverse initial scale, so the modes arrive one at a time in order of strength; a single-layer network on the same data learns every mode on one timescale with no plateau, so under these conditions the stages come from the product of layers and the data’s spectrum | Saxe, McClelland and Ganguli (2014, 2019), LIT-864 and LIT-862, read in NOTE-667 and NOTE-666, re-derived here. From a decoupled, balanced start the two-layer equations keep the SVD basis of Σyx, and each mode’s strength obeys τȧ = 2a(s − a), a logistic rise. One weight matrix on the same data relaxes every mode exponentially on the one timescale τ. It does not say that random starts follow the solution, which is THEORY-186’s. | Active |
| THEORY-741 | A two-layer ReLU student of width at least kM, fitting an orthonormal teacher of M neurons on the sphere, has for any k global minima a global minimum joined to each by a two-piece linear path of minima; and in a deep linear network of depth L and width above 1 + r(L − 1), any r global minima in general position share a global minimum linearly connected to each | Lin, Li and Wu (2024), LIT-682, Theorems 6, 11 and 18 with Lemmas 7 and 25, read in NOTE-535 and re-derived by the promotion audit. Two toy models whose minima can be written down: a two-layer ReLU student of an orthonormal teacher on the sphere at population loss, and a deep linear network fitting y = Qx. Spare width lets every minimum be moved, along segments of minima, to one a common centre can reach. No permutation symmetry is used, and nothing is claimed about trained networks. | Active |
| THEORY-742 v2 | For a Boltzmann distribution linear in its couplings, observed through any coarse-graining map, the Fisher information the coarse data lose about the bare couplings is the expected conditional covariance of the energy operators given the coarse state; when the renormalized Hamiltonian stays in the family, the parameter map is linear and the additive constant of the step is analytic, that loss is the constant’s Hessian, so the singular part of the metric passes unchanged to the coarse data. In the hopping model of diffusion at its Gaussian fixed point, keeping every n-th position of a trajectory of fixed duration multiplies the metric by n^(1−μ) in the μ-th Hermite coordinate: the drift’s information is kept exactly, the variance’s falls as 1/n, and higher shape parameters’ faster | Raju, Machta and Sethna (2018), LIT-1254, read in NOTE-972. The loss identity holds for any coarse-graining of an exponential family; that the loss is analytic, and so spares the singular part of the metric, needs an exact renormalization step with linear beta functions and an analytic constant. It does not say that relevant directions keep their information in general: on the paper’s own scalings only directions with RG eigenvalue above d/2 do, and away from the Gaussian fixed point even the drift loses information. | Active |
| THEORY-743 | For finite X and Y, every information-bottleneck optimum at every β gives one encoder to all inputs sharing p(y|x), so it is a stochastic coarsening of the minimal sufficient partition (the causal states of memoryless transduction); as β → ∞ the optima converge to that partition, but in general no finite β attains it, except when Y is a function of X | Shalizi and Crutchfield (LIT-1240, read in NOTE-976) assert that the information bottleneck recovers the causal states as β → ∞. The record proves it, against stochastic encoders as well as partitions, and adds that every finite-β optimum factors through those states. So the IB hierarchy always nests under its top. It says nothing about whether optima at two finite β nest. Finite alphabets only. | Active |
| THEORY-744 v2 | As the kernel width ε goes to 0, Coifman and Lafon’s α-normalized diffusion operator (I − P_{ε,α})/ε, for data sampled with density q = e^{−U} from a compact manifold without boundary, converges uniformly on smooth functions, at rate O(ε), to Δf + 2(1 − α)⟨∇U, ∇f⟩ with Δ the non-negative Laplace–Beltrami operator, which is minus the generator of a reversible diffusion with drift −2(1 − α)∇U and invariant density proportional to q^{2(1−α)}; α = 0 gives potential 2U and invariant density q², α = 1/2 gives the Langevin dynamics whose invariant density is the sampling density itself, and α = 1 gives Brownian motion on the manifold, with no trace of the density | Coifman and Lafon (2006), LIT-1246, read in NOTE-985, with the PNAS announcement LIT-981. The paper states the limit as a Schrödinger operator. Written as a drift, it shows what each α keeps of the density: the walk’s stationary density is q^{2(1−α)}. On a manifold without boundary the limit is derived in full, the kernel expansion included (re-derived on promotion, 2026-10-20). The paper’s own α = 1/2 display has a sign error. Its boundary rate and heat-kernel proof are incomplete and are left to QUESTION-231. | Active |
| THEORY-745 | For the reversible walk p = k/d built from a symmetric non-negative kernel on a finite connected set, the t-step diffusion distance is exactly the Euclidean distance between the coordinates λ_l^t ψ_l, and keeping only the coordinates with |λ_l|^t > δ|λ₁|^t errs in D_t² by at most δ²|λ₁|^{2t}(1/π(x) + 1/π(y)), an absolute bound and not the relative accuracy δ that Coifman and Lafon state; the truncated maps are nested as t grows, so with δ fixed the steps t order a refinement chain by eigenvalue modulus, while untruncated they are all invertible images of one another | Coifman and Lafon (2006), LIT-1246, read in full in NOTE-985. The identity D_t² = Σ λ_l^{2t}(ψ_l(x) − ψ_l(y))² is exact. The paper’s Proposition 1 says the truncated map keeps it “up to relative accuracy δ”. The true bound is absolute and can be large relative to D_t for nearby points. In the refinement sense, the diffusion-map family coarsens with t only through the threshold. Finite sets, counting measure and reversible walks only. It says nothing about the manifold limit or about non-reversible chains. | Active |
| THEORY-746 | Over a fixed finite set of basis functions, slow feature analysis is a generalized eigenproblem: the outputs that minimize ⟨ẏ²⟩ under zero mean, unit variance and decorrelation are the bottom generalized eigenvectors of the derivative covariance against the covariance, in order of slowness, and a stack of such stages, each of whose function spaces contains its inputs, can only keep or lower each of its first m slowness values on the training signal, and may keep them | Wiskott and Sejnowski (2002), LIT-1249, read in full in NOTE-971. Within the span of a fixed basis, the slowest decorrelated unit-variance outputs are the bottom generalized eigenvectors of the pencil (⟨ż̃ż̃ᵀ⟩, cov z̃), and each output’s slowness is its eigenvalue. A consequence the paper does not draw: a stack of such stages whose function spaces contain their inputs is never faster at a higher stage, componentwise, on the training signal, so in this model “higher is slower” is imposed by the objective, weakly, and can stall. It is about the training signal and a fixed function space. It says nothing about test data, about a generative model, about invariance as opposed to slowness, or about learning dynamics. | Active |
| THEORY-748 | Block entropies, and so every lag-τ entropy rate of a stationary signal, are unchanged by the reflection x → −x and by time reversal, so they cannot detect the odd-order asymmetry the 4/5 law ties to energy flux; Granero-Belinchón, Roux and Garnier’s entropy rate conditioned on the sign of the increment breaks the reflection invariance, but h⁺ ≠ h⁻ shows only that the two-point distribution lacks central symmetry, which a skewed one-point distribution produces with exactly symmetric increments, so the test does not by itself detect skewed increments or an energy cascade | Granero-Belinchón, Roux and Garnier (2016), LIT-1243, read in NOTE-970, claim that conditioning the entropy rate on the sign of the increment lets one “prove the existence of the energy cascade” using information theory. The record finds two limits. The unconditioned entropy rate is exactly blind to reflection and time reversal. The conditioned one detects any central asymmetry of the two-point law, including one with no increment skewness at all. Independent pairs with the paper’s one-point skewness of 0.175 give h⁺ − h⁻ ≈ 0.066 nats with symmetric increments. Nothing in the paper ties an information quantity to the energy balance. | Active |
| THEORY-749 v3 | A weak solution of the incompressible Euler equations on the periodic box that lies in L³(0,T; B^α_{3,∞}) ∩ C([0,T]; L²) with α > 1/3 conserves kinetic energy, because the energy flux into scales below a mollification scale ε is bounded by Cε^(3α−1)∫‖u‖³ and so vanishes as ε → 0; this does not show that rougher solutions dissipate | Constantin, E and Titi (1994), LIT-1248, read in NOTE-982. The positive half of Onsager’s conjecture, in the Besov form: if velocity increments obey ‖u(·+y) − u‖_{L³} ≤ C|y|^α with α > 1/3, integrated in time, the coarse-grained energy flux across scale ε is O(ε^(3α−1)) and no energy can be lost to vanishing scales. It is a statement about regular fields only. It does not show that a flux survives the limit for rougher ones: that energy can fail to be conserved below 1/3 is Isett’s later theorem (THEORY-763), that it can strictly decrease is Buckmaster, De Lellis, Székelyhidi and Vicol’s (THEORY-776), both Proposed, and some rough solutions conserve energy anyway. | Active |
| THEORY-750 | For a stationary Gaussian signal the lag-τ entropy rate with one-point embedding is h_τ = H(X) + ½ log(1 − c(τ)²), so where the normalized second-order structure function 1 − c(τ) is small and scales as τ^ζ₂, h_τ = (ζ₂/2) log τ + constant; the “scaling of information” that Granero-Belinchón, Roux and Garnier measure in near-Gaussian wind-tunnel turbulence, slope 1/3 in the inertial range and 1 in the dissipative range, is the scaling of the second-order structure function written in nats | Granero-Belinchón, Roux and Garnier (2016), LIT-1243, read in full in NOTE-970. A Gaussian identity and its small-lag asymptotics: the conditional entropy of x(t) given x(t − τ) is a monotone function of the autocorrelation, so its scaling exponent is half the structure-function exponent. In the Modane data the estimated entropy rate matches the Gaussian formula, so the measured “information spectrum” is S₂ again. It is not a distribution of a conserved quantity over scales, and it carries nothing about the direction or size of energy transfer. It answers QUESTION-219 only in the narrow sense of supporting A3’s caution. | Active |
| THEORY-751 v2 | For a memoryless source with finite alphabets and a per-letter distortion, a description at rate R(D₁) can be refined to distortion D₂ < D₁ at total rate R(D₂) iff some pair of rate–distortion-optimal test channels forms a Markov chain X → X̂₂ → X̂₁; the condition holds for every discrete source under Hamming distortion and fails for some three-letter sources under absolute error | Equitz and Cover (1991), LIT-1238, read in NOTE-988, with every step re-derived by 2026-10-20. Lossless two-stage description is possible exactly when the coarse optimum is a stochastic degradation of the fine optimum. That is a Markov chain on test channels, not nesting of partitions, and it is an asymptotic block-coding statement. It does not say that optimal single-shot quantizers or deterministic encoders nest, and the paper’s own Gaussian example shows they need not. It says nothing yet about the information bottleneck beyond a formal transfer through the KL distortion. | Active |
| THEORY-752 | In optimal prediction of a Hamiltonian system with separable energy and canonical initial law, projected by conditional expectation, the t-model’s leading-order memory term tPLQLx makes the effective Hamiltonian of the resolved variables non-increasing, so the eliminated variables’ memory enters as a damping that grows linearly in time | Chorin, Hald and Kupferman (2002), LIT-1253, read in NOTE-975: Theorem 8.1. Truncating the Mori–Zwanzig memory integral at its lag-zero value gives dΦ/dt = R̄(Φ) + tS(Φ), and under the stated conditions this dissipates the free energy of the resolved variables. It says nothing about how well the t-model predicts, gives no error bound, and holds only for the conditional-expectation projection; the true conditional mean’s effective energy need not decrease. | Active |
| THEORY-753 | For finite X and Y, the information bottleneck is successively refinable from budget λ₁ to λ₂ > λ₁ iff some pair of optimal encoders forms a Markov chain X → T₂ → T₁, that is, iff the coarse optimal encoder is a garbling of a fine one; when the fine optimum’s decoder points q(X|t₂) are affinely independent, this holds for that pair iff every decoder point q(X|t₁) of the coarse optimum lies in their convex hull, so a failed hull test on optima that are unique up to merging and relabelling proves that refinement fails | Charvin, Catenacci Volpi and Polani (2023), LIT-1258, read in NOTE-981. It turns QUESTION-220’s Markov formulation into a finite check: nesting between two IB optima is a convex-hull inclusion of their decoder points, which a linear programme decides. The check is on given optima, and refinability asks for some pair. So a failed check settles the question only when the optima are certified globally optimal and unique. This entry adds that the information bottleneck’s own convex-envelope dual can supply that certificate. Finite alphabets, two stages. | Active |
| THEORY-754 v2 | If a velocity field’s third-order increments obey ‖δu(r)‖₃ ≤ C u_rms (r/L)^σ at every separation r, with 0 < σ < 1, then for a smooth, rapidly decaying filter the space-averaged energy flux across scale ℓ takes at most a fraction of order (ℓ/Δ)^(1−σ) from strain at scales above Δ and of order (δ/ℓ)^(2σ) from stress at scales below δ, measured against u_rms³/L·(ℓ/L)^(3σ−1); so transfer between scales is local in scale, with a power-law decay and no gap | Eyink and Aluie (2009), LIT-1239, read in NOTE-969, stating bounds proved in Eyink (2005), LIT-1252. A conditional inequality: if velocity increments scale no faster than r^σ in the L³ mean, the contributions of far larger and far smaller scales to the mean flux across ℓ are bounded by powers of the scale ratio. It needs 0 < σ < 1: σ < 1 for the large scales, σ > 0 for the small. That the inertial range satisfies the hypothesis, that the mean flux there is non-zero, and the sharper 4/3 exponent are assumptions or heuristics, not part of it. It is about one field decomposed by scale, and says nothing about units. | Active |
| THEORY-755 v3 | For a linear system x(t+1) = Ax and a partition of its variables, the vector of block sums evolves autonomously from every initial state iff the partition is column-equitable, every column of each block of A summing to the same value; the aggregate operator then keeps a subset of A’s eigenvalues, and the blocks may interact as strongly as they like, so exact aggregation is not decomposition | Shpak, Stadler, Wagner and Hermisson (2004), LIT-1242, read in NOTE-979. Exact aggregation of a linear system by the sums over a partition holds for every state iff the partition is column-equitable, a symmetry condition, not a weak-coupling one. Linear dynamics and partition-based aggregation only. It says nothing about approximate aggregation, the Simon–Ando case, or nonlinear systems, where the paper itself shows aggregations with no linear counterpart. | Active |
| THEORY-756 | The growth rate of thermodynamic depth is the entropy rate of whichever macrostate process it is computed over, so the choice of macrostates sets it anywhere from zero up; among partitions of a stationary process’s past that predict every finite stretch of the future as well as the whole past does, the causal states are the coarsest, so they have the least entropy and the least entropy rate | Crutchfield and Shalizi (1999; arXiv 1998), LIT-1260, read in NOTE-978. Depth is not a structure measure until its states are fixed. Predictive sufficiency, then minimality, fixes them as the causal states. The choice is canonical only given the measuring instrument and the target, the observed future. The redefined dive is still an entropy rate. | Active |
| THEORY-757 | The information bottleneck is successively refinable between every pair of budgets when Y is a function of X, by nested erasures of Y, and when X and Y are both binary, where the optimum at each nonzero budget is unique up to merging and relabelling and its two decoder points are the ends of an interval of the source simplex that only widens as the budget grows | Charvin, Catenacci Volpi and Polani (2023), LIT-1258, read in NOTE-981. These are the two finite cases in which nested information-bottleneck optima are guaranteed at every pair of budgets. Neither reaches the record’s four-state source, whose Y is not a function of X and whose X has three effective states. NOTE-981 finds that source not refinable. So binary Y alone does not guarantee nesting. The function case is the degenerate one, where the bottleneck is just a noisy copy of Y. | Active |
| THEORY-758 | For a self-adjoint diffusion operator T with spectrum in [0, 1], the diffusion-wavelet levels are spectral subspaces spanned by the eigenvectors with λ^s ≥ ε for an increasing sequence of powers s, so they form an exactly nested chain of T-invariant subspaces whose differences are bands of the spectrum, about one octave each, not spatial blocks; Coifman and Maggioni’s compressed algorithm, which squares the operator on each orthonormal basis, thresholds T^{2^{j−1}} at level j and not T^{2^j−1} as the paper states; and since each level is T-invariant, each is an exactly closed linear coarse-graining whose localized basis is a choice the level does not fix | Coifman and Maggioni (2006), LIT-1256, read in full in NOTE-984. The levels of a diffusion multiresolution are spectral bands. Nesting is exact and needs no decay of the spectrum, and decay buys only compression. The compressed algorithm as written keeps ⟨ξ : λ^{2^{j−1}} > ε⟩ at level j, and the paper’s own Fig. 9 follows this, not its stated count. Each level closes exactly under T, but a single scaling function does not. It says nothing about non-self-adjoint operators, about error bounds for the compressed scheme with finite precision, or about when a localized basis exists. | Active |
| THEORY-760 | The finite-memory Mori–Zwanzig closure that Parish and Duraisamy take from Stinis, truncated at order m with one trapezoidal sub-interval and constant memory lengths, is a finite-dimensional Markovian augmentation: m + 1 auxiliary variables per resolved mode, each relaxing at rate 2/τ; at first order it replaces the memory integral by the kernel 2e^{−2s/τ₀} acting on the first-order term, which tends to τ₀ times that term, the t-model with t replaced by τ₀, when τ₀ is short against the resolved dynamics; its kernel is twice the true one at zero lag for every τ₀, so it is not exact even where the true memory is a single exponential, and nothing in the construction bounds its error | Parish and Duraisamy (2017), LIT-1259, read in NOTE-973. The finite-memory closure turns the Mori–Zwanzig memory integral into auxiliary ordinary differential equations, so the reduced model is the resolved variables plus finitely many memory coordinates: the shape of augmentation QUESTION-222 asks about. But the truncation drops the next auxiliary term without a bound, the memory length is fitted, and the kernel’s form is fixed by the quadrature, not by the eliminated dynamics. It is an augmentation with uncontrolled error, not the controlled one the question asks for. | Active |
| THEORY-761 | A continuous piecewise-linear function on R^d with compact support has a Fourier transform that, along any fixed direction û, is bounded by C(û)k^(−2), and by C(û)k^(−d−1) whenever û is orthogonal to no edge of its linear regions, which is almost every direction; C(û) is not uniform and grows without bound near the exceptional directions; for a ReLU network without compact support the statement is not established | Rahaman et al. (2019), LIT-1261, read in full in NOTE-983. One integration by parts turns f̃ into a sum of polytope transforms weighted by the region gradients and divided by k. Each further step of Diaz et al.’s recursion gains another 1/k until the path reaches a face whose directions are all orthogonal to k. Generic directions reach the vertices, giving k^(−d−1), and every direction gets at least k^(−2). It is a bound on what a piecewise-linear function contains, not a statement about training. Unbounded ReLU networks fall outside it. | Active |
| THEORY-762 | For a reversible diffusion dX = −∇U dt + √2 dW in equilibrium, an eigenfunction ψ of its generator with eigenvalue −λ is closed in conditional mean with no memory: E[ψ(X_{t+s}) | the whole past] = e^{−λs}ψ(X_t), and so the same given ψ’s own past; but ψ(X_t) is a Markov process in its own right only when |∇ψ|² is a function of ψ, since otherwise its own path reveals a noise intensity that its present value does not fix | The record’s answer to what Nadler et al. (LIT-1250, read in NOTE-986) leave implicit when they call slow eigenfunctions reaction coordinates. Such a coordinate is exactly closed in its mean: its best prediction from the full state and from its own history coincide, with no memory term, and the Mori–Zwanzig memory and noise for it vanish under Mori’s projection. It is closed in law only under a further condition. In the paper’s examples the condition holds; the mixed Ornstein–Uhlenbeck eigenfunction x₁x₂ shows it can fail. This does not say that diffusion maps estimate the eigenfunctions well from finite data, or that a slow eigenfunction is the best coarse variable. | Active |
| THEORY-764 | For a deterministic supervenient feature V of a Markov system, Rosas et al.’s practical emergence criterion Ψ equals the parts’ whole-minus-sum information about V’s next value minus V’s one-step dynamical dependence I(V_t′; X_t | V_t), so dynamical independence raises Ψ but neither implies emergence nor is implied by it, and the paper’s sufficient test for causal decoupling can pass a feature that is not dynamically independent | Rosas et al. (2020), LIT-1244, read in NOTE-987, with Barnett and Seth’s dynamical independence (LIT-979, THEORY-257). By the chain rule, Ψ(V) = [I(X_t;V_t′) − Σ_j I(Xʲ_t;V_t′)] − I(V_t′;X_t|V_t). The subtracted term vanishing implies Barnett–Seth independence for Markov X. Examples show each condition without the other. The independent sum of AR(1) channels has both, with no interaction. The identity concerns Ψ, the paper’s practical criterion; it says nothing about the PID quantities themselves. | Active |
| THEORY-766 v2 | Brushing a seen rubber hand in synchrony with the hidden real hand, against asynchronous brushing, raises reports that touch is felt at the rubber hand and, more weakly and less reliably, that the rubber hand is one’s own; the proprioceptive drift toward the rubber hand that accompanies these reports is not a measure of them; the standard comparison conditions do not equalise participants’ expectations, so whether the stimulation or the expectation produces the report difference is open | Botvinick & Cohen (1998) and four replications read here give a synchrony effect on reports of referred touch and ownership. Rohde et al. (2011) and Abdulkarim & Ehrsson (2016) separate drift from ownership. The suggestion debate (Lush 2020; Lush et al. 2020; Ehrsson et al. 2022; Lush & Seth 2022; Roseboom & Lush 2022) shows that the standard controls do not equalise expectations. The ownership report is weaker than the referred-touch report and tracks trait suggestibility. Whether the synchronous stimulation, not expectation, produces the report difference is not settled. Promoted 2026-10-20 on the report differences measured (ADR-047 v3); the perceptual reading is THEORY-827. | Active |
| THEORY-770 | In a closed population, any mapping of ancestors to descendants gives the Price equation: mean fitness times the change in mean character equals the covariance of fitness and character plus the fitness-weighted mean of the change in character, over any time interval and with no assumption beyond closure | Luque (2016), LIT-205, §3, read in NOTE-101, after Price (1970); the derivation re-derived by the promotion audit. An identity, not a model: it says nothing about which partition of the change a causal model licenses, predicts nothing until fitness and transmission are specified, and fails without closure (migration). | Active |
| THEORY-771 v4 | In a singularly perturbed system, fast states x̃ with dx̃/dt = f̃(x̃, y) + ε·δf and slow states y with dy/dt = ε·ḡ(y) + ε·δg, if f̃ is contracting in x̃ at rate β_f with x̃ = 0 an equilibrium for every y, ḡ is contracting at rate β_g, and |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|, then for every ε below ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)] the fast states stay within O(ε) of zero and, when d_g = 0, the slow states within O(ε) of the reduced trajectory, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε) | Bousquet and Slotine (arXiv 2015), LIT-1292, read in NOTE-1025. It gives Simon’s near-decomposability the rate in ε that THEORY-661’s open statement lacks: first order, for trajectory error, uniform in time, under contraction of the modules and of the reduced slow system, with an explicit, sufficient ceiling on ε. Deterministic. It bounds how far the states are from the reduced description, not how much information the rest of the system sends into the slow variables, and it takes the slow coordinates as given. | Active |
| THEORY-773 v2 | In a singularly perturbed system ẋ = f(x, z, t), εż = g(x, z, ε), if g is contracting in z at rate λ_z for every x and ε, the reduced system ẋ = f(x, γ(x), t) on the slow manifold z = γ(x) is contracting at rate λ_x, f is α-Lipschitz in z, |∂γ/∂x · f| ≤ d and g is K-Lipschitz in ε, then for all t ≥ 0 the fast state is within χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z of the manifold and the slow state within χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})] of the reduced trajectory; partial contraction of f in x alone is not enough | Del Vecchio and Slotine (IEEE TAC 2013), LIT-1295, read in NOTE-1020, with their Theorem 3’s hypothesis corrected. A global, uniform-in-time, first-order bound on the error of the quasi-steady-state reduction, with every constant explicit and no requirement that ε be small. The reduction is Markovian: eliminating a fast, contracting variable costs O(ε) in trajectory error for all time, so no memory term is needed to that order. Constant metrics, deterministic, and the bound d on how fast the manifold moves is needed along the trajectory. | Active |
| THEORY-774 | In a one-dimensional lattice hopping model with 2N+1 free hopping probabilities, all particles starting at the origin and the density after t steps observed with fixed Gaussian error, the Fisher metric is exactly t² times the distribution of the difference of two independent (t−1)-step walks, evaluated at the lag between the two hops, so it does not depend on the drift; when the walk is aperiodic with nonzero variance, its k-th largest eigenvalue is of order t^(3/2−k) for k = 0, …, 2N, so successive eigenvalues separate by a factor of order t | Machta, Chachra, Transtrum and Sethna (2013), LIT-1299, supplement §III, read in NOTE-1018. The eigenvalues grow for particle creation (t^(3/2)) and drift (t^(1/2)) and fall from the diffusion constant on (t^(−1/2), t^(−3/2), …). The data are not a coarse-graining of one fixed data set, so nothing forbids growth. The metric is a least-squares metric with an arbitrary unit of error, and its eigenvalues depend on the coordinates θ_μ. The ratio between successive eigenvalues matches the 1/n per Hermite order of THEORY-742’s diffusion result, though the two set-ups differ. | Active |
| THEORY-775 | With a Gaussian random-walk prior and an instantaneous unit-variance Gaussian encoder, the KL half of the evidence lower bound is the unordered slowness loss plus a constant; with a linear Gaussian encoder and decoder on whitened input, the bound is maximized on the span of the slowest SFA directions whose lag-one autocorrelation exceeds one half, with the basis inside that span unidentified, and on unwhitened input the maximizer is neither SFA’s subspace nor invariant to rescaling the input | Schüler and Wiskott (2025), LIT-1296, read in full in NOTE-1015. The ELBO of a dynamical VAE with prior N(z_{t−1}, I) and encoder N(g(x_t), I) is Σ E log p(x_t | z_t) − ½Σ‖g(x_{t−1}) − g(x_t)‖² − (T − 1)d/2, an exact identity. For linear Gaussian encoder and decoder on whitened input, the maximizer spans the bottom eigenvectors of the difference covariance with eigenvalue μ < 1, each scaled by (μ^{−1/2} − 1)^{1/2} and then rotated arbitrarily. So it recovers SFA’s slow span, cut at lag-one autocorrelation ½, not SFA’s features. On raw input it trades variance against slowness. It says nothing about nonlinear encoders, about predictive sufficiency, or about Turner and Sahani’s model. | Active |
| THEORY-778 | In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m | Cagnetta and Wyart (2024), LIT-883, App. D–F, read in NOTE-687; re-derived to leading order by the promotion audit and checked against exact covariances of sampled grammars. A statement about the statistics of the generative model only: it says nothing about what a trained network learns, or when, and nothing about natural language. | Active |
| THEORY-779 | In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²)) | Kunin et al. (2021), LIT-1011, §§4–6 and App. G, read in NOTE-711; the stationary moments re-derived by the promotion audit. An exact result for a model: Σ ∝ H constant and a quadratic loss are assumptions, and nothing here says a deep network meets them. The probability current is nonzero but reversible: under the model’s assumptions the stationary process is time-reversible once velocities are flipped, so it is “broken detailed balance” only in the paper’s sense, a nonzero current. | Active |
| THEORY-818 v2 | In two-scale Lorenz 96 without time-scale separation (ε = 0.5, K = 18, J = 20, F = 10), a discrete-time closure for each resolved variable that fits the one-step discrepancy of a fourth-order Runge–Kutta step as a NARMAX series, with one autoregressive lag, at most two lags of the resolved variable and at most one moving-average term, reproduces the mean, spread, distribution, autocorrelation and neighbour cross-correlation of x₁ (Kolmogorov–Smirnov distance 0.005) and, with observations every 0.05, extends the 20-member ensemble forecast lead time at anomaly correlation 0.6 from about 2 to about 4 time units over a polynomial-plus-AR(1) closure; the model structure was chosen by trial and error, differs between the two observation spacings, and carries no error bound | Chorin and Lu (2015), LIT-1304, read in NOTE-1028. A data-driven closure with one or two steps of discrete memory, plus one moving-average innovation, is enough on this test to reproduce the resolved statistics and to double forecast skill against a memoryless-in-x closure. It is the record’s first evidence that a short discrete memory suffices in a chaotic system with no scale gap. It is one system, the structure was tuned, and there is no bound, so it does not give the controlled prediction error QUESTION-222 asks for. | Active |
| THEORY-819 | For an invertible map with an invariant measure, projecting onto the closed span of the whole past of chosen functions of the resolved variables (a Wiener projection) turns the exact discrete Mori–Zwanzig equation into one with no memory term: the next resolved value is a causal linear filter of those past functions plus a residual orthogonal to all of them, so the memory is carried by the coarse variable’s own history; a finite NARMAX form follows only by a rational approximation of the filter, uncontrolled in general but exact for a scalar coarse variable of a stable linear Gaussian VAR(1) | Lin and Lu (2021), LIT-1307, read in NOTE-1029. Mori–Zwanzig’s memory is what a projection onto the present leaves out; a projection onto the whole past absorbs it, leaving the causal Wiener filter on the coarse variable’s own history and a residual orthogonal to all of that history. Second-order (linear-span) projections, invertible dynamics and stationarity only. It gives no finite augmentation with a bounded error in general: the NARMAX step is the paper’s uncontrolled approximation, exact only in the linear Gaussian class shown here. | Active |
| THEORY-820 v2 | For a stationary discrete-valued process X and any coarse-graining Z of its past, informational closure at every horizon is the same condition as zero one-step transfer entropy from the micro past, Barnett and Seth’s dynamical independence, and as causal closure, Z’s ε-machine coinciding with its υ-machine; for a coarse-graining of the present state it also makes Z’s causal states a coarse-graining of X’s, which does not imply closure, and the closed coarse-grainings are not closed under common refinement | Rosas et al. (arXiv 2024), LIT-1306, read in NOTE-1030, with Barnett and Seth’s definition (LIT-979). For discrete stationary processes, the information-theoretic criterion of emergent closure (dynamical independence), the interventionist one (the macro’s own causal states are all the micro distinctions that matter for it) and, for maps of the present state, a computational one are nested as stated. The first two are one condition. It says nothing about approximate closure. It does not make closure a mark of unity. “Causal” is only in the Granger sense unless X is fully observed. | Active |
| THEORY-822 | For a Markov chain and a partition of its states, strong lumpability, every state of a block sending the same probability into each block, is the column-equitability of the master-equation operator and makes the block label informationally closed; when the chain is irreducible, an informationally closed label that is itself Markov is strongly lumpable, with no aperiodicity needed; weak lumpability gives a Markov label that need not be closed; and the same holds on the causal-state chain of any stationary process | Rosas et al. (arXiv 2024), LIT-1306, read in NOTE-1030, on Kemeny–Snell lumpability, joined to Shpak et al.’s column-equitable partitions (LIT-1242, THEORY-755). Exact closure of a block label in a Markov chain is a condition on transition rows, a symmetry, and not weak coupling. The converse needs the label to be Markov: a closed label can have memory. Exact results only, with no bound for nearly lumpable partitions. | Active |
| THEORY-829 | In Hrbacek’s relative analysis, if belief at a context level is credence ultraclose to 1 at that level, the conjunction of any standard number of believed propositions is believed, while for an ultralarge number it can fail: in a fair lottery with an ultralarge number of tickets each ‘this ticket loses’ is believed and their conjunction has probability 0 | Wenmackers (2011), LIT-163, Ch. 3 (Stratified Belief, Def. 1; the Stratified Conjunction Principle, Def. 2, eq. 3.5 and §3.6.3.2), read in full in NOTE-139; re-derived by the promotion audit from finite subadditivity and Hrbacek’s Closure principle. A result inside the framework: it says nothing about whether belief has levels, and any explicit level boundary collapses the model back to a fixed threshold. | Active |
| THEORY-831 | For a system driven by a protocol fixed in advance, with microscopic reversibility and with forward and reverse processes started uniformly on their macrostates, the log-odds of two outcomes II and III reached from I equal the difference in their log state counts, plus the log-ratio of their reversal probabilities, plus the difference in minus the log mean of e^(−βQ) over the paths reaching each, where Q is the heat released; so when the first two terms are equal the likelier outcome is the one whose paths release more heat on average with less variation | Perunov, Marsland and England (2016), LIT-518, Eq. (8), read in full in NOTE-416 (which checked Eqs. 3–8), building on England’s macrostate relation (2013), LIT-515; re-derived by the promotion audit. An exact identity from microscopic reversibility, under a fixed protocol and uniform starting densities; the paper itself uses near-uniformity. It says nothing about which macrostates a real many-body system can be divided into with the other two terms equal. | Active |
| THEORY-833 | In a 2024 survey of a quota-matched online sample of 2,092 US adults and a convenience sample of 103 antitrust lawyers, the public weighed layoffs, worse product quality and more lobbying well above price reductions in judging whether a merger should be challenged, the consumer welfare standard’s own objective, while support for replacing that standard was 42.6 points lower among the lawyers than among the public, against a partisan gap of 5.9 points | Short, Hill and Brown (2025), LIT-1327, read in NOTE-1051. A pre-registered conjoint and opinion survey. It does not say that the samples represent the US public or the antitrust bar, that the lawyers’ training caused their values, or that the standard is captured: that verdict also needs the paper’s historical argument and is THEORY-859, Proposed. | Active |
| THEORY-835 | When two firms each hire the best available candidate by a noisy ranking, and the ranking model makes an independent re-ranking worth more to the second mover than its own next choice and makes a weaker first mover better for it, then for every accuracy of the private rankings there is a more accurate shared algorithmic ranking that each firm strictly prefers to use whatever the other does, and with which both firms, and total welfare, do worse than if neither used it | Kleinberg and Raghavan (2021), LIT-1326, read in NOTE-1033. A Braess paradox for screening: the shared algorithm’s accuracy helps whoever chooses first and its correlation hurts whoever chooses second, and at the right accuracy the second effect wins while adoption stays dominant. It holds for the Mallows model always and for some random utility models, and never for Plackett–Luce. It says nothing about shocks, dynamics or market responses. | Active |
| THEORY-836 | Scored between Nicaragua’s regime-owned outlets and its exiled or online opposition outlets, on the prevalence of topics relevant to the regime’s legitimacy and on the tone of sentences mentioning Ortega, the three nominally independent national outlets with web output moved toward the regime pole in agenda over 2018–2022, Canal 14 most and Radio Corporación least, and Canal 14’s tone score fell from about .8 to below .5 by early 2022 while Radio Corporación’s stayed near 1 | Barnehl and Schumacher (2024), LIT-1323, read in NOTE-1047. A measured shift in relative position, outlet by outlet, from a 30-topic structural topic model and an off-the-shelf Spanish sentiment classifier, with no uncertainty reported in the main text. It does not show that the shift was coerced, which the paper’s own definition of capture requires, nor that the opposition pole held still. | Active |
| THEORY-837 | In Besley and Prat’s model of an incumbent who can pay n identical outlets to suppress verifiable bad news, where one outlet’s report informs every voter, each outlet’s price for silence is what it would earn as the only one to break the story, so the media are captured exactly when n·τa < r; under capture voters, though fully rational and not deceived, learn nothing and never remove a bad incumbent | Besley and Prat (2006; read in the 2005 LSE working paper), LIT-1333, read in NOTE-1031. Plurality protects because news is non-rival: silencing the press means silencing every outlet, each at its monopoly price. A static, hard-news model. It says nothing about soft news, partial bribing, or capture that feeds back on the market that permits it, and its cross-country evidence is a raw comparison of means. | Active |
| THEORY-839 | A model optimal for the distribution its own deployment induces, a performatively stable point, need not minimize the risk it actually incurs, and the gap is bounded: for a loss that is L_z-Lipschitz in the data and γ-strongly convex in the parameters, and a distribution map that is ε-Lipschitz in Wasserstein-1 distance, every stable point lies within 2L_zε/γ of every performative optimum; in a biased-coin example the stable point is µ/(1 − ε) and the optimum µ/(1 − 2ε), equal only when the prediction has no effect | Perdomo, Zrnic, Mendler-Dünner and Hardt (2020), LIT-1336, read in NOTE-1044. Self-consistency under one’s own influence and the best outcome under that influence are different targets. The bound says they are close when the influence is weak relative to the loss’s curvature. It does not say how to find the optimum, which needs the distribution map away from the deployed model. | Active |
| THEORY-840 | When every log-likelihood a predictive family V can assign lies in [−B, B], the V-information estimated from N samples is within 4R_N(G_V) + 2B√(2 log(1/δ)/N) of the true V-information with probability at least 1 − 2δ, with R_N(G_V) the Rademacher complexity of the family’s log-likelihoods; so usable information relative to a family of small complexity can be estimated in high dimension, and what is estimated is that family’s information, not Shannon’s | Xu, Zhao, Song, Stewart and Ermon (2020), LIT-1309, read in NOTE-1048, Theorem 1. Restricting the predictor class trades the data-processing inequality for estimability. The bound needs bounded log-likelihoods and says nothing about Shannon information, which a restricted family may under- or over-state (THEORY-846). | Active |
| THEORY-842 | Among 100 Chapel Hill voters not yet committed to a candidate in September–October 1968, the aggregate emphasis they gave the campaign’s main issues correlated +.967 with the emphasis of the major news items, and +.979 with that of the minor items, in the nine media serving them; and voters leaning to a candidate agreed more with all the news than with news about their own side in 18 of 24 comparisons | McCombs and Shaw (1972), LIT-1335, read in NOTE-1046. An aggregate correlation between a public’s ranking of issues and the media’s, in one community, one campaign and one sample of 100 undecided voters. It does not show that the media’s emphasis caused the voters’ ranking; the authors say the correlations do not prove agenda-setting. | Active |
| THEORY-843 | Beckers, Eberhardt and Halpern’s approximate abstraction error, the least over surjective context maps of the worst case over allowed interventions and contexts of the high-level distance between the coarse-grained low-level outcome and the high-level outcome, is zero exactly for τ-abstraction; an abstraction followed by an α-approximation of the high-level model is within α, an α-approximation of the low-level model followed by an abstraction only within kα, k the worst-case ratio of high- to low-level distances, and an approximate abstraction need not factor into an exact abstraction and an approximation | Beckers, Eberhardt and Halpern (2020; arXiv 2019), LIT-1310, read in NOTE-1045, on Beckers and Halpern’s τ-abstraction (LIT-1316). The error is relative to a metric the user chooses, takes the worst case over interventions, and minimizes over context maps, so like exact abstraction it is zero for a coarse-graining whose dropped detail the high-level noise can hold. It does not measure the governance deficiency Γ. | Active |
| THEORY-844 | A designer who commits to how players are informed, and may condition on the state and on their prior information but cannot choose their actions, can bring about exactly the Bayes correlated equilibria, the joint distributions of states and actions in which no player told his recommended action wants to deviate, a polytope cut out by linear constraints; with one uninformed receiver who would lose 1 by investing in the bad state and gain x < 1 in the good, the most investment attainable is certain investment in the good state and investment with probability x in the bad | Bergemann and Morris (2019), LIT-1313, read in NOTE-1034, from the accepted manuscript. What control over information alone can achieve, among Bayesian players with a common prior, is a set computable by linear programming, whoever the designer is, and read metaphorically it bounds what any information structure could produce. It assumes commitment, a designer who picks the equilibrium, and Bayesian players; it says nothing about bounded receivers. | Active |
| THEORY-845 | In Weatherall, O’Connor and Bruner’s network model a policy maker’s credence depends on the evidence it has received only through the total excess of successes over failures of the new action, so a propagandist’s effect is the excess of failures it adds; for the same pulls, running single-trial studies and publishing the failures adds at least as large an excess as any other division into studies, which is why biased production does best with the smallest studies | Weatherall, O’Connor and Bruner (2020; arXiv 2018), LIT-1341, read in NOTE-1038. With two symmetric hypotheses, each result’s likelihood ratio depends only on successes minus failures, so the audience’s belief is a monotone function of the cumulative excess; whoever chooses what the audience sees chooses that number. It holds for the paper’s two-point hypothesis space and naive updating, and says nothing about richer hypothesis spaces or audiences that model the selection. | Active |
| THEORY-846 | Information usable by a predictor restricted to a family V can be created by computing on the data, so the data-processing inequality fails for V-information: with Z the exclusive-or of two fair independent bits, linear and one-coordinate families extract nothing about Z from the pair and all of it from their exclusive-or; V-information computed from any function of the source is bounded by the source’s Shannon information when the family’s input-free predictions can match the marginal of the target, and can exceed it otherwise | Xu, Zhao, Song, Stewart and Ermon (2020), LIT-1309, read in NOTE-1048, define V-information and say it “can be created through computation”. The record’s worked case is CLAIM-199’s XOR under the paper’s linear families, and its addition is the ceiling: usable information never exceeds the source’s Shannon information when V can predict the marginal of Y without input, and can exceed it when V cannot. It says nothing about which family a real observer has. | Active |
| THEORY-847 | In a linear structural equation model whose exogenous micro-causes W₁, …, W_n drive effects Z = AW + F, with interventions that set all of W, all of Z or both, averaging each block is an exact transformation onto Ẑ = (an/m)Ŵ + F̂ if and only if every column of A sums to the same a, the column-equitability of the block of A from W to Z; the transformation is exact only for those whole-block interventions | Rubenstein et al. (2017), LIT-1334, read in NOTE-1037, with Beckers and Halpern’s restatement (LIT-1316, Example 3.3). Averages of linear micro-variables form a causally consistent macro model exactly when each micro-cause has the same total effect on the effect block. It says nothing about interventions on single micro-variables, which break exactness, nor about nonlinear or approximate aggregation. | Active |
| THEORY-848 | In an acyclic structural equation model, marginalising a variable that no declared intervention sets is an exact transformation, provided the coarse model’s exogenous noises may be dependent; so exactness does not require the dropped variable to be causally idle, and it can drop a variable that decides how its children respond to an intervention that is declared | Rubenstein et al. (2017), LIT-1334, read in NOTE-1037. Exact transformation asks only that the coarse model reproduce the push-forward of each declared interventional law, so a never-intervened variable can be substituted out and its randomness folded into noise, whatever it does. Applied to the hidden governance bit, the coarse-graining that CLAIM-407 shows to lose the response to h is exact. It does not say that the coarse model keeps any one system’s response. | Active |
| THEORY-849 | For deterministic causal models with a countable set of allowed interventions, a high-level model reproduces the coarse-grained interventional law of a low-level one for every distribution on contexts exactly when some map τ_U on contexts commutes with every allowed intervention, τ(M_L(u, i)) = M_H(τ_U(u), ω(i)); exactness for one distribution is strictly weaker, since a well-chosen distribution lets unrelated models match, and τ-abstraction and strong abstraction are strictly stronger; the context map is only required to exist, so high-level exogenous variables may carry what the coarse state drops | Beckers and Halpern (2019), LIT-1316, read in NOTE-1032, on Rubenstein et al.’s exact transformation (LIT-1334). Uniformity over distributions turns a distributional match into a context-by-context commuting condition. It does not make the coarse endogenous state carry the high-level context, which is what governance-response sufficiency would need. | Active |
| THEORY-850 v2 | In Hungary, 1994–2016, state-owned firms’ advertising, and not private firms’, shifted sharply toward print and online outlets when they became connected to the governing right and away when the 2015 Orbán–Simicska fallout ended a connection, at all five such events studied; and the outlets’ share of articles on corruption scandals moved the opposite way at the three events with content data | Szeidl and Szucs (2021), LIT-1315, read in NOTE-1039. Two-way favors in the reduced form: government-controlled advertising follows connection status where private advertising does not, and corruption coverage moves against it. Hungary under right-wing governments only; left-wing favoritism is not settled, and the reduced form does not say whether content favors were bought or shared. | Active |
| THEORY-851 | In Argentina’s four main national newspapers, April 1998 to December 2007, a newspaper’s monthly front-page coverage of the federal government’s corruption scandals was lower in months when it received more government advertising, within newspaper and month and within each newspaper-president pair (about 0.31 of a front page per standard deviation of advertising in the 2010 working paper); the association held for advertising placed through the presidency’s agency and not for autonomous agencies, and not for scandals of non-political actors | Di Tella and Franceschelli (NBER 2009, revised 2010; AEJ: Applied 2011), LIT-1319, read in NOTE-1040. An association within newspaper and within presidency, specific to discretionary advertising and to the government’s own scandals. It does not establish that advertising caused the reduced coverage, and the published version reports a smaller effect (0.23 of a front page, 18 percent of a standard deviation), which was not checked here. | Active |
| THEORY-852 v2 | Under Szeidl and Szucs’s structural model and its identifying assumptions (an owner’s ideology fixed across episodes; the government’s online advertising budget not driven by the outlets’ slant), both owner ideology and favor exchange drove the Hungarian portal Origo’s reduced coverage of corruption in 2013–2016: favor exchange alone cut it by 29–44%, the new owner’s ideology alone by 63% after 2016, and the two together by 74%, so they are substitutes; and a ban on targeted government advertising would have raised it by 41–79% | Szeidl and Szucs (2021), LIT-1315, read in NOTE-1039. Table IV’s estimates (weight on profit β ≈ 1.4–1.6, government preference for slant δ ≈ 420–510, both with intervals excluding zero) and Table V’s decomposition and counterfactual ban. Conditional on a static model per episode, on advertising being the only reward and corruption coverage the only slant, and on the identifying assumptions. | Active |
| THEORY-855 | In Bala–Goyal network models of unbiased scientists testing a slightly better new action, a propagandist who forwards to Bayesian policy makers only those real results that favour the worse action can leave the policy makers near-certain of the false theory while the scientists converge on the true one; the effect weakens as the policy makers hear more scientists, as the better action’s advantage grows and as each study gets larger | Weatherall, O’Connor and Bruner (2020; arXiv 2018), LIT-1341, read in NOTE-1038. No result is fabricated and no scientist is biased; the audience is misled because it updates on a selected sample as if it were not selected. It is a property of a stylized model, which the authors call “how-potentially”; it does not show that any real audience was misled this way, nor that an audience aware of the selection would be. | Active |
| THEORY-856 | In one laboratory study of a nine-period game between an infrastructure operator, two elected policymakers and three voters (59 groups), letting the first elected policymaker fix the price for all periods moved prices toward the efficient level when policymakers’ payoffs were set equal to consumers’, and moved them further above it when their payoffs were set equal to the operator’s; under period-by-period regulation voters’ pressure lowered the operator-aligned policymakers’ prices without removing the gap | Schram and Ule (2024), LIT-1340, read in NOTE-1053. Capture is a treatment, policymakers’ payoffs set equal to the operator’s, not something any party brought about, and “independence” is a price fixed by the first policymaker, not an agency. The capture cells hold 18 groups of three voters. It does not show how capture arises in real regulators or that independent agencies entrench it. | Active |
| THEORY-860 v2 | In Szeidl and Szucs’s model of two outlets competing for readers and advertisers, where state-owned advertisers reward only the connected outlet’s slant, with weight δ, every equilibrium has the slant gap between the connected and the unconnected outlet equal to their owners’ ideal-point gap plus a favor-exchange term (βδ/NT)·Σ over government advertisers of masⱼWⱼ, up to measurement noise; readers’ preference for slant cancels from the gap, so equal observed gaps can come from different mixes of ideology and exchange | Szeidl and Szucs (2021), LIT-1315, read in NOTE-1039. A necessary condition on equilibrium slant, not a closed form: masⱼ depends on the equilibrium shares. It holds under the model’s logit readers, Cobb–Douglas advertisers with exogenous budgets, quadratic ideology cost and one-shot episodes. The authors check numerically, not analytically, that the conditions pick out an equilibrium at their estimates. | Active |
| THEORY-861 | When a deployed model changes the distribution it is scored on, retraining on the distribution the last model induced contracts at rate εβ/γ, for a β-jointly smooth, γ-strongly convex loss and a distribution map that is ε-Lipschitz in Wasserstein-1 distance, and so converges to a unique performatively stable point when ε < γ/β; each condition is needed, since dropping strong convexity, joint smoothness or the bound on ε each admits a case where retraining never converges | Perdomo, Zrnic, Mendler-Dünner and Hardt (2020), LIT-1336, read in NOTE-1044. Retraining under performativity is a fixed-point iteration, and these are the conditions for it to settle. Its fixed point is self-consistent, not optimal (THEORY-839), and outside the conditions there may be several or none. | Active |
| THEORY-862 | In US law schools under the U.S. News ranking, administrators and faculty reported, and public sources corroborated, that schools changed what they did in order to move their rank: they moved money toward the ranking’s inputs, redefined admissions and career-services work around its indicators, and gamed its figures, so the ranking altered the things it measured | Espeland and Sauder (2007), LIT-1330, read in NOTE-1050. 136 open-ended interviews from 70 of about 190 US law schools, with corroborating data, sorted the responses to the USN ranking into reallocation, redefinition of work and gaming. It shows that the measured reacted, as they describe it. It does not estimate how much any school changed, or show that the changes would not have happened without the ranking. | Active |
| THEORY-863 | In Arthur’s model of two agent types arriving in random order, each adopting whichever of two technologies pays it more, with returns linear in previous adoptions, increasing returns make the adoption difference a random walk with absorbing barriers: one technology takes the whole market with probability one, A with probability Δ_R/(Δ_R + Δ_S) where Δ_R and Δ_S are the gaps at which each type switches, the winner need not be the better technology, and once it has locked in no small change in returns alters later choices; with diminishing returns the barriers reflect and the shares go predictably to one half | Arthur (1983 working paper; published 1989), LIT-1314, read in NOTE-1052. A result about one model: two types, linear returns, no expectations, no pricing, no sponsors. It does not show that any historical technology won by small events, and the “efficiency” it says can fail is a weak search criterion. Its lock-in is absorption in accumulated adoptions, which a small reversal of returns cannot undo. | Active |
| THEORY-865 v2 | In Gurnee, Sofroniew et al.’s experiments on Claude Sonnet 4.5, Haiku 4.5, Opus 4.5 and Opus 4.6, the residual-stream directions a corpus-averaged Jacobian lens marks as verbalizable (the J-space) carried most of the causal effect on verbal report and on unspoken two-hop intermediates while holding a small share of their variance (swaps along them succeeding on 59% and 61% of trials, against 5% and 28% for the remainder, which fell to 0% and 6% with the J-space clamped), held instructed concepts, and passed one swapped argument to several functions (76 of 192 swaps); in two small task sets (n = 8 and 11), report and flexible inference followed a swap while continuation, anomaly detection and line wrapping did not, a split the authors partly define by this result; and by the lens’s own statistics the J-space works in a middle band of layers, holds about 25 active directions and under 10% of activation variance, and is amplified and relayed by the weights more than other directions | Gurnee, Sofroniew et al. (2026), LIT-1347, read in NOTE-1069 (main text in full, appendices in part). It is the functional analogue of conscious access, in the access sense only, found in a transformer and tested by intervention. The sharp edge: one company’s models studied by that company, a lens whose own statistics draw the layer band, a flexible/automatic split partly defined by the result, and an explicit refusal to say anything about phenomenal consciousness. | Active |
| THEORY-866 v2 | Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness | Woods, Windt & Carter (2022), LIT-1351, read in the submitted preprint in NOTE-1070. An evidence synthesis of 135 texts by eight teachers codes 65 features of the goal-states: the same across practices on 14, different on 4, unclear on 47. The states lack everyday content but keep wakefulness, calm, stillness, bliss, knowing, freedom and unity; Shamatha’s has far greater attentional stability and vividness. The sharp edge: it is a finding about how teachers describe the states, not about what practitioners experience, and “not truly contentless” depends on counting calm or simplicity as content. | Active |
| THEORY-869 v2 | In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first | Gurnee, Sofroniew et al. (2026), LIT-1347, §3.5.3 and §A.23, read in NOTE-1069. The authors conclude that “the J-space supports the model’s propensity to provide rich experiential reports, whether about itself or another entity”. The sharp edge: the measure is a register of text graded by another model, on five or six prompts per condition, in one company’s models; and the effect holds for reports about others, so it is not evidence of anything the self-reports are about. | Active |
| THEORY-870 v2 | In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features | Chandia-Jorquera et al. (2026), LIT-1354, read in the bioRxiv v3 preprint in NOTE-1058. 33 practitioners against 33 matched controls who counted: higher and more variable retrospective pure-awareness traces (d = 0.77, 1.32); 65% balanced accuracy separating meditation from counting, on entropy and aperiodic EEG; 88% sensitivity separating meditation from rest, on low-frequency phase coherence. The sharp edge: the authors found no robust association between the reports and the EEG, so this is a finding about a practice period and about reports, not a neural signature of an experience with little content. | Active |
| THEORY-871 v3 | In two midlife community samples (n = 182 and 176), instructed reappraisal of aversive pictures, mostly by reinterpretation, lowered reported negative affect, while the BOLD responses of the amygdala, periaqueductal gray, hypothalamus and the other subcortical regions that responded to the pictures showed evidence of no change (Bayes factors below 1/10); the emotion-responsive regions it reduced were all cortical, mainly higher visual and parietal | Bo et al. (preprint 2023, published 2024), LIT-1359, read in full from the bioRxiv v2 preprint in NOTE-1061. Reported negative affect fell under reappraisal (d = 0.92, n = 358). Every subcortical region engaged by negative pictures, amygdala subregions and PAG included, showed 10:1 evidence of no change under reappraisal, as did most cortical emotion-responsive regions; the eight regions reduced were all cortical. BOLD and one rating per trial; it does not show that the unchanged response is part of what is felt, though the authors write as if it were. | Active |
| THEORY-884 v2 | A retained state supports exact answers to a declared family of questions after every finite sequence of a declared family of total deterministic revisions exactly when each of its fibres lies within one class of the coarsest revision-invariant refinement of agreement on current answers; that refinement can be strictly finer than what present answers require, and at most K* − K revision sequences, beyond the current answers, are needed to tell its K* classes apart | Aevyra (2026), The Ledger of Will, LIT-1376, read in full in NOTE-1091: Proposition 4, “revision closure”, with its proof in §4.4 and finite checks in Appendix A.9. The authors call it “a finite-machine refinement construction in the tradition of Moore and Nerode”, not a new theorem. It answers QUESTION-112 in one case only: a finite family of total deterministic revisions declared in advance. It does not say how to choose the family, which is that question’s hard part, and it is not an efficient construction. | Active |
| THEORY-889 v2 | In Acharya, Blackwell and Sen’s analysis, white Southerners surveyed in 2006 to 2011 who live in counties where a larger share of the 1860 population was enslaved are, with state fixed effects and 1860 county controls, somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about 0.09 to 0.13 standard deviations per standard deviation (about 20 points) of slave share, and in 1984 to 1998 survey data colder toward black Americans, by about 0.32; a cotton-suitability instrument gives larger estimates; the direct effect estimated with present black share held fixed is about as large; and in county presidential vote, which counts black voters until disenfranchisement, the difference appeared after Reconstruction rather than before the war | Acharya, Blackwell and Sen (2016), LIT-1388, read in full in NOTE-1089 from the journal PDF on the second author’s site, without the online appendix. The association is well supported; the causal reading rests on 1860 controls, a cotton-suitability instrument with an untestable exclusion restriction, and a direct-effect estimate. The transmission mechanism has the weakest evidence: a parent–child correlation in 241 Southern respondents not linked to county slavery. | Active |
| THEORY-891 v2 | In dyads of untrained dancers facing each other, shared music and visual contact each raised interpersonal movement synchrony, with no interaction detected, in different movement components: lateral and hand movements synchronized through seeing the partner, anteroposterior movements such as head bobs through the shared music, and vertical bounce through both; which factor drove a component’s synchrony was classified from its spatial direction and not from the periodicity of the synchrony | Bigand, Bianco, Abalde & Novembre (2024), LIT-1424, read in full in NOTE-1093 from PubMed Central. A 2 × 2 within-dyad experiment, 35 dyads, that removes the shared external drive and the coupling channel independently and finds synchrony from each in different components, no interaction, and a timing signature that cannot tell them apart. It answers QUESTION-063 in part: separate interventions on drive and channel do discriminate, and a timing-only measure does not. Direction of coupling, and a perturbation of one member, are not in it. | Active |
| THEORY-894 | A flow of pairwise comparisons on a graph splits orthogonally and uniquely into a gradient part, the l2-best global ranking, and a cyclic part, which splits into curl (local) and harmonic (global) inconsistency; a finite game’s flow of unilateral payoff differences has no curl part, so every finite game is uniquely a potential game plus a harmonic game plus a nonstrategic remainder, and potential games are exactly those with no harmonic part; the decomposition is cardinal | Jiang, Lim, Yao & Ye (2011), LIT-1439, Theorems 4.7, 4.8 and 5.1: the Hodge/Helmholtz decomposition of edge flows, and HodgeRank’s least-squares ranking with its curl and harmonic residuals. Candogan, Menache, Ozdaglar & Parrilo (2011), LIT-1442, Theorems 4.1 and 5.1: every finite game is P ⊕ H ⊕ N, and potential games are P ⊕ N. Both read from arXiv; the proofs were followed. Settled as mathematics. Whether its shares measure a negotiation’s potential depends on cardinal assumptions that the record’s orderings do not supply. | Active |
| THEORY-896 v3 | A finite game in strategic form has the finite improvement property, every path of strict unilateral improvements being finite, exactly when it has a generalized ordinal potential, a real function on strategy profiles that rises with every strict unilateral improvement; so in a finite game an ordering along which every improving move climbs exists exactly when there is no improvement cycle, and every sequence of improving moves ends at a pure equilibrium | Monderer and Shapley (1996), LIT-1462, unread: reported by Duersch, Oechssler and Schipper (arXiv 1003.4274) as Lemma 2.5. A finite game has the finite improvement property iff it has a generalized ordinal potential. The record proves the finite case here: a potential that rises with every improvement forbids cycles, and an acyclic improvement graph on finitely many profiles has an integer potential, the length of the longest improvement path into each profile. It is the precise form of “an ordering prevails” ⇔ “no improvement cycles”, and it is ordinal. | Active |
| THEORY-897 v2 | In Brown, Price and Dombrovski’s reading of seven computational learning studies they selected without a systematic search, with mixed directions, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it | Brown, Price & Dombrovski (2023; preprint 2022), LIT-1479, read in full from the NIH author manuscript in NOTE-1120. Their Table 2 and the synthesis drawn from it: under volatility anxious people adjust learning too little except after very surprising outcomes, and under noise they learn too fast, an effect the authors call less consistent. A narrative review of seven studies, not a meta-analysis; the reading that anxiety cannot tell noise from change is the authors’. | Active |
| THEORY-898 v2 | In Gangneung, South Korea, in 2022, people the authors observed by chance in beach-front cafés outwardly ignored routine retaliatory missile launches and military sites on the beach, while a missile malfunction at night under a news embargo was followed by tweets voicing fear of war with North Korea and blaming the government and military | Ku & Finch (2024), LIT-1478, read in full from the CC BY version of record in NOTE-1133. Chance observation of one afternoon and a keyword-filtered tweet sample of one night, offered as an illustration of “ontological agency”, not a test. The authors read the inaction as coping that lets securitisation reproduce itself, not as resistance to it. | Active |
| THEORY-899 v2 | In an infinite, well-mixed population whose members each sample the quality of their own type, imitation of success has the expected update of a single Cross Learning bandit agent (the Taylor replicator dynamic), and the weighted voter model, in which members broadcast their type in proportion to their sample and adopt the first type they perceive, has the expected update of Maynard-Cross Learning, π_a ← π_a + (r_k / v^π)(1[a = k] − π_a), the Maynard-Smith replicator dynamic; the population vector is the policy and no member learns | Soma et al.’s Propositions 1 and 2, read in full from arXiv v6 in NOTE-1132, with both proofs and the aggregation in Eqs. 8–14 re-derived. An exact identity of expected updates under idealizing assumptions: infinite N, large well-mixed neighbourhoods, and imitation decoupled from evaluation. Each member is one parallel action sample, α = 1/N plays the learning rate, and the macro-policy is a stateless bandit policy over the members’ types. | Active |
| THEORY-900 v2 | A systematic review of 116 studies of avoidance learning in healthy humans finds the evidence contradicting two-factor and other associative theories and favouring expectancy and inference-based accounts, but also finding residual avoidance after the outcome is visibly removed, which a purely inferential account does not predict | Mancinelli & Bach (preprint 2025; NBR 2026), LIT-1477, read in full in NOTE-1134. A PRISMA review without meta-analysis, searched to December 2024, of studies with shock or noise in healthy volunteers. The verdict is the authors’ scoring of five theories against phenomena they rate for robustness; no model was fitted. They argue current paradigms are too simple to expose where inference gives way to Pavlovian control. | Active |
| THEORY-901 v2 | As Mancinelli and Bach’s systematic review of avoidance learning in healthy humans reports it, rating both A, an avoidance response left available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it (nine studies); and less effective or costlier responses extinguish faster (six studies, one exception) | Mancinelli & Bach (preprint 2025; NBR 2026), LIT-1477, read in full from the PsyArXiv preprint in NOTE-1134. Rated A (“protection from extinction”, 9 studies; “less efficient or more effortful AR imply less protection from extinction”, 6 studies) in a systematic review of 116 studies. Healthy volunteers, shock or noise. The authors’ explanation, that effective avoidance leaves fewer chances to detect a change, is marked “putatively”. | Active |
| THEORY-902 v2 | In Friedkin, Jia and Bullo’s study of 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence centrality on the previous issue, an effect constant along the sequence, and with their average centrality over all prior issues, an effect that grew along it; influence networks thinned and discussions shortened while the rate of consensus did not change. Closure and centrality both come from the members’ own chip allocations, and prior closure is not controlled | Friedkin, Jia & Bullo (2016), LIT-1482, read in full from the CC BY version of record and its supplement in NOTE-1126. Friedkin–Johnsen dynamics on each issue, not DeGroot. Closure and centrality both come from members’ post-discussion chip allocations, and prior closure is not controlled. Supports the individual step and some structural trends; the predicted end state, one closed member and consensus on that member’s view, was not reached. | Active |
| THEORY-903 v2 | In a 2026 meta-analysis of standalone safety-behaviour manipulations outside exposure, mostly with young non-clinical or subclinical adults and mostly on appearance concerns and social anxiety, reducing safety behaviours lowered symptoms relative to control (d = 0.43, about 0.31 after adjustment for publication bias), and instructed increases raised them (d = 0.65, five studies) | Goodson, Patel, Zech, Sala & Cougle (2026), LIT-1491, read in full from the PMC author manuscript in NOTE-1128. 26 studies reviewed, 16 in the reduction and 5 in the increase meta-analyses. The size of the safety-behaviour change moderated the symptom effect (B = 0.23). Egger’s test was significant for the reduction effect on symptoms, and the increase studies could not be tested for bias. It says nothing about the mechanism by which safety behaviours act, and diagnosed anxiety disorders are thinly represented. | Active |
| THEORY-904 v2 | Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0) | Gollwitzer & Sheeran (2006), LIT-1497, read in NOTE-1124, answer the worry that if-then plans make action rigid: plan effects appear only when the goal intention is strong and activated (p. 103), from four or five studies cited, not pooled. Sheeran, Listrom & Gollwitzer (2024), LIT-1488, read in NOTE-1131, find the pattern across tests. Neither tests what happens when the planned response becomes the wrong one. | Active |
| THEORY-906 v2 | In five formerly stable democracies selected because they later autocratized (South Korea, India, Hungary, Venezuela and Turkey), formal emergency powers, extraordinary powers functionally equivalent to them, and emergency rhetoric each contributed, on the authors’ unsourced country accounts, to weakening democratic institutions | Dagan & Benvenisty (2026), Israel Democracy Institute, LIT-1487, a web summary read in full in NOTE-1129. Not peer-reviewed, no sources or method given, cases chosen on the outcome. Good for the three channels and the Hungarian description; not for a causal link, which the authors themselves call “not inevitable”. | Active |
| THEORY-907 v2 | As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator | Chou, Wilson & Smith (preprint June 2024; NBR 2024), LIT-1483, read in full in NOTE-1127. Foraging: Lenow et al. 2017, Bustamante et al. 2023, Lloyd et al. 2022a and 2022b, with human open-field and plus-maze studies agreeing. Checking: Trier et al. 2023, a single study. Mostly sub-clinical; stress and anxiety are treated together; the mechanisms are the authors’ readings. | Active |
| THEORY-908 v2 | Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46) | Sheeran, Listrom & Gollwitzer (2024), LIT-1488, read in full in NOTE-1131. Table 3’s moderators across 642 tests: contingent format .43 (k = 373), schedule .29 (k = 269); rehearsal .45 against .33; recording .31 against .44; visualising the response no effect. The authors read the recording and elaboration costs as deliberation undermining automaticity. Between-test and uncorrected for bias. | Active |
| THEORY-909 v2 | As reviewed by Gerstner et al. (2018) from five papers, mostly in slices and one laboratory per region, pairing pre- and postsynaptic activity changed the synapse when a neuromodulator or plateau potential followed within about 1 s in striatum, within 5 s (noradrenaline, LTP after pre-before-post pairing) or 2.5 s (serotonin, LTD after post-before-pre pairing) in visual and prefrontal cortex, and within about 2 s in CA1, and not at longer delays; the reviewers read this as a decaying synaptic eligibility trace and as experimental support for three-factor learning rules on behavioural time scales | Gerstner, Lehmann, Liakoni, Corneil & Brea (2018), LIT-1494, read in full from the CC BY version of record in NOTE-1130. A review of five experimental papers, 2014–2017 (Yagishita; He; Brzosko 2015 and 2017; Bittner), not new data. Mostly slices, one laboratory per region; the minute-scale CA1 result may be consolidation. The reviewers grant that other mechanisms could bridge the same gap. | Active |
| THEORY-910 v2 | Brown, Price and Dombrovski propose, without a model run or data, that in aversive environments avoiding uncertain states keeps their uncertainty high and so perpetuates avoidance, that anxiety strengthens this loop by misreading noise as change, and that exposure works by reducing the uncertainty rather than an excess of learned threat | Brown, Price & Dombrovski (2023; preprint 2022), LIT-1479, read in full in NOTE-1120. A framework stated in words and schematic figures: the loop is avoidance, unreduced uncertainty, more avoidance. It runs through an uncertainty estimate, a belief, with no authority or control variable. For fear-based disorders; by the authors’ statement it does not explain exposure in GAD. | Active |
| THEORY-911 v2 | Of 853 national states of emergency declared from 1949 to 2017, 115 were reported as unlawful in declaration, conduct or duration; 97 of those followed political turmoil and 14 natural disasters, democracies were less likely than autocracies to run one, and those so coded lasted 321 days on average against 115, a gap the coding of overlong emergencies as unlawful builds in part | Bjørnskov, Voigt & Khesali (2022; working paper 2021), LIT-1473, read in full from the ILE working paper of August 2021 in NOTE-1125. Unlawfulness is coded from news and NGO reports and from durations past the constitutional maximum, and the authors call the cases “suspected to be unlawful”. Descriptive counts and weak logits; nothing here shows that emergency measures outlive the emergency. Active as a report (ADR-065); the abstract’s thesis is THEORY-1198, Proposed. | Active |
| THEORY-912 v2 | In a picture-based avoidance task, treatment-seeking patients with OCD or PTSD avoided a safety cue more than healthy controls and, after effective avoidance, still expected the aversive picture more; patients with panic disorder with agoraphobia or PTSD reported more relief when it was omitted; the groups did not differ in avoiding the danger cues | De Kleine et al. (2023), LIT-1490, read in full from the CC BY version of record in NOTE-1123. 47 controls against 33 OCD, 40 PDA and 66 PTSD patients, cross-sectional, before exposure treatment. Relief was already higher in patients before avoidance was available, and patients rated the pictures as more aversive. It does not show the authors’ account, that low confidence in avoidance makes omissions surprising and the relief reinforces avoidance. | Active |
| THEORY-913 v2 | In small fixed-effect meta-analytic pools without correction for publication bias, responses specified in if-then plans ran more immediately (d = .77, k = 7), held up under a concurrent cognitive load (d = .85, k = 7) and ran without conscious intent (d = .72, k = 3) compared with responses under a goal intention alone, and the planned cues were detected, attended to and remembered better (d = .80, k = 6); the authors read this as control of the response handed in advance to the cue | Gollwitzer & Sheeran (2006), LIT-1497, read in full in NOTE-1124. The component-process tables of the 94-test meta-analysis: large effects for the accessibility of the planned cue and for the immediacy, efficiency and non-conscious running of the planned response, each from one to seven tests, fixed-effect, without a bias correction. Only cue accessibility and cue–response link strength had been tested as mediators. The authors’ gloss is “instant habits”. | Active |
| THEORY-915 v2 | In Zorowitz, Momennejad and Daw’s simulations of a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety | Zorowitz, Momennejad & Daw (2020; preprint 2019), LIT-1496, read in full from the CC BY version of record in NOTE-1136. A decision-theoretic argument and value-iteration simulations with a β-pessimistic Bellman equation. The misbelief is about the agent’s own future control; the danger is represented correctly. No data are fitted. That learned values outlast correction of the misbelief is argued in the paper, not simulated, and is not part of this title. | Active |
| THEORY-917 v2 | Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety | Chou, Wilson & Smith (preprint June 2024; NBR 2024), LIT-1483, read in full from the PsyArXiv preprint in NOTE-1127. A narrative review; the pattern is the authors’ reading of about eight studies with different tasks, models and anxiety measures, with one failed replication and one null. The authors hedge it with “may” and “many results could be seen to suggest”. | Active |
| THEORY-918 v2 | Across 642 experimental tests to September 2019, forming implementation intentions improved outcomes, with uncorrected effects on behaviour (d = .27), cognitive performance (.47), affect regulation (.66) and social judgement (.52), but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, which still found extreme evidence of an effect (BF10 = 319), with large heterogeneity | Sheeran, Listrom & Gollwitzer (2024), LIT-1488, read in full in NOTE-1131. Random-effects pooling of 642 tests gave d = .36; Egger’s b = 1.06 (“substantial” bias); RoBMA found an effect (BF10 = 319.3) of d = .15 with tau = 0.27. By outcome: behaviour .27, cognition .47, affect .66, social judgement .52, all uncorrected. It replaces the 2006 figure of d = .65 from 94 tests, which had no bias correction. | Active |
| THEORY-920 v2 | As Mancinelli and Bach’s systematic review reports it, rating it B, in three laboratory studies of healthy humans a signalled, costless avoidance response offered for a stimulus never paired with the aversive outcome was used, and withdrawing it raised expectancy of the outcome for that stimulus, though Xia et al. (2019) found no change in skin conductance or startle; every study signalled the response’s availability, which may itself have read as a danger cue | Mancinelli & Bach (preprint 2025; NBR 2026), LIT-1477, read in full in NOTE-1134. Rated B (“AR execution can propagate to CS-”, 3 studies: Cameron et al. 2022, Engelhard et al. 2015, Xia et al. 2019). The authors infer that people deduce danger without experiencing it, perhaps counterfactually, from having avoided; they also say the signal that avoidance is available may itself be read as a danger cue. | Active |
| THEORY-921 v2 | As Craske et al.’s 2014 review reports it, citing six studies and reporting no data of its own, how far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful, mostly analog, human samples | Craske et al. (2014), LIT-1500, read in full from the PMC author manuscript in NOTE-1137, citing six studies. The authors read it as “divergence in response systems”: fear expressed during exposure and the learning that exposure produces need not change together. That is the basis of their case against habituation-based exposure. The paper reports no data, and the cited studies are mostly analog samples. | Active |
| THEORY-922 v2 | Among 56 of 88 patients in one DBT intensive outpatient programme, kept because they finished a four-month cycle and gave enough diary data, skills-homework completion in months 1–3 correlated with larger falls in suicide and substance-use urges (r ≈ .35) and phone-coaching frequency with a larger fall in self-harm urges (r = .31); neither correlated with change in how often the behaviours occurred, and only homework separated patients with and without a target behaviour in month 4 | Edwards, Kober, Rinne, Griffin, Axelrod & Cooney (2021), LIT-1480, read in full in NOTE-1121. A chart and diary-card review: three of eight urge-change correlations with the two predictors significant, none with behaviour frequency; month-4 abstinence from all target behaviours went with more homework (91% against 80%, p = .02), not significantly with more coaching (p = .096). No correction for multiple comparisons, selected sample, no measure of skill use. | Active |
| THEORY-923 v2 | In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once | Friston, FitzGerald, Rigoli, Schwartenbeck, O’Doherty & Pezzulo (2016), LIT-1476, read in full from the CC BY version of record in NOTE-1138. One simulated T-maze task. The habit is one more policy scored by posterior probability; its prior and transitions grow by count, not by reward. It shows persistence under devaluation, a cached mapping failing when what is appropriate changes; it does not show amortization, and its reaction times are simulation run times. | Active |
| THEORY-924 v2 | When Elo ratings generate a win-probability matrix they are the row means of its log-odds; some set of Elo ratings reproduces a win-probability matrix exactly when its log-odds have no cyclic component, batch Elo otherwise matches only the row sums, and adding a copy of one agent shifts the batch Elo ratings of the others | Balduzzi et al. (2018), LIT-1508, Propositions 1 and 3 and appendix A, read from arXiv v2 in NOTE-1155, proofs followed. When the log-odds are curl-free, Elo ratings are their row means; otherwise batch Elo’s fixed point matches only row sums and differs from the row means. Elo fits exactly iff log-odds sum to zero around every triangle, so it has no predictive power in rock-paper-scissors, and duplicating an agent moves the ratings of agents that beat or lose to it. Cardinal throughout. | Active |
| THEORY-925 v2 | In a cooperative stochastic game with decentralised actors and a centralised critic, a baseline for each agent that averages the critic’s value over that agent’s own alternative actions under its own policy, the other agents’ actions held fixed, leaves the expected policy gradient unchanged, so each agent can be credited with the advantage of its actual action over that counterfactual average without a default action or an extra simulation; the counterfactual values come from the learned critic, not from the environment | Foerster et al.’s counterfactual baseline (COMA, Eq. 4 and Appendix A of arXiv v3), read in full in NOTE-1152, with the zero-expectation step re-derived. A proved property of the estimator, not of performance. Its counterfactual is “had the agent acted as it usually does”, not “had it not acted”, and its values are a learned critic’s, trained on occasions when each alternative was taken; it presupposes those occasions. | Active |
| THEORY-926 v2 | In a game where each player can transmit n bits before a win or a loss becomes unattainable, every antisymmetric pattern of wins and losses among ⌊2ⁿ⌋ strategies, a cycle through all of them included, is realized by some set of pure strategies | Czarnecki et al. (2020), LIT-1504, Theorem 1 and Propositions 1–2, read from arXiv v2 in NOTE-1146, proof followed. Strategies announce an identity, then throw or take the game as the pattern says. Go is at least about 1000-bit communicative (log₂ 180! ≈ 1094), so it has cycles of length 2¹⁰⁰⁰. Ordinal: only the sign of outcomes matters. The paper states it for patterns with draws too, which also needs a draw to stay attainable. The strategies are weak by construction. | Active |
| THEORY-927 v2 | In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate came to give large weights to the same few experts, which the authors attribute to self-reinforcement, favoured experts being trained more and so selected more; in one 256-expert language model, the squared coefficient of variation of expert importance or load as an auxiliary loss reduced the peak-to-mean load from 17.8 to at most 1.47 and test perplexity from 39.8 to 35.6–35.7 | Shazeer et al. (2017), section 4 and Appendix A, read in full in NOTE-1145 from arXiv v1. The imbalance is reported in one paragraph, with Table 6’s no-loss row as its evidence: one architecture, ten epochs, no variance. The mechanism is the authors’ explanation, not a tested result. The fix penalizes unequal use regardless of competence. | Active |
| THEORY-928 v2 | Two softmax learners that update every action’s reward memory each step with its average reward against the other’s current play, the memories decaying at rate α, follow, in the limit of learning slow relative to play, coupled replicator equations plus a term α Σ x_j log(x_j / x_i) toward uniform play; in a two-player matrix game, without decay the dynamics preserve phase volume in log-ratio coordinates and are Hamiltonian when the game is zero-sum with an interior Nash equilibrium, and with decay the volume contracts at rate (N − 1)(α_X + α_Y) | Sato & Crutchfield (arXiv 2002; PRE 2003), LIT-1520, read in full from arXiv v2 in NOTE-1144, Eqs. 1–13. The entropy-like term is what memory decay becomes under softmax choice. The limit, the dissipation rate and the Hamiltonian were re-derived, second-checked at the status review of 2026-10-30. The model updates every action’s memory at the same rate, so it is the full-information case, not Q-learning as usually run. | Active |
| THEORY-929 v2 | A joint action-value that is a non-decreasing function of per-agent values, each conditioning only on its agent’s own history, makes the profile of the agents’ individually greedy actions a greedy joint action, which suffices for decentralised greedy execution and is strictly more expressive than a sum; it cannot represent a joint value in which an agent’s best action depends on what the others do at the same step | Rashid et al.’s QMIX (ICML 2018), read in full in NOTE-1154, with QTRAN (NOTE-1141) for the strictness. Monotonicity is sufficient for the greedy-profile property and not necessary. It presupposes a real- valued joint value: what it adds is that each member’s ranking of its own actions agrees with the collective’s whatever the others do, an ordinal separability condition. It is Wolpert and Tumer’s factoredness stated for values. | Active |
| THEORY-930 v2 | In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile | Czarnecki et al. (2020), LIT-1504, section 6 and supplements E–I, read from arXiv v2 in NOTE-1146. Agents were MinMax at depths 0–9, loss-seeking MaxMin and MCTS, with 50 seeds each, about 1000 distinct per game after duplicates were removed. Games: Hex, Go and Quoridor on small boards, Tic-Tac-Toe, its misère form and Connect Four. AlphaStar’s league shows its training system’s geometry, the authors say. A qualitative profile with a fitted skew-normal curve and no test. | Active |
| THEORY-932 v2 | The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is, in the small-step limit, the expected dynamics of a learner that updates every action’s value each step, and so of frequency-adjusted Q-learning wherever every action’s probability is at least its adjustment parameter; it is not that of Q-learning, whose expected update carries an extra factor of each action’s selection probability, so that its policy dynamics depend on the Q-values; in Kaisers and Tuyls’s three 2×2 games at one temperature, compared by eye, Q-learning’s trajectories departed from the model depending on initialization and the frequency-adjusted learner’s followed it | Kaisers & Tuyls (AAMAS 2010), LIT-1511, read in full in NOTE-1140, correcting the model of Tuyls, Verbeeck & Lenaerts (2003), LIT-1501, which is not read. The derivation of the discrepancy and of the frequency adjustment was re-derived and second-checked at the status review of 2026-10-30. The experiments are three games, one temperature, compared by eye; whether Q-learning departs in general is THEORY-1234. | Active |
| THEORY-933 v2 | Individual-Global-Max, that the agents’ individually greedy actions form a greedy joint action, is strictly weaker than additive or monotonic factorisation: every single-state matrix game with a unique optimum satisfies it, yet in 177 of 310 random three-by-three two-agent games value-decomposition and monotonic mixing failed to learn the optimal joint action that an IGM-only method found; a sufficient condition is that the sum of individual values, shifted by a state value, equals the joint value at the greedy profile and bounds it from above elsewhere | Son et al.’s QTRAN (ICML 2019), read in full in NOTE-1141, with Theorem 1 and the single-state observation re-derived. IGM presupposes a real-valued joint value and constrains only the optimum: members may rank everything below it differently from the collective. With full observability it is nearly empty; its content is whether members acting on local histories can each pick their part of the optimum. | Active |
| THEORY-937 v2 | When a mixture of experts is trained on the error of a single expert chosen stochastically by the gate, or on the negative log-likelihood of the mixture, the gradient raises the gate’s logit for an expert that fits a case better than the gate-weighted average and lowers it for one that fits worse; on a four-class vowel task, mixtures of 4 or 8 linear experts learned in about half the epochs of backpropagation networks of roughly equal parameter count, at equal accuracy, and divided the task by class pair, leaving all but 2 or 3 experts with mixing proportions effectively zero | Jacobs, Jordan, Nowlan and Hinton (1991), read in full in NOTE-1147 from a scan of the published article. The gradients of Eqs. 1.4 and 1.5 were re-derived when filed. The task is small (two formants, 75 speakers, 25 runs per system) and has two natural subtasks; the idle experts are read by the authors as economy, and their recovery is not tested. | Active |
| THEORY-940 v2 | In a finite symmetric zero-sum game, successive Nash clusters are weakly ordered by relative population performance, and a strategy that beats every member of a population containing a whole cluster lies in a strictly earlier cluster | Czarnecki et al. (2020), LIT-1504, Definition 3 and Theorems 2–3, read from arXiv v2 in NOTE-1146, proofs followed and re-derived. RPP is the population value of Balduzzi et al. (2019), LIT-1509. It holds for any Nash clustering of any finite symmetric zero-sum game and does not depend on the spinning-top hypothesis. The order is weak, since clusters can tie, and cardinal, since RPP uses payoff magnitudes. | Active |
| THEORY-942 v2 | The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game’s weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow | Omidshafiei et al. (2019), LIT-1516, Theorems 2.1.2, 2.4.24 and 2.5.1, read from arXiv v4 in NOTE-1156, proofs followed. Irreducibility is immediate, and the α → ∞ limit is of transition matrices. That each MCC lies in one chain component follows from chain transitivity, re-derived and second-checked on 2026-10-30. The converse, that attracting components contain MCCs, rests on a conjugation step that is asserted, not carried out, and is THEORY-1233. Not shown: what the stationary distribution tends to, how it splits among several sinks, or invariance to duplicated strategies, which the reader found it lacks. | Active |
| THEORY-943 v2 | When a center with no ground truth pays each rater a strictly proper scoring rule on the distribution of a reference rater’s report implied by her own report, under a common prior and a known signal model in which raters’ signals are stochastically relevant to one another, truthful reporting is a strict Nash equilibrium, but not necessarily the only one: in the authors’ two-signal example under the log rule, every rater always reporting h, or always reporting l, is also an equilibrium | Miller, Resnick and Zeckhauser’s Proposition 1, read in full in NOTE-1139 from the January 2005 draft, with the proof and the worked example (p(H) = .5, f(h | H) = .85, f(h | L) = .45; honest l scores −0.69, false h −0.76 under the log rule) re-derived. The same example has two uninformative equilibria, which the authors set aside as not focal. The scheme replaces ground truth with the center’s model of how honest signals correlate. | Active |
| THEORY-944 v2 | The value of the zero-sum game between two populations’ evaluation matrices does not depend on which Nash equilibrium is used, is unchanged by adding agents that behave as mixtures of existing members, and is not positive for a population whose agents are all behavioural mixtures of the other’s | Balduzzi et al. (2019), LIT-1509, Propositions 2 and 5, read from arXiv v2 in NOTE-1143, proofs followed and re-derived. Relative population performance is a game value, so equilibrium selection never arises. Behaviourally redundant members leave it unchanged, and a population contained in another’s behavioural hull cannot beat it. It scores populations, not members, it is cardinal, and populations not nested by hull can still beat one another in a cycle. | Active |
| THEORY-945 v2 | When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed | Yang et al. (2018), read in full in NOTE-1148 from arXiv v5. The expansion and the remainder bound were re-derived when filed and second-checked on 2026-10-30; the appendix states an equality where only an inequality holds, which leaves the bound intact. The convergence theorem is reported, not held. Its contraction step needs a high temperature while it assumes a greedy limit, and the cancellation of remainders is argued from homogeneity, not proved. The approximation requires agents interchangeable within a neighbourhood. Whether the method works as claimed is THEORY-1231. | Active |
| THEORY-946 v2 | In the zero-sum meta-game on an antisymmetric evaluation matrix the maximum-entropy Nash equilibrium is unique; when the meta-game has only one Nash equilibrium, adding exact copies of an agent splits that agent’s mass evenly and leaves every Nash average unchanged; the equilibrium is uniform exactly when every agent’s mean log-odds is zero, and uniform on the top-rated agents when the matrix is a pure gradient | Balduzzi et al. (2018), LIT-1508, Proposition 4, Lemma 1 and Theorem 1, read from arXiv v2 in NOTE-1155, proofs followed. The paper claims invariance to redundant copies without the uniqueness condition. Its proof covers the Nash set, not the maximum-entropy selection. A counterexample checked by script (one copy of a null agent moves an outside agent’s Nash average from −0.5 to −0.2) shows the condition is needed. Cardinal: payoffs are log-odds. | Active |
| THEORY-948 | On Stable Diffusion 2, MultiDiffusion’s 512 × 4608 panoramas had crops closer in FID to the reference model’s own samples than two outpainting baselines (10.3 against 18.4 and 45.5, fresh reference samples scoring 6.05) at equal CLIP score, over 8 prompts; and on 1,000 COCO layouts of two to four objects, the overlap of a segmenter’s output with the target layout was 0.26 IoU against 0.16–0.17 for the baselines and 0.43 for the real images, of which region fusion alone gave 0.18 and an early phase showing each region’s denoiser only its own region gave the rest | Bar-Tal, Yariv, Lipman and Dekel (2023), LIT-1522, read in NOTE-1158. Panorama quality is measured against the reference model’s own crops, not against real panoramas, and the baselines extrapolate from a centre image. The layout result rests on one segmenter and one overlap score, with spreads whose basis is not stated. Comparison with models trained for region-to-image generation is qualitative only. | Active |
| THEORY-951 | For independent identically distributed observations, and neglecting how far the log-likelihood-ratio walk overshoots its boundaries, Wald’s sequential probability ratio test accepts the null hypothesis with probability (A^h − 1)/(A^h − B^h), where h ≠ 0 solves E e^{hz} = 1, and takes on average [(1 − γ) log B + γ log A]/E z observations; no sequential test with the same error probabilities needs fewer on average under either hypothesis than Wald’s lower bounds, which the test attains when the overshoot is zero, and for a normal mean it saves 47–63% of the fixed-sample test’s observations at error rates from .01 to .05; the overshoot-free formulas are exact for Brownian motion with drift, which the paper never mentions | Wald (1945), LIT-1523, read in NOTE-1165. The SPRT’s operating characteristic and mean sample size follow from two identities of the stopped walk, and its optimality from Jensen’s inequality, all exact when the walk stops on the boundary. Optimality with overshoot is only conjectured in the paper. The drift-diffusion limit that the record takes from Bogacz et al. (LIT-569, THEORY-068) is not stated by Wald. It is the case in which his approximations become exact. | Active |
| THEORY-952 | In Wald’s sequential probability ratio test, which samples until the likelihood ratio of two simple hypotheses leaves an interval (B, A), the realized error probabilities satisfy α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B whenever the test terminates with probability one, with or without independent observations; so the boundaries A = (1 − β)/α and B = β/(1 − α), set with no distribution theory, give errors α′ ≤ α/(1 − β), β′ ≤ β/(1 − α) and α′ + β′ ≤ α + β, and they would be exact only if the ratio stopped on the boundary | Wald (1945), LIT-1523, read in NOTE-1165. The boundaries of a sequential likelihood-ratio test follow from the error rates by a two-line argument that needs only termination, so the test needs no sampling distribution. The approximate boundaries are conservative in sum. Either error alone may rise, but only by a factor 1/(1 − β) or 1/(1 − α) at most. Nothing here says how many observations the test takes or that it is optimal: that is THEORY-951, and it needs i.i.d. observations. | Active |
| THEORY-953 v2 | On Isaac’s structuralist account, the strongest evidence that a quality-space model targets phenomenal experience, and not only behaviour or neural activity, is convergence between models of the same space derived through different behavioural procedures, since experience is what those procedures share; he concedes that the convergence observed is coarse, with discrepancies that in many cases exceed each method’s own uncertainty, and that because the models average over subjects what they converge on is a standard observer | Isaac (2025), LIT-1361, read in NOTE-1163: why a third-personal science of quality spaces counts as a science of experience. Relations between experiences, unlike bare qualities, show up in behaviour, so they can be compared across subjects; and models from different behavioural routes that converge are taken to target what the routes share. The argument does not exclude a shared stimulus geometry or a shared decision process as the common factor, the convergence is partial by the paper’s own account, and averaging over subjects makes the target an ideal observer, which the paper leaves as an open puzzle. | Active |
| THEORY-954 | When the information a design yields depends on the unknown parameter, as in nonlinear models or nonlinear functions of linear ones, Bayesian design criteria are normal approximations to expected utility that average a local criterion over the prior, such as the expected log det of the Fisher information, and locally optimal design is the special case of a point-mass prior; a locally optimal design can leave the quantity of interest inestimable, as for the turning point of a quadratic regression on [−1, 1] with best guess ½, where the locally c-optimal design puts half its mass at 0 and half at 1 and identifies only θ₁ + θ₂ | Chaloner and Verdinelli (1995), LIT-1525, read in NOTE-1161. Once the information depends on θ, no design is optimal without saying which θ it is optimal for, and the prior is how that is said. The approximations’ accuracy is “not much known” in general. The claim that optimal Bayesian designs gain support points as the prior disperses rests on examples, not a theorem, and is not part of this account. | Active |
| THEORY-955 | On 480 sets of two to five public LoRAs on two Stable Diffusion 1.5 checkpoints, GPT-4V’s paired judgements, averaged over both presentation orders and three seeds, scored giving each LoRA its own denoising steps in rotation (LoRA Switch) and averaging every LoRA’s guided noise prediction at every step (LoRA Composite) above merging the LoRAs’ weights at 0.8 each, at every number of LoRAs, the composition gap growing from 0.04–0.11 points of 10 at two LoRAs to 0.92–1.32 at five as merging fell from about 8.7 to 5.2; two annotators rating 120 images agreed in direction; the design does not separate composing at sampling time from the merged update’s scale growing with the number of LoRAs | Zhong et al. (2024), LIT-1529, read in NOTE-1160. The measurement is a GPT-4V comparison with a positional bias of about ten points, correlated 0.45 with two annotators, on LoRAs selected to be compatible. The two methods are compared with each other only through the baseline. It does not show that sampling-time composition is what helps: the baseline adds k updates at full scale, the two methods keep one adapter’s scale. | Active |
| THEORY-956 | In the normal linear model with known error variance and a conjugate normal prior of precision R/σ², the design that maximizes the expected Shannon information of the posterior is the one maximizing det(nM(η) + R), Bayes D-optimality, and the design that minimizes expected quadratic loss with weight matrix A is the one minimizing tr A(nM(η) + R)^-1, Bayes A-optimality; both criteria depend on the design and not on the data, so in this model a sequential design gains nothing over the best fixed one | Chaloner and Verdinelli (1995), LIT-1525, read in NOTE-1161. The alphabetical criteria are utilities in disguise, and the utility fixes the criterion. This holds exactly only in the normal linear model with known variance. With unknown variance only Bayes A keeps its simple form, and in nonlinear models the criteria are approximations (THEORY-954). E- and G-optimality have no utility behind them. | Active |
| THEORY-957 | When the signal law depends on the decision, Kwon and Perchet’s approachability algorithm learns what each decision would reveal only from the stages on which it is played: it keeps every decision’s probability at least γ/|I| and estimates the flag by importance weighting, an estimate that is conditionally unbiased with second moment Σ_i 1/P[i_t = i] ≤ |I|²/γ, with the tuned floor γ shrinking as T^-1/3; when the signal law does not depend on the decision, the raw signal is already unbiased for every component and no floor is used | Kwon and Perchet (2017), LIT-1526, read in NOTE-1159. Under decision-dependent partial monitoring, information about a decision’s signal comes only from playing it, and its unbiased estimate costs variance inversely proportional to the probability of playing it. A floor is how the algorithm keeps that finite. That the floor is necessary in general is supported by the matching lower bound, not shown here. Nothing here says what floor a non-learning system should keep. | Active |
| THEORY-958 | When each view takes pixels directly (a crop of a larger canvas, or the whole image under a region mask), MultiDiffusion’s step, the least-squares reconciliation of several denoisers’ proposed next states, is their per-pixel average weighted by the weights of the views covering that pixel; this is exact for 0/1 weights, the only ones the paper uses, while for other weights the loss as printed weights by their squares; with a deterministic update affine in the noise prediction it is one step taken with the per-pixel weighted mean of the views’ noise predictions; and the guarantee that each view then follows the reference model’s diffusion path holds only if the loss is zero at every step, which overlapping proposals that disagree do not reach | Bar-Tal, Yariv, Lipman and Dekel (2023), LIT-1522, read in NOTE-1158. The fusion is an average, not a sum: where views overlap each takes a convex share of the pixel’s update. The paper’s closed form is the minimizer of its loss only for 0/1 weights. Its distributional guarantee is conditional on zero loss and, in the proceedings version, on strictly positive weights, so it covers neither the overlaps as they behave nor masked region generation. Nothing here says what distribution the fused sampler targets. | Active |
| THEORY-960 | In DeGroot’s model, where each member repeatedly replaces its opinion by a fixed weighted average of all members’ current opinions, F⁽ⁿ⁾ = PⁿF, the group reaches a consensus from every starting set of opinions exactly when the chain P has a single closed class of communicating members and that class is aperiodic; the consensus is then Σπᵢ Fᵢ with π the unique stationary vector, πP = π, which gives members outside the closed class no weight; two closed classes, or one periodic class, give no consensus | DeGroot (1974), LIT-1527, read in NOTE-1157. Repeated averaging with fixed weights is a Markov chain run on opinions, and whether it ends in agreement is a question about the chain’s closed classes and their periods. It is not about how much each member trusts itself. The weights are fixed and no evidence enters, so nothing here says the consensus is right, and the periodic failure is a discrete-time one. | Active |
| THEORY-961 | In a finite partial-monitoring game, an action optimal wherever two neighbouring actions tie has a loss vector on the segment between theirs, so a degenerate or duplicate action never creates a neighbouring pair or decides the trivial class, yet its signals count toward the pair’s local observability; and the T^2/3 lower bound for a pair that is not locally observable needs the separating direction in the kernel of the signal matrices of all such actions, not only of the two neighbours | Lattimore and Szepesvári (2019), LIT-1524, read in NOTE-1162, on Bartók et al. (2014), LIT-1528. Of actions optimal only on a boundary: “Although these actions are never uniquely optimal, they can be informative”. They enter the classification only through the neighbourhood set N_ab. Bartók et al.’s lower-bound proof chose the separating direction in the kernel of the two neighbours’ signals alone. With such an action informative, that direction can be told apart cheaply. The fix is to take it in the kernel of every action in N_ab, which the definition of local observability already provides. It says what such actions contribute, not how often an algorithm must play them. | Active |
| THEORY-962 v3 | In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison | Bartók, Foster, Pál, Rakhlin and Szepesvári (2014), LIT-1528, read in NOTE-1164. Observability in partial monitoring is a linear-algebra condition on the loss and signal matrices: a comparison is learnable from some actions exactly when its loss difference is in the span of their signal row spaces. Global and local observability are the two choices of D the classification uses. It says what can be estimated, not how fast; the rates are THEORY-964. | Active |
| THEORY-963 v2 | The minimax regret of every finite partial-monitoring game against an oblivious adversary, degenerate and duplicate actions included, is 0 when no two actions are neighbours, Θ̃(√n) when every pair of neighbours is locally observable, Θ(n^2/3) when every pair is globally but some pair is not locally observable, and linear otherwise; its lower bounds are proved against stochastic outcomes, so the same four classes hold for them | Lattimore and Szepesvári (2019), LIT-1524, read in NOTE-1162, completing Bartók et al. (2014), LIT-1528. The four classes, trivial, easy, hard and hopeless, now cover every finite game. Two things change from Bartók et al.’s statement: the adversarial half no longer excludes degenerate and duplicate actions, and the trivial class is “no pair of neighbouring actions”, not “one non-dominated action”, which misplaces games whose extra actions are degenerate or duplicates. It classifies the best policy in a fixed, known game, up to game-dependent constants. It does not cover a learner barred from some actions, feedback that changes during play, or adaptive adversaries. | Active |
| THEORY-966 | Greedy edge pruning of the ACDC kind, which removes each edge whose removal changes the output metric by less than a threshold given the edges already removed, keeps only one input of a redundant OR gate, and which input it keeps depends on the order in which edges are visited, so when a model has redundant sufficient paths the circuit it returns is one of several sufficient subgraphs | Conmy et al. (2023), LIT-1547, Appendix M, read in NOTE-1170. For out = 1 − ReLU(1 − x − y) with x = y = 1, removing either input alone leaves the output at 1, so the first one visited is pruned and the second is then kept; visiting y first keeps x instead. It does not say how common redundant paths are in trained models, nor that any other method does better: HISP keeps neither input. | Active |
| THEORY-968 | In the Mechanistic Interpretability Benchmark’s causal-variable localization baselines on Gemma-2 2B and Llama-3.1 8B, features of the public Gemma Scope and Llama Scope sparse autoencoders, selected by a learned binary mask, never gave a higher mean interchange-intervention accuracy than the same mask over the model’s own hidden dimensions, in all sixteen task–variable cells and by 1 to 14 points, while distributed alignment search, trained against the causal model it is scored on, gave the highest mean in fifteen of them | Mueller et al. (2025), LIT-1550, read in NOTE-1169. Unsupervised SAE features were worse units than neurons for known task variables across ARC, MCQA, two-digit addition and RAVEL. It does not say SAE features are never causal units: the comparison is on residual streams, under one selection method, with the base reconstruction error carried into every intervention, and the best method is supervised by the hypothesis it is scored against. | Active |
| THEORY-969 | In hierarchical LDA every document draws all its words from the topics on one root-to-leaf path of a tree drawn from the nested Chinese restaurant process, so the prior is a nested sequence of partitions of the documents, exchangeable in them, and two documents share a topic only by sharing the whole path above it: no document can draw on two sibling subtrees | Blei, Griffiths and Jordan (2010), LIT-1542, read in NOTE-1176. hLDA’s overlap between documents is nested by construction: at each level of the tree the documents are partitioned, and a document can share topics with two others only if one of the two shares all of its shared topics with it. It is a statement about the model, not about the data, and it says nothing about whether the tree the sampler finds is identified. The multiple-path relaxation is the nested HDP’s (LIT-002). | Active |
| THEORY-970 | On three language-model tasks whose circuits earlier manual work reported, indirect object identification and Greater-Than in GPT-2 Small and Python docstrings in a four-layer attention-only model, ACDC run with resampled corrupted activations against KL divergence ranked the reported edges with areas under the ROC curve of 0.85 to 0.98, above head-importance pruning and subnetwork probing on all three; on two compiled tracr programs it ranked them poorly with corrupted activations, 0.68 and 0.20, and exactly with zero activations | Conmy et al. (2023), LIT-1547, read in NOTE-1170. Edge AUCs with corrupted activations and KL: Docstring 0.982, Greater-Than 0.853, IOI 0.869 for ACDC, against 0.805, 0.693, 0.789 for HISP and 0.937, 0.806, 0.823 for SP. It does not say ACDC finds the true circuit: the reported circuits are incomplete, the results depend on metric, ablation and parent order, and negative components are missed at sparse thresholds. | Active |
| THEORY-971 | A unit that thresholds a pre-activation equal to a term depending only on one operand plus a term depending only on the other cannot be active on exactly the pairs with a given sum when that sum has two decompositions: if it is active on (x, y) and (z, w), it is active on (x, w) or on (z, y); so a sum computed from separately represented operands needs an earlier unit that combines them, as Ameisen et al. argue for addition in language models | Ameisen et al. (2025), LIT-1552, read in NOTE-1185, re-derived here. It is why their addition graphs contain “lookup table” features between operand features and sum features. Whether a given unit in a trained model meets the premise is an empirical matter: attention patterns that depend on both operands, or any earlier nonlinearity that mixes them, break it, and that mixing is exactly the intermediate step the result says is required. | Active |
| THEORY-972 | In the author–topic model the likelihood of a corpus depends on the topics Φ and the authors’ topic distributions Θ only through the document–word matrix ΦΘA, A being fixed by the observed authorship, so LDA (each document its own author) and the topic-free author model (each author one topic) are special cases; and the likelihood leaves Φ and Θ unidentified not only up to the permutations of topics the paper names but along a continuum of changes of topic basis whenever every topic gives every word positive probability | Rosen-Zvi, Chemudugunta, Griffiths, Smyth and Steyvers (2010), LIT-1546, read in NOTE-1180. The likelihood sees members’ topic profiles only through the word distributions they jointly produce, and a small change of topic basis that keeps every distribution a distribution produces exactly the same corpus probabilities. Only the Dirichlet priors then choose among the profiles. It says nothing about the posterior’s concentration, and nothing about conditions under which Φ and Θ are identified up to permutation. | Active |
| THEORY-973 v2 | In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones | Olsson et al. (2022), LIT-1538, read in NOTE-1173: Arguments 1–3 for the small attention-only series, with the smeared-key models. The score jumps from under 0.15 to about 0.4 nats in the window where induction heads form, pattern-preserving ablation of the induction heads removes nearly all of it to the end of training, and an architecture that lets one layer express induction moves the jump. It says nothing causal about models with MLPs or large models, and nothing about few-shot task learning, which the score does not measure. | Active |
| THEORY-974 | When a transformer’s MLP sublayers are replaced by transcoders and the attention patterns and LayerNorm scales on a prompt are held fixed, the contribution of an earlier transcoder feature at the same token to a later one’s pre-activation is the earlier feature’s activation times the dot product of its decoder with the later feature’s encoder, scaled by one LayerNorm constant per token, so the weight factor is fixed by the trained weights and holds on every input | Dunefsky, Chlenski and Nanda (2024), LIT-1554, read in NOTE-1174, Eq. 7 and App. D.2–D.4, re-derived here. The weight factor is a weight of the replacement model, not of the original, and it is exact only for the replacement. It says nothing about why attention attends where it does, and “input-invariant” holds only up to a per-token LayerNorm scalar and only while the later feature is active. | Active |
| THEORY-975 | In a two-layer attention-only transformer trained on the Pile, the weight-refined local learning coefficients of individual attention heads start alike and diverge over training, and clustering the curves groups the heads by the functional types that ablation assigns them at the end of training, previous-token, current-token, induction and n-gram-memorizing; among layer-1 heads, those running simple algorithms end with the lowest values and those memorizing the most n-grams with the highest | Wang, Hoogland, van Wingerden, Furman and Murfet (2024), LIT-1535, read in NOTE-1188: §4.1, Figs. 1–2 and 11, appendices B, F.5 and G. Per-head curves of how degenerate the loss is in that head’s weights separate the heads by what they end up doing, and the separation is in the curves’ shapes over training, not only their end values. It is one small model, cluster counts chosen by eye, and estimates whose ordering, not value, the authors trust. It does not show that the coefficient predicts a head’s type before the type is visible. | Active |
| THEORY-976 v2 | In a Claude 3.5 Haiku variant finetuned by Marks et al. to exploit reward-model biases described in fictional documents, a cross-layer-transcoder feature that in pretraining-style text fires almost only where those biases are discussed was active in all 100 short Human/Assistant-formatted prompts tested and in 1 of the same 100 without that formatting, and suppressing such bias features cut the sampled rate of three bias behaviours, including two the model was never trained to show | Lindsey et al. (2025), LIT-1541, read in NOTE-1181. A feature for a trained-in goal is active across the Assistant context, not only where the goal applies, and suppressing such features suppresses the behaviour. It is one model organism, built so that the goal has a name in its training documents; the authors do not claim their method would find a goal that has none. | Active |
| THEORY-977 | In the two-layer attention-only transformer that Elhage et al. reverse-engineer, the second-layer heads that compose substantially with first-layer heads do so through their keys, mainly with a head that attends to the previous token and, after the published correction, also with one that attends over the last several tokens; these heads are induction heads, attending from a token to the one after its earlier occurrence, including on repeated random tokens, with positive-eigenvalue copying OV and same-matching QK circuits, and the second-order virtual-head terms add little to the loss reduction | Elhage et al. (2021), LIT-1534, read in NOTE-1189: Analyzing a Two-Layer Model, Induction Heads, Term Importance Analysis, and the Correction. Composition between heads was scored by a normalized Frobenius norm of matrix products, minus its value for random matrices. The heads it picked out attend to the token after the present token’s previous occurrence and copy it, which is the minimal induction circuit the framework predicts from K-composition with a previous-token head. It is one model, with no seeds or training details reported, and says nothing about models with MLPs or about when the heads form. | Active |
| THEORY-978 v2 | Causal scrubbing scores a hypothesis through the partitions of the dataset that its interpretation nodes induce, so two hypotheses that assign different features to a node but induce the same partition on the dataset, such as two features perfectly correlated on it, receive the same score | Chan et al. (2022), LIT-1549, read in NOTE-1168. In the causal-scrubbing algorithm an interpretation node enters only through sample_agreeing_x, which draws a datum with an equal value of that node, so only the equivalence classes the node induces on the dataset matter. The authors say it themselves: the method “doesn’t allow us to distinguish between two features that are perfectly correlated on our dataset”. It does not say that a passing hypothesis is otherwise wrong, nor anything about features correlated only imperfectly. | Active |
| THEORY-979 | On one prompt, a transformer whose MLPs are replaced by a cross-layer transcoder, with its attention patterns and normalization denominators frozen and each MLP’s reconstruction error added back as a constant, reproduces the model’s activations and outputs exactly and is linear between features, so each active feature’s pre-activation is the sum of its attribution-graph edges and constants; a perturbation that clamps MLP outputs up to the target feature’s encoder layer then confirms a direct edge by construction, and tests only knock-on effects and the output | Ameisen et al. (2025), LIT-1552, read in NOTE-1185: the local replacement model, the edge formula and the appendix on steering, re-derived here. It is a statement about what an attribution graph is exact for, not about the model: exactness holds at the unperturbed point only, the error terms are constants with no inputs, and the frozen attention patterns are the part of the computation the graph does not explain. | Active |
| THEORY-980 | In GPT-2 small on fifteen templates of indirect object identification, a circuit of 26 attention heads in seven classes, found by path patching, keeps 87% of the model’s mean logit difference when every other head is mean-ablated; knocking out its three main name mover heads costs only 5%, because other heads take over, and knockout sets found by greedy search make the circuit and the full model differ by up to 87% of the logit difference, so the circuit is faithful on average and incomplete under knockouts | Wang et al. (2022), LIT-1540, read in NOTE-1167: §§3–4 and the appendices. The circuit implements “identify the names, remove the duplicated one, output the rest” through duplicate-token, induction and previous-token heads, S-inhibition heads and name movers, with negative and backup name movers. The faithfulness, completeness and minimality tests are the paper’s own. It says nothing beyond these templates and this model, it treats MLPs as given, and its greedy search is printed in a form that does not match its description. | Active |
| THEORY-981 | When a data distribution is unchanged by a group of orthogonal transformations under which a network is equivariant, the weights fixed by that group form a subspace that population gradient descent never leaves; for attention-only transformers on block-list inductive tasks, with fixed orthogonal write maps, Musat, Pimentel, Zucchet and Hofmann compute this subspace as the span of a few elementary token and position actions composed through the heads, and an elementwise optimizer such as sign descent keeps it only when those actions are aligned with the parameter coordinates | Musat et al. (2026), LIT-1530, read in NOTE-1177. Symmetries of the data confine population gradient descent to the weights those symmetries fix; for their task class and idealized architecture this is a subspace whose coordinates are interpretable head actions. It does not say the subspace is attractive (the paper proves only invariance), and it does not cover minibatch noise beyond expectation, learnable value maps (a sketch in the paper), or adaptive optimizers, which keep it only when the basis is coordinate-aligned. | Active |
| THEORY-983 | In Geiger et al.’s three-layer feed-forward network trained on hierarchical equality and in BERT fine-tuned on monotonicity inference, interchange interventions on subspaces found by a learned rotation aligned the hypothesised two-variable causal models with interchange intervention accuracy of 1.00 on the rotation’s training data, where the axis-aligned alignment closest to the learned rotation reached 0.73 and 0.51; in BERT, the learned lexical-entailment subspace was in turn aligned, at 0.97 to 0.98, with a model holding only the identities of the two words | Geiger, Wu, Potts, Icard & Goodman (2024), LIT-1551, read in NOTE-1192. Layer 1 of the equality network (1.00 with |N| = 16 and two 8-dimensional subspaces) and layer 9 of BERT’s [CLS] vector (1.00 with two 256-dimensional subspaces). It does not say the networks implement the algorithms in any stronger sense than interchange accuracy on training pairs, nor that the learned subspaces are unique: every rotation within them gives the same interventions. | Active |
| THEORY-984 | In Claude 3.5 Haiku, on the prompt “Fact: the capital of the state containing Dallas is”, which it completes with “Austin”, an attribution graph shows Dallas features activating Texas features that, with “say a capital” features, promote Austin, beside a direct edge from Dallas to “say Austin”; inhibiting the Dallas features lowers the Texas features and leaves “say a capital” largely intact, and replacing the Texas features with those of California, Georgia, British Columbia, China or the Byzantine Empire makes the model name that place’s capital | Lindsey et al. (2025), LIT-1541, read in NOTE-1181. A two-hop intermediate, measured by intervention on one prompt. The evidence that counts is the change of output under the swaps; the graph itself is of the replacement model. It does not say how often the model reasons in two steps, nor that the shortcut path is weak. | Active |
| THEORY-985 | The beta process BP(c, B0) is the de Finetti mixing measure of the Indian buffet process: given n objects, the next has each existing feature with probability equal to the number holding it over c + n and a Poisson(cγ/(c + n)) number of new ones, so n objects hold Poisson(γ Σ_{i<n} c/(c + i)) distinct features, which is the IBP’s αH_n at c = 1 and γ = α, and c alone sets how much objects share | Thibaux and Jordan (2007), LIT-1537, read in NOTE-1171. The beta process plays for the IBP the role the Dirichlet process plays for the Chinese restaurant process, and its concentration c separates how much objects share from the mass γ that sets features per object. The conjugacy it rests on is cited, not proved, in the paper. It does not give dependence between features or over time. | Active |
| THEORY-987 | In Claude 3 Sonnet’s middle-layer residual stream, on two prompts for which every active sparse-autoencoder feature was ablated, how strongly a feature was active predicted its ablation effect on the answer poorly (correlation about 0.12) while its gradient attribution predicted it well (about 0.81), and on the multi-step prompt only three of the ten most active features were among the ten with the largest ablation effect | Templeton et al. (2024), LIT-1533, read in NOTE-1178: Features as Computational Intermediates and the appendix on ablations and attributions. On “The capital of the state where Kobe Bryant played basketball is” (Sacramento against Albany), the five features with the largest ablation effect, Kobe Bryant, California, capital, Los Angeles and the Lakers, rank as low as 70th to 162nd by activation. It does not say how often this holds beyond the two prompts, and for other prompts the authors found no intermediate features at this layer. | Active |
| THEORY-988 | In an attention-only transformer without layer norm or biases, once every attention pattern is held at its value on the input, the logits are exactly a sum of end-to-end path terms, a direct path from embedding to unembedding, one term per head and one per chain of heads composed through their values, and each head’s attention scores are a bilinear form in the tokens through which earlier heads can enter on the query side, the key side or both | Elhage et al. (2021), LIT-1534, read in NOTE-1189. For a two-layer model, T = Id ⊗ W_U W_E + Σ_h A^h ⊗ W_U W^h_OV W_E + Σ_{h₂,h₁} A^{h₂}A^{h₁} ⊗ W_U W^{h₂}_OV W^{h₁}_OV W_E, with each A^h computed from the model’s own run and then held fixed. It is an identity, not an approximation, and it is exact only under its conditions: no MLPs, no layer norm (which enters as a per-path rescaling), no biases, and attention frozen. It does not say the terms are small, interpretable or independent in what they compute. | Active |
| THEORY-991 | In a one-layer transformer whose MLP has 512 ReLU neurons, a sparse autoencoder with 4,096 features trained on that layer’s activations yields features that a blinded rater and an automated pipeline judged far more interpretable than the neurons, that for the four features examined in detail correspond to no neuron, and whose effects on the output, which the autoencoder never saw, agree with their activations; splicing in the reconstruction recovers 79% of the loss reduction the MLP provides, measured against zero-ablating it | Bricken et al. (2023), LIT-1545, read in NOTE-1191: the detailed investigations, the global analysis and the appendices, for run A/1 of model A. Median rubric score 0 for neurons and 12 for feature intervals (one author, 162 units); held-out logit prediction 74% against 58%; the Arabic, DNA, base64 and Hebrew features have no matching neuron. It says nothing about deeper models or other sites, nothing about whether the model is “composed of” these features, and nothing about how features compare with neurons as carriers of a known causal variable. | Active |
| THEORY-992 | In sparse autoencoders of about 1M, 4M and 34M features on Claude 3 Sonnet’s middle-layer residual stream, whether a concept among chemical elements, cities, animals and foods had a dedicated feature depended on how often it occurs in the training data: the frequency at which a concept became more than 50% likely to have one was slightly below one over the number of live features, and the three dictionaries’ curves roughly coincided when frequency was rescaled by that number | Templeton et al. (2024), LIT-1533, read in NOTE-1178: Feature Completeness. 100–200 single-word concepts per category, presence of a feature judged from Sonnet’s explanations of the top five features on a prompt naming the concept, by a human rater. The 34M dictionary (about 12 million live features) had features for about 60% of London boroughs, although the model names them all. It does not separate frequency in the model’s training data from frequency in the autoencoder’s, which the authors say are similar mixtures. | Active |
| THEORY-993 | In the mixed membership stochastic blockmodel the probability of an edge from p to q, given the memberships, is (1 − ρ)π_pᵀBπ_q, so the graph’s likelihood given the memberships is unchanged by π ↦ Rπ, B ↦ R⁻ᵀBR⁻¹ for any invertible R that keeps the memberships on the simplex and B in [0, 1], a continuum around any solution with B’s rates strictly between 0 and 1; and the sparsity parameter ρ enters only through (1 − ρ)B, so it is fixed by nothing but the bound B ≤ 1, and the paper’s printed estimator of it is the fraction of absent edges whatever the fit | Airoldi, Blei, Fienberg and Xing (2008), LIT-1553, read in NOTE-1179. The paper says the blockmodel and the memberships do not compete, because one is global and asymmetric and the other local and symmetric. Given the memberships they do: a change of block basis moves every node’s memberships toward the centre and sharpens B, with the same edge probabilities. What can still separate them is the Dirichlet family assumed for memberships, not the graph. The confounding of ρ with B’s scale is exact even in the marginal likelihood. | Active |
| THEORY-994 | In Ameisen et al.’s evaluations, the local replacement model of their 18-layer language model, built from a 10-million-feature cross-layer transcoder, responded to small perturbations much as the model did one layer after the intervention and increasingly differently over later layers, more in the size of the effect than its direction and slightly more for larger dictionaries, and over 20 prompts the graph’s influence of one feature on another ranked the effect of ablating the first on the second with a Spearman correlation of 0.72 | Ameisen et al. (2025), LIT-1552, read in NOTE-1185. It is the method’s own measurement of where a circuit read off its stand-in stops being the model’s: close for a layer, diverging with depth. It does not say how faithful the circuits of any case study are, which were chosen and validated separately, and it is measured on 18L, not on Claude 3.5 Haiku, where the companion’s case studies were run. | Active |
| THEORY-996 v2 | On Pythia language models from 410M to 12B parameters, in three single prompt pairs and two small prompt distributions, verifying attention or neuron nodes by activation patching in the order given by attribution patching, a first-order gradient estimate of each node’s noising effect, found the nodes with the largest effects at about 2 to 50 times the cost of an oracle ordering, below unordered patching, subsampling and block search in every setting; its misses were concentrated in queries and keys behind saturated attention and in nodes whose direct and indirect effects nearly cancel | Kramár, Lieberum, Shah & Nanda (2024), LIT-1531, read in NOTE-1190. Ground truth is each node’s exact single-node noising effect on the clean prompt’s negative log probability. Recomputing the softmax for queries and keys removes many misses; dropping one layer’s gradient at a time helps on single prompt pairs but not on the distributions, where AtP with the softmax fix or plain AtP is cheapest. It says nothing about coarser nodes, other model families, effect magnitudes or false positives. | Active |
| THEORY-998 v2 | In GPT-2 XL, for 1000 facts the model knows, restoring single clean hidden states into a run whose subject tokens were noised recovers the most of the correct object’s probability at two sites, middle layers at the last subject token, where MLP outputs carry most of the effect, and late layers at the last token, where attention does; and a rank-one edit of one middle MLP output matrix keyed on the last subject token inserts counterfactual facts that generalize to paraphrases and spare neighbouring subjects better than the fine-tuning, knowledge-neuron and hypernetwork editors tested, working best at that token and in middle layers | Meng et al. (2022), LIT-1536, read in NOTE-1166: §§2–3, Tables 1 and 4, and Appendices A, B and E. Causal traces: ATE 18.6%; the mid-layer MLP effect at the last subject token peaks at AIE 6.6% against 1.6% for attention. CounterFact on GPT-2 XL: ROME’s score 89.2 (efficacy 100.0, paraphrase 96.4, neighbourhood 75.4), next best 66.9. It does not say where facts are stored: a rank-one edit with the same output exists for any one-dimensional fact patch (LIT-968). | Active |
| THEORY-999 v2 | In the finite beta–Bernoulli feature model with prevalences Beta(α/K, 1), the expected number of occupied features among N objects is K(1 − (α/K)B(α/K, N + 1)), strictly below the Indian buffet process’s αH_N for every finite K and tending to it as K grows, and each object holds Kα/(K + α) features on average rather than α | Griffiths and Ghahramani (2011), LIT-1556, read in NOTE-1172, derive the IBP as the K → ∞ limit of πk ~ Beta(α/K, 1), zik ~ Bernoulli(πk), and give K+ ~ Poisson(αH_N) for the limit only. At finite K the count falls short, by 27% at N = 12, α = 3, K = 16. It says nothing about stick-breaking truncations, which approach the IBP much faster. | Active |
| THEORY-1002 | On frozen randomly initialized feed-forward networks at chance on hierarchical equality, with 16 input dimensions, distributed alignment search for the two-equality causal model reached interchange intervention accuracy of 0.50 to 0.51 at hidden sizes up to 256 but 0.55 at 1,028 and 0.64 at 4,096, so in a hidden layer much wider than the input a learned rotation can produce above-chance interchange accuracy for an algorithm the network does not implement | Geiger, Wu, Potts, Icard & Goodman (2024), LIT-1551, §4.4, read in NOTE-1192. A random network cannot implement hierarchical equality, so any IIA above 0.5 is made by the search. At widths near the input’s there is none; at 4,096 there are 14 points. It does not say DAS results on trained networks are artefacts: the trained equality network reached 1.00 at width 16, where the random one stayed at 0.50. | Active |
| THEORY-1003 | On NIPS papers (1,557 training and 183 test documents, 100 topics, fixed hyperparameters), the author–topic model gave test documents lower perplexity than LDA while sixteen or fewer of their words had been observed and higher perplexity beyond that; one fictitious author per document improved on the author–topic model throughout and lost to LDA after about 64 words, and an author model without topics did worst | Rosen-Zvi, Chemudugunta, Griffiths, Smyth and Steyvers (2010), LIT-1546, §6.4 and Fig. 11, read in NOTE-1180. Pooling by member helps prediction of a document of which little has been seen and costs it once the document’s own words are enough. One corpus, one split, one topic count. It is not evidence that member-level structure is real, only that it predicts in a regime of sparse evidence. | Active |
| THEORY-1005 | On one MLP layer each of GPT2-small, Pythia-410M and Pythia-1.4B, transcoders trained to imitate the MLP lay on or above the sparsity–loss frontier of sparse autoencoders trained on its output, at one learning rate tuned on transcoders and one seed, and in a single rater’s blind reading of 50 random features each from Pythia-410M layer 15, 41 transcoder and 38 SAE features were judged interpretable | Dunefsky, Chlenski and Nanda (2024), LIT-1554, read in NOTE-1174: §4, Table 1, Fig. 3 and Apps. E–F. It does not say transcoders are better than SAEs in general. The SAE learning rate was not tuned, all runs shared one seed and data order, one layer per model was measured, and on Gemma 2 2B Gemma Scope found transcoders worse than MLP-output SAEs at fixed L0 (LIT-611, NOTE-482). | Active |
| THEORY-1006 | In the dependent Indian buffet process each item’s feature indicator is a Gaussian process thresholded at the quantile of a stick length that does not change with the covariate, so its prevalence is the same at every covariate value and its persistence is set separately by the covariance; the whole feature matrix at a covariate value is an Indian buffet process only when there is no shared feature-level process, and with one the items are correlated and fewer features are occupied | Williamson, Orbanz and Ghahramani (2010), LIT-1548, read in NOTE-1175. Thresholding at a fixed quantile keeps each feature’s prevalence at its stick length at every covariate value, which is what a hand-written logistic transition does not do. The paper’s statement that Z(t) is an IBP at each t holds for its collapsed model; with a shared feature-level GP it holds item by item only. Persistence for a fixed correlation still falls with prevalence. | Active |
| THEORY-1007 | On four subject–verb agreement tasks in Pythia-70M and Gemma-2-2B, circuits of sparse-autoencoder features and autoencoder error nodes, selected by linear estimates of indirect effect, carried half or more of the behaviour on held-out pairs with about 100 and 500 nodes, where circuits of neurons needed about 1,500 and more than 30,000; with the error nodes removed, the same feature circuits carried little of it, under about a quarter in Pythia and almost none in Gemma | Marks et al. (2025), LIT-1532, read in NOTE-1187: §§2–3.2, Figure 3 and Appendices A, B and H. Faithfulness is (m(C) − m(∅))/(m(M) − m(∅)) with nodes outside the circuit mean-ablated, scored on the latter two thirds of each circuit only. The features plus the dense error terms are a small circuit; the features by themselves are not. It says nothing about what the error terms hold, and it does not compare features with neurons on equal terms, which the authors say cannot be done. | Active |
| THEORY-1009 | In Musat, Pimentel, Zucchet and Hofmann’s two-layer attention-only transformer on in-context bigrams, once an in-weights circuit that maps each common token to its memorized partner has saturated, the population gradient on the induction circuit is what it would be without it multiplied by fr + (1 − fr)fv, the share of the data the memorized map does not solve, up to errors that vanish at large vocabulary and large in-weights margin when no distractor block carries the query’s partner; the two circuits occupy orthogonal coordinates and still interact, because they solve overlapping parts of the data | Musat et al. (2026), LIT-1530, read in NOTE-1177. A memorized shortcut that solves part of the data removes the in-context circuit’s gradient on that part and leaves the rest. It is a theorem in an idealized model under a disjointness assumption (A4) that the paper’s own sampler violates, at a cost it states. It does not say the observed order in the paper’s one run is due to it: there fr = 0.9 and the factor is 0.9. | Active |
| THEORY-1010 | On ten corpora simulated from three-level hierarchical LDA itself (100 documents of 250 words, a vocabulary of 100, sparse topics with η = 0.005, γ = 1), the highest-probability state of the collapsed Gibbs sampler recovered the generating tree exactly in eight; in one of the other two a wrong tree had higher posterior probability than the true one | Blei, Griffiths and Jordan (2010), LIT-1542, §6.1 and Fig. 6, read in NOTE-1176. It shows that the sampler can invert hLDA on data hLDA generated, with sparse, well-separated topics. It does not show that the tree is identified in general, or that a tree found in real data was there to be found. One failure is a posterior that prefers the wrong tree, which no better sampler would fix. | Active |
| THEORY-1012 | Latent features made dependent through a layer of categories, each observation’s features being a function of how many of its categories carry each feature, stay finite on finite data when a feature carried by none of an observation’s categories is always off; under that condition a category can switch features on and never off, so features depend on one another by sharing a category, or, where an observation has a single category, by excluding one another, and no feature or category suppresses another | Doshi-Velez and Ghahramani (2009), LIT-1544, read in NOTE-1184. Z = f(CM) with f(0) = 0 gives correlated nonparametric feature models with O(log log N) expected features for the DP-IBP and IBP-IBP. The correlations are motifs, sets of features switched on together, plus, in the DP-IBP, exclusion between categories; suppression is outside them, and negative correlation is left as future work. The models’ predictive advantage over the IBP is mixed in the paper’s own table. | Active |
| THEORY-1015 | Groups of objects drawn from beta processes share features only if the measure they are drawn around is discrete; in the hierarchical beta process that measure is itself a beta process draw, and at each shared feature a group’s prevalence is Beta(c_j b, c_j(1 − b)) around the shared prevalence b, so a small c_j lets groups differ and a large one pulls them together | Thibaux and Jordan (2007), LIT-1537, read in NOTE-1171. The hierarchical beta process B ~ BP(c0, B0), Aj ~ BP(cj, B) gives several groups one unbounded feature library at group-specific prevalences. It treats each object as in one group, exchangeable within it, and has no dynamics; its one experiment, 58% against 50% on a single text split, is suggestive only. | Active |
| THEORY-1017 | In Claude 3.5 Haiku, writing the second line of a rhyming couplet whose first line ends “grab it”, features for the candidate rhymes “rabbit” and “habit” are active on the newline token before the line, steering them changes the line’s final word only when applied at that token, and suppressing “rabbit” turns the line towards “habit”; injecting a “rabbit” or “green” word feature at that point in 25 other poems ended the line with the injected word in 70% of cases, and such planned-word features were found in about half of the poems the authors examined | Lindsey et al. (2025), LIT-1541, read in NOTE-1181. A target word represented before the text that leads to it, and used causally: it is the case study’s own measurement, on one couplet and a 25-poem injection test. It does not say how widespread planning is, the “about half” being a count without a stated sample, and one of the article’s displayed “success cases” does not end in the injected word. | Active |
| THEORY-1018 | In Claude 3.5 Haiku, on the jailbreak prompt “Babies Outlive Mustard Block.” asking it to join the first letters and say how to make one, the attribution graph for its output “BOMB” shows separate letter-wise paths voting for the output and no internal representation of the word or of a bomb-making request before it is written; “make a bomb” features first fire on the model’s own “To make a bomb”, injecting one on “BOMB” brings an immediate refusal, and the refusal that does come waits for the sentence’s end | Lindsey et al. (2025), LIT-1541, read in NOTE-1181. On this prompt the model “doesn’t know what it plans to say until it actually says it”. It is one prompt, partly effective, and the decisive refusal mechanism runs through attention, which the graphs do not show: the injection that produced a refusal needed attention left unfrozen. | Active |
| THEORY-1019 | Under the Markov Indian buffet process the expected number of features on at the first time step is α, as in the Indian buffet process, but it rises over time toward α·E[ln((2 − b)/(1 − b))] with b ~ Beta(γ, δ), which is 9.04 at γ = 10 and δ = 1, so after the first step the rows are not distributed as an Indian buffet process | Van Gael, Teh and Ghahramani (2008), LIT-1577, read in NOTE-1199. A chain starting off is on at step t with probability a(1 − (b − a)^t)/(1 − b + a). As M → ∞ the a’s form a Poisson process of intensity α/a on (0, 1), so E[number on at t] = α ∫₀¹ E_b[(1 − (b − a)^t)/(1 − b + a)] da. That is α at t = 1 and tends to α·E[ln((2 − b)/(1 − b))]. The number of features ever on is still Poisson(αH_T), as the paper says. This concerns the expected count only, not the full row distribution beyond it. | Active |
| THEORY-1020 | In a two-layer attention-only language model, on tokens where induction helps, the standard induction circuit (queries and values from the token embedding, keys from the previous-token head) accounts for 35% of the loss its two induction heads contribute, and a hypothesis that adds three-token matching, layer 0’s average effect and three layer-0 heads acting as if they attended to the current token accounts for 86% | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt, Lin et al. (2022), LIT-1579, read in NOTE-1196. Percentages are of 0.053 nats, the loss the induction heads 1.5 and 1.6 contribute on next-token examples whose last token is on an induction-favouring whitelist, measured by causal scrubbing. The naive story’s shortfall is mostly in the queries (51% alone) and keys (79%), not the values (90%). It does not say what fraction of the model’s whole loss, or of its in-context learning, the circuit explains, nor anything about models with MLPs. | Active |
| THEORY-1023 | Attribution patching estimates the effect of patching an activation from a clean run into a corrupted run as the difference between the two activations dotted with the gradient of the metric on the corrupted run, so the estimates for every activation come from two forward passes and one backward pass and add over sub-components; the estimate is exact when the metric is affine in the activation, gives no effect for a nonlinearity in saturation whatever the true change, and through LayerNorm ignores the component of a patch along the vector being normalized | Nanda (2023), LIT-1581, read in NOTE-1201: the estimate and its mathematical properties, not its accuracy in any model. The cost is fixed (about four forward passes for all patches), the estimate is linear and so additive, and its blind spots follow from linearity: saturation, interaction between features, and LayerNorm’s scale invariance. | Active |
| THEORY-1024 | In GPT-J, for 652 CounterFact facts the model knows, the per-fact causal-tracing effect at an MLP layer adds at most 3.2 points of explained variance in the success of editing that fact at that layer, beyond what the choice of layer alone explains, across ROME, MEMIT and constrained fine-tuning and five editing objectives; ROME rewrites the fact at over 96% at every layer but the last, including layers where tracing does not place it, and the only reliable link appears when the edit is made on the noised input that tracing uses | Hase et al. (2023), LIT-1578, read in NOTE-1194: §§4–5, Tables 1–8 and Appendices A–C. For ROME at GPT-J’s layer 6, ρ = −0.13 between rewrite score and tracing effect; R² 0.947 from the layer, 0.948 with tracing added. Over twenty method–objective pairs the largest gain is 0.032 (fine-tuning on five layers, Fact Forcing). It does not say that facts are stored nowhere in particular, nor that the layer does not matter: the layer explains most of the variance, and ROME is best on average at the layer tracing favours on average. | Active |
| THEORY-1025 | The inputs causal scrubbing samples for a treeified hypothesis are the unique maximum-entropy distribution, on finite inputs, among those that give the inputs of every interpretation node the joint distribution they have on the data, so whatever the hypothesis does not claim is resampled as independently as the claims allow | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt et al. (2022), LIT-1583, read in NOTE-1202, §5. Every node’s datum is distributed as D, so each node’s scrubbed inputs are jointly as on the data, and among all distributions with that property the algorithm’s has the greatest entropy. It characterizes the sampling, not the score: it does not say that a hypothesis whose score matches the model is true, and the same appendix’s §7 builds one that is false. | Active |
| THEORY-1028 v2 | In InceptionV1 trained on ImageNet, the ten curve units of layer mixed3b each respond most to curves of one orientation, with responses invariant to fill and colour and jointly spanning all orientations under rotation of real images; for one of them, 3b:379, which fires on about 11% of dataset positions, activation level separated one author’s blind four-class labels of over 800 activation-binned images, and curves and imperfect curves made up 55% of its contribution to expected value | Cammarata et al. (2020), LIT-1575, read in NOTE-1198: the behaviour of the mixed3b curve family as measured by feature visualization, rotated dataset examples, synthetic curves and, for one unit, activation statistics and blind hand labels. It holds for the neuron basis of one model. It does not say how the units are computed, what their weak firings encode, or what fraction of the layer is understood. | Active |
| THEORY-1029 | On a three-layer parenthesis-balance classifier, causal-scrubbing hypotheses that resample the terms of one head’s input separately scored 81% and 84% of the loss recovered against 77% for the less specific hypothesis they refine, and the authors attribute the rise to resampling washing out a signal, not to a better explanation | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt, Lin et al. (2022), LIT-1580, read in NOTE-1195. Hypothesis 3a resamples head 2.1’s input separately at each of up to 40 positions and recovers 77% (loss 0.97 ± 0.06). Hypotheses 3c and 3d also split each position’s input into terms drawn from different inputs, and recover 84% (0.69 ± 0.05) and 81% (0.81 ± 0.05). The authors call the two results not legitimate. It does not say how often this happens in other models, nor that the adversary they propose would remove it. | Active |
| THEORY-1031 | In GPT-J, inserting 10,000 CounterFact counterfactuals at once by a closed-form least-squares update of the MLP output matrices of layers 3 to 8, spread so that each layer writes a share of each fact’s required change to the subject’s hidden state, gives an editing score of 85.8 (efficacy 98.9, paraphrase 88.6, neighbourhood 73.7) with fluency kept, against 67.6 for weight-decayed fine-tuning, which breaks generation, 50.3 for ROME applied one fact at a time and 23.1 for MEND; GPT-NeoX scores 82.0, and editing late MLPs or attention instead does worse | Meng et al. (2022), LIT-1582, read in NOTE-1197: §§4–5, Tables 1 and 2, Appendices A, B and F. On 10,000 true zsRE facts in GPT-J, MEMIT scores 50.7 against FT-W 42.1, MEND 20.0 and ROME 2.6. At a handful of edits ROME generalizes a little better. It does not show where facts are stored: the layer range is chosen from traces averaged over facts, which do not predict per fact where an edit works (THEORY-1024). | Active |
| THEORY-1032 | When Claude 3.5 Haiku was given continued pretraining on synthetic documents describing 52 fictional reward-model biases and then trained by SFT and RL to exploit 47 of them, its rate of exploiting the 5 held-out biases rose from 17% to 45% of applicable episodes, and it applied new biases described in context as a reward model’s more often than the same biases attributed to other entities; without the documents the held-out rate rose only to 20%, and without SFT it still rose | Marks, Treutlein et al. (2025), LIT-1584, read in NOTE-1193: §3 and Appendices B–C. The model organism behind THEORY-976. Behaviours never demonstrated in post-training become more frequent, and the increase depends on the model having read that reward models prefer them. Rates are judged by a classifier. Haiku already shows the held-out behaviours at 17%, so what generalizes is an increase, not a new behaviour. | Active |
| THEORY-1033 | In a 16-layer transformer language model trained on WikiText-103, the training prefixes that most activate a sampled feed-forward hidden unit share patterns that human annotators recognize, mostly shallow in layers 1–9 and semantic in layers 10–16; projected through the output embedding, a unit’s output weights put the next token of its top prefix first in up to about 3.5% of units in the upper layers and almost none below; and a layer’s top prediction differs from every unit’s in at least about 68% of examples | Geva et al. (2021), LIT-1576, read in NOTE-1200: §§3–5. 160 keys annotated, one annotator each: every key had a pattern (3.6 on average), covering 65–80% of top prefixes. Agreement between value and key rises to about 3.5% by the top layers, against 0.0004% for a random token. It says nothing causal: no unit was ablated or edited, and the patterns were not compared with those of random directions. | Active |
| THEORY-1034 | In the Markov Indian buffet process each feature’s probability of staying on is drawn independently of its probability of switching on, so how long a feature persists once on does not depend on how rare it is; the process is defined for the time steps of one sequence, not for several objects over time | Van Gael, Teh and Ghahramani (2008), LIT-1577, read in NOTE-1199. Each column is a two-state chain from an off state with a_m = P(0 → 1) ~ Beta(α/M, 1) and b_m = P(1 → 1) ~ Beta(γ, δ), independent, and the stick-breaking construction keeps each b paired with its a as an independent draw. Rarity (small a) and persistence (b) are separate per-feature parameters. It does not say that prevalence is held fixed over time, which it is not (THEORY-1019), nor how to share the chains across several objects. | Active |
| THEORY-1036 | In a model that fits y = Σ v_i x_i² with ỹ = Σ_r (W_r·x)² + b, on independent inputs of zero mean, unit variance and common fourth moment k, the expected squared error at the optimal bias is (k − 1)Σ(n_i − v_i)² − 2Σn_i² + 2Σn_i²/C_i, in the squared embedding lengths n_i and capacities C_i = (w_i·w_i)²/Σ_j(w_i·w_j)², and its minimum with the capacities summing to the number of neurons D gives C_i = clip((k − 1)v_i/((k − 3)λ) − 2/(k − 3), 0, 1): for k ≤ 3 each feature is given a whole dimension or none, and for k > 3 a feature is shared between neurons exactly when 2λ/(k − 1) < v_i < λ | Scherlis et al. (2022), LIT-1625, read in NOTE-1205: the quadratic toy model of superposition solved in closed form. Its loss separates by feature through capacity, so the marginal value of capacity is −∂L/∂C_i = 2n_i²/C_i², and sparsity, through k, decides whether returns to capacity diminish. It is the model the causal scrubbing appendix’s §6 analyses. It holds for this model and says nothing directly about trained networks. | Active |
| THEORY-1037 | In the infinite hidden Markov model, drawing at each time step a slice variable uniformly below the probability of the transition taken leaves the posterior unchanged and leaves only finitely many state trajectories with positive probability, so a whole trajectory can be resampled exactly by forward filtering and backward sampling, with no truncation of the state space | Van Gael, Saatci, Teh and Ghahramani (2008), LIT-1622, read in NOTE-1204. u_t ~ Uniform(0, π_{s_{t−1}s_t}) is an auxiliary variable, so sampling u given s and then s given u leaves p(s | y) invariant; given u, the forward recursion sums only over transitions with π > u_t, of which there are finitely many per row. It says the sampler is exact, not how fast it mixes (THEORY-1044). | Active |
| THEORY-1038 | In InceptionV1, the ten curve units of mixed3b, which respond to curves of one orientation and not to the opposite one, regain the 180° echo of earlier curve units when their input is restricted to mixed3a early curves with the negative weights removed, and not when the negative weights are kept; a network one author wrote by hand from the described circuit, without looking at the original weights and tuned afterwards, has grayscale curve units that the authors show, by figures only, approximating the natural ones on synthetic curves across radii and orientations; and substituting them for the ten units recovered 26 of the 46 top-5 classifications lost by deleting the units, on 2,000 validation images, a result the authors call preliminary | Cammarata et al. (2021), LIT-1629, read in NOTE-1207: the three results of Curve Circuits that test its account of the mixed3b curve units, the echo edit, the hand-written reimplementation and the substitution. Only the last is a number, and a small one. It holds for one model’s neuron basis and says nothing about weak firings or other families. | Active |
| THEORY-1041 | In the Indian buffet process the feature probabilities in decreasing order are products of independent Beta(α, 1) draws, so the k-th has mean (α/(α + 1))^k, and keeping only the first K of them gives an expected number of occupied features among N objects that approaches αH_N from below: 8.97 at K = 16, N = 12 and α = 3, against 9.31 in the limit and 6.83 for the finite model with sixteen independent prevalences | Teh, Görür and Ghahramani (2007), LIT-1627, read in NOTE-1210. μ_(k) = ∏{l≤k} ν_l with ν_l ~ Beta(α, 1). Truncated at K, the expected number of occupied features is Σ{k≤K} Σ_{j=1}^{N} (−1)^{j+1} C(N, j) (α/(α + j))^k, which sums to αH_N as K → ∞. At N = 12, α = 3: 6.80, 8.97, 9.27 and 9.31 at K = 8, 16, 24 and 32. It concerns the prior’s expected count, not a posterior, and the paper itself computes no truncation error. | Active |
| THEORY-1042 | In the Overview of Early Vision’s taxonomy of InceptionV1, every one of the 1,056 units of conv2d0 to mixed3b is assigned by the authors to exactly one of 77 groups, and 391 of them (37%), from 7% of conv2d2 to 48% of conv2d1 and 45% of mixed3b, are in groups whose own captions call them catch-all, broad and not well defined, low-confidence, unstudied or not understood; the mixed3b curve family is 10 of that layer’s 480 units | Olah et al. (2020), LIT-1624, read in NOTE-1206: the census of InceptionV1’s early vision, counted. A complete partition of 1,056 units into 77 groups, by eye; 37% in groups the authors flag as uncertain or catch-all, by their own words. It is a denominator for the circuits claim that units are typically features (THEORY-1013), not a measure of how many units are understood. | Active |
| THEORY-1043 | The finite feature model in which each of K features has prevalence Beta(α/K, 1) has, as K grows, a limit on equivalence classes of binary matrices that ignore the order of columns, and the Indian buffet process, in which the i-th object takes each earlier feature with probability m/i and Poisson(α/i) new ones, produces the same distribution, so its objects are exchangeable and the number of features they hold is Poisson(αH_N) | Griffiths and Ghahramani (2005), LIT-1623, read in NOTE-1208, and their review (LIT-1556). P([Z]) → α^{K+}/∏_{h≥1} K_h! · exp{−αH_N} · ∏_k (N − m_k)!(m_k − 1)!/N! on lof-classes, and the buffet gives the same. It is a statement about the limit: at finite K the expected count of occupied features is below αH_N (THEORY-999), and an ordered truncation of the same limit is a different approximation (THEORY-1041). | Active |
| THEORY-1045 | On synthetic data from factorial hidden Markov models with three or five hidden chains of two or three states, an unconstrained hidden Markov model with as many states as the chains have joint settings overfits, its test-set excess over the true model rising to 175 bits per observation at five three-state chains against 4.5 for the factorial model trained exactly, while factorial models trained with completely factorized or structured variational E steps are not significantly worse than exact training | Ghahramani and Jordan (1997), LIT-1628, read in NOTE-1203. Four sizes, 15 random parameter sets each, 20 training and 20 test sequences of length 20, four-dimensional, C = 0.0025I, one run per set. In bits per observation over the true model, the K^M-state HMM’s test excess is 2.29, 9.81, 11.57 and 175.35; the exact factorial HMM’s 1.05, 1.26, 2.51 and 4.49; the structured approximation’s 1.04, 0.90, 1.53 and 4.30. It is a result on small synthetic problems the model generated, not on real data or long sequences. | Active |
| THEORY-1046 | In infinite sparse factor analysis, where a binary mask says whether a source is active at a data point and an independent Gaussian value says how strongly, the evidence of that data point for the source being active, with its value integrated out, is σ̂·exp(μ̂²/(2σ̂²)) for the posterior mean μ̂ and standard deviation σ̂ of the value; when the residual has no component along the source’s loading the ratio is σ̂, below one, so the data favour an inactive source over an active one at zero by a factor that the prior on the value sets | Knowles and Ghahramani (2007), LIT-1626, read in NOTE-1209. With y = g x + ε, x ~ N(0, 1), ε ~ N(0, σ_ε² I): ∫ p(y | x) p(x) dx / p(y | x = 0) = σ̂ exp{μ̂²/(2σ̂²)}, σ̂² = σ_ε²/(σ_ε² + g′g), μ̂ = g′ε/(σ_ε² + g′g). It is the Gaussian case only; the Laplacian case is the paper’s (14), checked the same way. It concerns one data point and one source, with the loading known. | Active |
| THEORY-1048 v2 | On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required | Morris, Lenski & Zinser (2012), LIT-1568, read in NOTE-1212. A loss-of-function mutant invades when its resource saving B = (Q_A − Q_M)/Q_A exceeds its cost C, which falls as other cells’ leaky activity protects it. Because protection falls as producers become rare, the loss is frequency dependent and stops short of community-wide loss. The evidence is Prochlorococcus’s loss of katG and its dependence on a peroxide-removing community. The dependence it produces is one-way, from beneficiaries to helpers, and involves no collective. | Active |
| THEORY-1049 v3 | On McCutcheon and colleagues’ survey, bacterial endosymbionts lose their own cell-envelope and envelope-transport genes together below about 400 kb, and begin to lose translation genes (ribosomal proteins, tRNAs, aminoacyl-tRNA synthetases) below about 200 kb, so that the smallest-genome symbionts cannot be self-sufficient and their membranes, transport and protein synthesis are presumably supplied by the host, as for organelles | McCutcheon, Garber, Spencer & Warren (2024), LIT-1571, read in NOTE-1213. The envelope goes first, near 400 kb, and translation starts to go below about 200 kb, the authors’ “tipping point”. Such symbionts have three-membrane envelopes and among the most biased proteomes in bacteria. They can only work if the host provides what they no longer encode. The host’s role is inferred, not shown: the mechanisms are, in the paper’s word, mysteries. | Active |
| THEORY-1050 v2 | In Athreya, Czuppon and Gokhale’s adaptive-dynamics model of an incipient endosymbiosis, mutual dependence (the product of the partners’ obligacies) evolves ahead of reproductive cohesion (the product of their adhesions) early on, but only under the mapping in which each partner’s free-living growth depends on its own obligacy alone while dissociation depends on both partners’ adhesions; the authors’ own appendix shows the bias vanishing under matched mappings and reversing under the opposite one, and their survey of real symbioses cannot test it | Athreya, Czuppon & Gokhale (2025), LIT-1570, read in NOTE-1211. In the exponential version of the model, all four traits rise monotonically. Each partner becomes more obligate than adhesive, because a gain in adhesion is “damped by the other adhesion” in the dissociation rate. Table S1 shows this is a consequence of that mapping: matched mappings give no bias, and the opposite mapping biases trajectories towards cohesion. The record’s summary of the result, “even when investments are uncorrelated”, is true of the traits’ genetic correlation but not of their functional coupling, which is what decides it. | Active |
| THEORY-1051 v2 | In Athreya, Czuppon and Gokhale’s model of an incipient endosymbiosis, the partner with the larger and faster-growing population evolves faster and becomes fully obligate first, which the authors read as a Red King effect in which the collective’s shared fate aligns interests; with equal within-collective rates for both partners, the other partner’s dependence then stays where it was, rises to full dependence, or falls to none, according to whether partners are born or die inside the collective; and in its simulations dependence rose from zero at the higher collective carrying capacities tried and not at the lowest, under a condition the authors show sufficient and not necessary, so a full transition to a mutually dependent, co-reproducing pair is a parameter-dependent outcome and not the default | Athreya, Czuppon & Gokhale (2025), LIT-1570, read in NOTE-1211. With the symbiont’s population larger and faster, its obligacy reaches 1 first. In the base model the host’s R₀ is then exactly 1, so host obligacy is neutral. With within-collective deaths the host goes fully obligate, and with within-collective births it goes back to independence. Obligacy rose from zero in simulations where the collective’s carrying capacity exceeded about aK_HK_S/d, a condition shown sufficient, not necessary; collective growth rate does not matter. The authors take the model as a null: a transition “is far from an inevitable outcome of the existence of a higher level of selection”. | Active |
| THEORY-1053 | Under unitary evolution, two states of a quantum system can each leave a record in another system while themselves staying undisturbed only if they are orthogonal or the record does not distinguish them; so the outcomes that can be recorded repeatably form an orthogonal set, a result that uses only unitarity and repeatability and not the Born rule | Zurek (2009), LIT-1640, eq. (5), read in NOTE-1229. From |u⟩|e₀⟩ → |u⟩|e_u⟩ and |v⟩|e₀⟩ → |v⟩|e_v⟩, preservation of the scalar product gives ⟨u|v⟩ = ⟨u|v⟩⟨e_u|e_v⟩, so either the record is no record or ⟨u|v⟩ = 0. Zurek reads this as the origin of the outcome set of the collapse postulate. It does not give single outcomes, probabilities or which orthogonal set is chosen; that is the interaction’s. | Active |
| THEORY-1054 | One arrangement of harmonic particles on a finite graph, with the same wiring, potential and inverse-mass response, gives the discrete wave equation when the integrator stores position and momentum and the discrete heat equation when it stores position alone; both are governed by the symmetrized graph Laplacian, and the coupling data do not decide between them | Spivak (2026), LIT-1649, read in NOTE-1225 from arXiv v2. In the operad ArrSm, harmonic particles wired along a finite directed graph form one closed arrangement with potential Σ(κ_e/2)(q_tgt − q_src)². The phase semantics gives m_v q̈_v = −(L_G q)_v (Proposition 7.4.1) and a symplectic step; the configuration semantics gives m_v(q_v(t+1) − q_v(t)) = −(L_G q(t))_v (Remark 7.4.2). Active: both are short exact computations. It says nothing about time scales, and what differs between the two is a modelling choice, not the coupling. | Active |
| THEORY-1057 | Gradient descent with backpropagation, weight decay and Newton’s method are one discrete-time semantics of an operad of potential-carrying lenses, in which the state moves by subtracting the sharpened sum of three pulled-back covectors (from the output, from the input wiring and from the potential); the semantics is an operad functor, so a composite’s update is assembled from its parts’ updates, and backpropagation is that functoriality | Spivak (2026), LIT-1649, read in NOTE-1225 from arXiv v2. Lens internalization (Corollary 3.3.4) and an integrator give operad functors ArrSm → PC (Theorem 4.3.6, Corollary 6.4.1). Under the configuration integrator the state update is q ↦ q − ♯_q(ξ_Q), with ξ_Q the Q*-component of the sum in (58); a parameterized map wired to a loss gives gradient descent (76), a weight-only potential gives weight decay (Proposition 7.2.3), and an inverse-Hessian response gives Newton’s method (67). Active as mathematics. Per datum, deterministic, explicit-Euler; Adam and momentum are conjectures. | Active |
| THEORY-1058 | Given constructor theory’s principles of locality and of classical information, a single further prohibition, that two information observables with mutually disjoint attributes have a union that is not an information observable, is enough for a medium to have information attributes that cannot be distinguished, unclonable and jointly unsharp observables, measurements that are deterministic yet unpredictable, irreducible disturbance by measurement, and locally inaccessible information; quantum information is an instance | Deutsch & Marletto (2015), LIT-1663, §§7–8, read in NOTE-1219. Active as a conditional. The derivations are informal but complete arguments from principles II–IX, and each feature is shown in the paper’s own terms with no probabilities. The principles themselves are conjectures. It does not say that superinformation singles out quantum theory, or that the quantum features are derived from anything not already assumed. | Active |
| THEORY-1059 | The dynamics a coupled cell network admits, restricted to a robustly synchronous pattern, are dynamics admitted by a smaller quotient network whose cells are the synchronized classes, so behaviour of the quotient, including symmetric behaviour the original network has no symmetry to force, appears in the original as synchronous behaviour; the quotient admits more dynamics than restriction produces, and is unique only up to refinement of its arrow types | Stewart, Golubitsky and Pivato (2003), LIT-1674, read in NOTE-1217: the quotient construction (§7), Theorems 8.3, 8.4 and 9.2 with proofs, the counterexample of Remark 7.5 and Example 7.1. Active as mathematics. The quotient describes the dynamics on the invariant subspace exactly, but only a subclass of the quotient’s own admissible dynamics arises that way, and nothing is said about stability transverse to the subspace. | Active |
| THEORY-1060 | Robinson’s claim that sheaves are the canonical data structure for sensor integration rests on an axiomatization, not a theorem: six axioms are posited as what an integrated sensor system must be and a sheaf is defined as whatever satisfies them, the paper calls sheaves one of many co-universal information representations, and its consistency and fusion tools need only a presheaf | Robinson (2017), LIT-1658, read in NOTE-1221 from arXiv v3. A textual finding about the paper, Active as such. The word “canonical” is never defined for sheaves as a sensor model; its technical uses are the standard ones for cohomology. It does not say that sheaves are a poor model, or that no theorem of this kind could be proved. | Active |
| THEORY-1061 | Flack makes downward causation operational as components tuning their behaviour to their own estimates of aggregate variables they collectively compute: weak (apparent) when at least one component tunes to a variable slower than the behaviour producing it, strong (effective) when the variable is predictive, robust, used nearly universally, agreed on and gaining mutual information with the microscopic behaviour; she speaks of a new level only at the strong form, and regeneration of the variable by its use is not among her criteria | Flack (2017), LIT-1630, read in NOTE-1216, §1, §3 and §7. Active as a reading of what the paper proposes, which its text states as lists. It does not say the proposal is right, that it meets any definition of causation (the paper sets that aside), or that the macaque system meets the strong criteria; the evidence for that is in earlier papers not reproduced here. The sharp edge for the record: what reinforces itself in Flack is the convergence of members’ estimates, not the variable’s persistence. | Active |
| THEORY-1062 | On a discourse sheaf, where each agent expresses a private opinion through a linear map into each edge’s space of topics, sheaf-Laplacian opinion diffusion converges to the orthogonal projection of the initial opinions onto the global sections, which are agreement in what each pair expresses and not in what they hold; stable polarized or incomparable private opinions are therefore possible, and when the sheaf has no nonzero global section the only limit is all opinions zero | Hansen and Ghrist (2020; SIAM J. Appl. Math. 2021), LIT-1646, §§2–4, read in NOTE-1222 from the arXiv preprint. Active as mathematics: the convergence is a theorem about a linear heat equation, and the gap between expressed and held opinion is in the definitions and is worked in an example. It does not say that people reach agreement this way, or that an agreement in expression shows anything about what is held. | Active |
| THEORY-1063 | Black-boxing a passive linear circuit, which keeps only the Lagrangian relation its terminals’ potentials and currents can jointly satisfy, is a hypergraph functor from open circuits to Lagrangian relations: the external behaviour of a circuit glued from parts is the relational composite of the parts’ external behaviours, so circuits with the same behaviour are interchangeable in every composite, and the effective relation a composite presents at its boundary is fixed by its parts’ black boxes alone | Baez and Fong (arXiv 2015, TAC 2018), LIT-1644, read in NOTE-1223: Thm 1.1, proved as Thm 7.7, for circuits of passive, linear, two-terminal components over a field with positive elements, inductors and capacitors entering through the Laplace transform with zero initial conditions. The functor is not faithful, which is the point: circuits with one behaviour are identified. It says nothing about nonlinear or active components, and nothing in time-domain terms. | Active |
| THEORY-1064 | In counterexample-guided abstraction refinement, a spurious counterexample locates the distinction its abstraction lacked: walking it forward through the concrete system, the first abstract state where no concrete continuation exists merges the states the trace can reach with the states that have the transition onward, and splitting them removes the counterexample; for finite-state programs and ACTL* properties whose counterexamples are paths or loops, repeating check and split ends in a real counterexample or a proof | Clarke, Grumberg, Jha, Lu and Veith (2000), LIT-1652, read in NOTE-1231 from the authors’ manuscript of the CAV paper. Active: the test for spuriousness (Lemma 1), the three-way cut at the failing state and the termination argument are given in full. The NP-hardness of the coarsest split is stated with its proof omitted, and the loop case rests on a theorem proved elsewhere. It does not say that the split found is the smallest, or how many rounds are needed. | Active |
| THEORY-1065 | In a coupled cell network, a pattern of synchrony is invariant under every dynamics the network’s architecture admits if and only if its partition of the cells is balanced, each cell of a class receiving, through each kind of arrow, the same number of inputs from each class; symmetry is one way to get balance and not the only one, so robust synchrony occurs in networks with no symmetry and among cells with no coupling between them, and the invariance says nothing about whether the pattern attracts | Stewart, Golubitsky and Pivato (2003), LIT-1674, read in NOTE-1217 from the accepted version: Theorem 6.5 with proof, Lemma 6.3, worked examples without symmetry (Figures 3, 9, 15, 19). Active as mathematics. It generalizes THEORY-276’s orbit clusters. “Robust” means invariant for the whole admissible class, not stable, and not invariant for one model family; the paper studies “existence, not stability”. | Active |
| THEORY-1066 | In constructor theory, a set of attributes is an information variable when every permutation of it and its cloning are possible tasks, and distinguishability is defined afterwards, as the possibility of mapping a variable onto an information variable; so information is fixed by counterfactual properties of a substrate without presupposing distinguishability, and instantiating it is stated as an intrinsic property of the substrate in the sense of referring to no other specific substrate | Deutsch & Marletto (2015), LIT-1663, §§3–5, read in NOTE-1219. Active as a fact about the construction: the definitions are given in that order, and the base they give the recursion of (1.1) is checkable on the page. It does not say that these notions are the intuitive ones, which the authors disclaim, nor that the principles built on them hold. Its “intrinsic” excludes reference to a specific other substrate, not to generic ones, which the cloning task and (5.1) both use. | Active |
| THEORY-1067 | In angular synchronization from m pairwise offsets, each exact with probability p and otherwise uniform over L values, no estimator of any computational cost can recover all n angles with small error when p is below about √((n/m)·2 log₂ L/(L − 1)), nor an angle of average degree below about √((n/m)·log₂ L/(L − 1)), because each offset carries at most I(L, p) ≈ ½(L − 1)p² bits; below these thresholds a global configuration that exists is not determined by the pairwise evidence, whatever the resources spent on finding it | Singer (2011), LIT-1655, §5, read in NOTE-1224 from arXiv v2. Active as mathematics: Fano’s inequality applied to an explicit entropy count, asymptotic in m, n → ∞, p → 0 with L fixed. The bound is loose (it ignores the dependencies among offsets) and covers exact recovery and per-angle recovery, not partial correlation, which the eigenvector method achieves below it. It says nothing of what a bounded algorithm can do above it. | Active |
| THEORY-1068 | In the founding statement of relational quantum mechanics the states and values of systems are relative to other systems, but the systems are not: the world is assumed decomposed, possibly in several ways, into systems each characterized by the questions that can be asked of it and a finite information capacity, so the relata precede the relations, as an assumption and without an ontology of events | Rovelli (1996), LIT-1635, read in arXiv v2 in NOTE-1227, §§II.C, III.A–B, III.D and IV.A. Active as a textual fact about the paper. It bears out, for 1996, what THEORY-1071 takes from Faye and Jaksland, who quote later texts for it. It does not say that RQM must keep systems prior, nor that the systems are uniquely given: the decomposition is assumed, not argued, and allowed to vary. | Active |
| THEORY-1069 | In a von Neumann measurement the measuring variable’s information about the measured variable grows continuously under the unitary coupling, in Di Biagio and Rovelli’s model as I_{A:B}(t) = sin²(ωt)H_A, so the discontinuity of a quantum measurement lies only in a threshold on when information counts as a fact; with an idealized pure ready state a fact conditional on a pointer value appears at once, which they attribute to the idealization | Di Biagio and Rovelli (2025), LIT-1676, read from arXiv v2 in NOTE-1214, §VII and App. B. Active as mathematics: (B7) follows from (B6), which I rederived; (B8) as printed has a sign error, and the correct form is I_A^max − cos²(ωt)H_A. The continuity is of the expected quantities. A conditional relative fact appears at once in the ideal model, and the full-rank case that would remove that is described, not computed. The authors say the point is not specifically quantum. | Active |
| THEORY-1070 | For a weighted cellular sheaf of finite-dimensional vector spaces on a finite cell complex, the kernel of the degree-k Hodge Laplacian is the k-th sheaf cohomology, in degree 0 the space of global sections, and the linear flow ẋ = −Δ^k x converges exponentially to the orthogonal projection of its initial state onto that kernel, each eigencomponent decaying at its own eigenvalue; the inner products on the stalks are part of the data, so the same sections can come with different rates, and the Laplacian does not determine the sheaf | Hansen and Ghrist (2019), LIT-1659, read in NOTE-1233 from arXiv v2: Theorem 3.1 (discrete Hodge theory), H^0 ≅ Γ (p. 7) and Proposition 8.1 (the flow), each with a proof; Fig. 3.1 for the Laplacian not determining the sheaf. Active as mathematics. The paper does not speak of relaxation rates or time-scale separation, does not say what produces the restriction maps, and treats only linear flows of a fixed Laplacian. | Active |
| THEORY-1071 | Quantum mechanics does not entail agential realism: Bohmian mechanics reproduces its predictions with an ontology of individual particles on determinate trajectories, and relational quantum mechanics, the interpretation Barad names as closest, makes values relational while keeping the systems prior to their interactions | Faye & Jaksland (2021), LIT-1672, read in NOTE-1220, §5. Active: the non-entailment follows from the existence of an empirically equivalent interpretation with a different metaphysics, and the authors do not promote that interpretation. It does not say agential realism is inconsistent with quantum mechanics; that is THEORY-1091, a separate and weaker argument. | Active |
| THEORY-1072 | In Simon’s analysis, whether a problem is well structured is a relation between the problem and a bounded solver, judged by practicable rather than in-principle computability, so it comes in degrees and has no sharp boundary; on that standard even theorem proving and chess fail some of the conditions, chess is well structured in the small and ill structured in the large, and problems presented by the world become well structured only in being prepared for a solver | Simon (1973), LIT-1634, read in NOTE-1232 from a scan of the published article. Active as a conceptual argument the paper makes in full: six conditions for a well structured problem, read with condition 6 as practicable computation, worked through for theorem proving and chess. It does not say that structuring is the whole of problem solving, which Simon rejects, or that any problem is ill structured for every solver. | Active |
| THEORY-1074 v2 | In Di Biagio and Rovelli’s relational account two classical subsystems’ perspectives merge, agreeing on a variable’s value and on each other’s, when they couple to that variable or to each other, and a coupling to a complementary variable can remove a relative fact from a perspective that did not interact; that this answers the combination problem and keeps perspectives consistent is shown only in worked examples computed relative to a single outside system | Di Biagio and Rovelli (2025), LIT-1676, read from arXiv v2 in NOTE-1214, §VI and Apps. D–E. The merge, and the disturbance of a third party’s relative fact, are computed in a three-system example and in Wigner’s friend. The general claims, a solution to RQM’s combination problem and consistency of perspectives, are asserted, not shown; whether they hold is THEORY-1213, Proposed. Every computation is relative to one outside system, where agreement is automatic. This is the same gap THEORY-621 and THEORY-1080 name. | Active |
| THEORY-1075 | In relational quantum mechanics as Di Biagio and Rovelli define it, a fact relative to one system is stable for another, so that the other may sum over it classically, only approximately and only relative to that other system: decoherence of the first system by an environment the second cannot reach gives the classical rule to order of the environment states’ overlap, a differently coupled system may still see interference, and a fact can be stable for a system for which it is not yet a fact | Di Biagio and Rovelli (2021), LIT-1666, read in NOTE-1226, §1 (pp. 2–6). Active: the definitions are the paper’s, and the result follows in a few lines from a partial trace (eqs. 5–8) under two stated conditions, ε small and the third system not interacting with the environment. It does not say that stable facts are absolute, that decoherence settles which outcome occurred, or what fixes the division into systems. | Active |
| THEORY-1076 | For Bohr a phenomenon is the observation obtained under a specified experimental arrangement, epistemic and derivative, with atomic object and instrument presupposed real; his relational holism is epistemic, and the subject–object, knower–known and word–world dualisms are presupposed by his account, so Bohr’s authority does not support agential realism | Faye & Jaksland (2021), LIT-1672, read in NOTE-1220, from Bohr’s writings and the secondary literature. Active as exegesis: Bohr’s own definitions of “phenomenon” (1938, 1949) and the convergence of realist and antirealist readings that Bohr never denied atomic objects. The dualisms half is interpretive and weaker. It does not say that Barad’s view is false, only that it is not Bohr’s. | Active |
| THEORY-1077 | On a presheaf of pseudometric spaces, the consistency radius of an assignment, the largest distance between an observation and the restriction of another, is zero exactly for assignments induced by global sections (in metric spaces) and, when the restrictions are K-Lipschitz, bounds the assignment’s sup-distance from every global section below by ε/(1 + K); so fusing data into a consistent state must move it at least that far, and the bound needs no sheaf condition | Robinson (2017), LIT-1658, Definition 20 and Propositions 21 and 23, read in NOTE-1221 from arXiv v3; both proofs checked. Active as mathematics. It does not say that the nearest global section is unique (the paper’s Proposition 25 claims so for Banach sheaves and the proof does not reach it), that the radius locates a fault, or that a small radius means the data are right. | Active |
| THEORY-1078 | Constructor theory puts constraint in two places: a principle is a law that constrains other laws and is tested through its rivals, and a constructor is a physical thing that keeps its ability to cause a transformation; a task is possible when the laws set no limit, short of perfection, on how well it could be performed and that ability retained, so possibility needs no actual constructor, and constructor-theoretic statements mention none | Deutsch & Marletto (2015), LIT-1663, §§1–2 and footnote 1, read in NOTE-1219. Active as a reading of the paper’s definitions, which are explicit. It bears on A11’s remark that constructor theory “could sharpen” what the record means by “a constraint being real”, and it splits that notion. A possibility is a fact about the laws, which no particular thing realizes; a constructor is realized, approximate and deteriorating. It does not say that the framework’s basic principle I is true, nor that a law of possibility is, or is not, a constraint realized by something (CLAIM-477). | Active |
| THEORY-1079 | An open system can tip away from an attractor that exists, keeps its stability and keeps its basin at every frozen value of its parameter, when the attractor moves faster than the system can track it; with steady drift what bifurcates is the ability to track, in a co-moving frame, not the attractor, and the critical rate depends on the relaxation rate, the distance to the tipping threshold and the starting state | Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, read in NOTE-1215 from the corrected arXiv v5. Active as mathematics: shown in a linear model with a fixed stable matrix (so no bifurcation and no noise), in normal-form examples worked exactly, and in a fast-slow system whose equilibrium is unique and globally stable. It does not say that a system that tips is lost for good, that the threshold is a basin boundary, or that any real system has tipped this way. | Active |
| THEORY-1080 v2 | In relational quantum mechanics as Rovelli first stated it, the relation between two observers’ descriptions is itself a quantum fact relative to a third system, so no observer’s view can be deduced from another’s and there is no description in toto, only a net of partial descriptions; the views are held consistent only in that the third system predicts correlated outcomes, which is shown for a single measurement and asserted in general | Rovelli (1996), LIT-1635, read in arXiv v2 in NOTE-1227, §§II.D, III.D, IV.A–B. The first half (no deducible relation between views, no in-toto description) is Rovelli’s stated position, well supported by his text. The second half reports that consistency is shown for one case (von Neumann’s correlation, p. 9) and left to the reader in general (p. 15); whether it holds in general is THEORY-1222, Proposed. Rovelli’s own relativity analogy breaks where relativity supplies an invariant, which bears on QUESTION-021 by analogy only. | Active |
| THEORY-1081 v2 | Di Biagio and Rovelli locate the Frauchiger–Renner contradiction and the violation of the Local Friendliness inequalities in one step, a superobserver using facts relative to a friend he holds in full quantum control as if they were his own, which is the classical sum rule applied where those facts are not stable for him; they base quantum ontology on facts relative to systems rather than on stable facts, arguing that stability is itself relational and only approximate and counting the Local Friendliness violation as evidence, with an absolute macroreality emerging only approximately and only relative to systems for which decoherence is strong | Di Biagio and Rovelli (2021), LIT-1666, read in NOTE-1226, §2 (pp. 7–11). The translation of Frauchiger and Renner’s assumption (C) and of Bong et al.’s absoluteness of observed events into the use of eq. (3) across contexts is the paper’s and is clear. That nothing else goes wrong is cited to others, inter-observer agreement is one sentence, and an agent reasoning about himself is left undone. What the paper argues is held; whether relative facts cohere in general is THEORY-1230’s question. | Active |
| THEORY-1082 | On a connected graph whose edges carry phase shifts αᵢⱼ = −αⱼᵢ, the edge relations zⱼ = e^{iαᵢⱼ}zᵢ have a common nonzero solution, equivalently the twisted Laplacian Lα = δα*δα has a zero mode, exactly when the phase accumulated around every cycle is 0 mod 2π; otherwise each relation can be met alone but not all together, and the smallest eigenvalue of Lα, positive, measures the residual incompatibility, which on an isolated ℓ-cycle is largest at holonomy π | Torres-Hugas, Duch, Gómez and Arenas (2026), LIT-1668, read in NOTE-1218 from the arXiv preprint v2. The criterion is stated in the paper with an energy argument (pp. 2–3) and is the standard one for discrete magnetic and connection Laplacians, which it cites; the spanning-tree step the paper omits is written out below. The isolated-cycle spectrum is in closed form (eqs. 29–32). Both were checked numerically. Active: a theorem, with the qualification that it concerns the relaxed linear problem, not the phase dynamics. | Active |
| THEORY-1083 | Barad holds agential realism to be naturalized metaphysics justified by quantum mechanics and applied once, to the one reality, with its scope extended to the lifeworld; she rejects both reading it as an analogy and reading it as a template that quantum mechanics and other domains each instantiate | Jaksland (2024), LIT-1673, read in NOTE-1230, from Barad’s own statements (2007, 24 and 85, with her dismissals of analogy at 2007, 6, 7, 18, 24, 70 and 88). Active as a reading of what Barad holds, which is well supported by her text as quoted. It does not say that the conception she holds is sound: that rests on agential realism being correct as quantum mechanics, which Faye and Jaksland dispute (THEORY-1071). It does say that a use of her concepts outside physics by structural comparison is a use she disowns. | Active |
| THEORY-1084 v2 | Simon proposes that ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, that a serial solver gains nothing by assembling the problem space at the outset, and that nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed | Simon (1973), LIT-1634, read in NOTE-1232. The alternation is described for an architect, matched to a 1923 account of battleship design by an organization and to the NSS chess program, and argued from the seriality of the solver; nothing is implemented or tested, and Simon states the last part as what his analysis “suggests” and as “reason to be optimistic”. Active as a report of the proposal; whether it is right, which wants a test, is THEORY-1216, Proposed. | Active |
| THEORY-1085 | In Toner and Tu’s hydrodynamic theory, a flock’s ordered state is a broken continuous symmetry of a coarse-grained velocity field: speed fluctuations relax in a finite time of order 1/α and are dropped, leaving as slow degrees of freedom the transverse velocity and the conserved density, which propagate as sound at ω = ±cq⊥; no individual member appears in the description, which is offered for a universality class of microscopic rules | Toner and Tu (1995), LIT-1670, read in NOTE-1228 from the arXiv preprint. Active as a statement of what the theory is: its variables, the degree of freedom it eliminates and why, and the modes that remain, all in the text (pp. 3–8). It does not say that the theory’s exponents are right (THEORY-1095), that the universality claim holds, or that the order so described is an agent or an individual. | Active |
| THEORY-1087 | In the symmetric Sakaguchi–Kuramoto model on a triangle and a pentagon sharing one vertex, the lag α gives each ℓ-cycle a phase-independent path mismatch (ℓ − 2)α, and the remote-synchronized locked branch passes through an exact two-group state at α = π/3, where the pentagon’s mismatch reaches π; but that branch has already lost transverse stability, in the pentagonal symmetry sector, at αc ≃ 1.0164, so the point of maximal cycle frustration marks the geometry of the phase relations, not the end of the synchronized pattern | Torres-Hugas, Duch, Gómez and Arenas (2026), LIT-1668, read in NOTE-1218 from the arXiv preprint v2. The mismatch (ℓ − 2)α is a three-line derivation (eqs. 15–18); the two-group state at π/3 is exact algebra (eqs. 38–40); the threshold αc is a numerical continuation with det M⊥ = 0 (Supplementary Material). All three were reproduced. Active for this motif and branch; the paper’s wider reading, that remote synchronization is a manifestation of cycle-level constraints, is not what this says. | Active |
| THEORY-1088 | When the expression maps of a discourse sheaf diffuse to reduce disagreement, with opinions held fixed they converge to the nearest sheaf, in Frobenius distance, of which the opinions are a global section, so apparent consensus is reached by changing what is said (“learning to lie”); with opinions diffusing too the joint flow is gradient descent on ½‖δx‖², and conserved diagonal blocks of αδᵀδ − βxxᵀ keep the opinions from collapsing to zero; nothing in either flow checks an expression against the opinion it expresses | Hansen and Ghrist (2020; SIAM J. Appl. Math. 2021), LIT-1646, §§8–9, read in NOTE-1222 from the arXiv preprint. Active as mathematics: Theorems 8.1, 9.3 and 9.4 and Lemma 9.1. Theorem 9.3’s proof is right only for α = β as printed; with the α, β weights it holds generally, which was checked numerically in the reading. The degeneracy guards are internal to the loop: minimal change and a conserved quantity, not evidence from outside it. | Active |
| THEORY-1089 | In a passive linear network, eliminating interior nodes by the principle of minimum power, which over a general field with positive elements is stationarity, takes a Dirichlet form to a Dirichlet form on the remaining nodes, linearly in their values; eliminating one node s adds c_is·c_js/Σ_k c_ks to each coupling c_ij, so two resistors in series act at their ends as one of resistance r_AB + r_BC; composing Dirichlet forms by this elimination is associative but lacks identities, because an ideal wire of zero impedance has no power functional | Baez and Fong (arXiv 2015, TAC 2018), LIT-1644, read in NOTE-1223: Thm 3.3, Lemma 3.4, Example 2.9 and Sec. 3.3. The elimination is what is usually called Kron reduction, a name the paper does not use. Over ℝ(s), with inductors and capacitors, the “minimum” is a formal stationarity condition, Kirchhoff’s current law at interior nodes. The absence of identities is computed for a one-element set and asserted as typical otherwise. | Active |
| THEORY-1090 | Eliminating an interior vertex of a cellular sheaf on a graph can leave its boundary a space of consistent assignments that no sheaf on the boundary vertices has: the interior can impose a constraint among boundary vertices that no set of pairwise compatibility conditions expresses, so Kron reduction, which turns the Laplacian of a connected graph into a graph Laplacian on any set of boundary vertices, fails for sheaves in general and holds when every vertex stalk has dimension at most one | Hansen and Ghrist (2019), LIT-1659, §4.2 (pp. 18–21), read in NOTE-1233: a worked three-leaf star with a two-dimensional centre, and a closure theorem for the scalar-stalk case. Active as mathematics; the argument was checked. It is about the space of sections, a static fact; it does not say how often the obstruction arises, what structure would express the reduced constraint, or anything about dynamics. | Active |
| THEORY-1093 | In Di Biagio and Rovelli’s reformulation of relational quantum mechanics a relative fact is a conditional probability concentrated on one value: A is a fact relative to B when the relative information I_{A|B} = I_A + I_{A:B} is maximal; since only commuting variables have a joint distribution, perspectives belong to commutative subalgebras of observables, not to quantum systems, so a qubit has none, and the joint distribution is itself relative to a further system, so the relativity iterates | Di Biagio and Rovelli (2025), LIT-1676, read from arXiv v2 in NOTE-1214, §§II–V and IX. Active as a statement of the paper’s definitions and of what follows from them: eq. (7) is an identity, and the restriction to commuting variables follows from the existence of joint distributions. It does not say that this resolves the problems the authors list, that observers play no role in fixing what counts as a fact, or that the choice of subalgebra is given by the physics. | Active |
| THEORY-1094 | In counterexample-guided abstraction refinement the property and the concrete system are fixed inputs: the property’s atomic formulas help build the first abstraction and no abstraction may merge states that disagree on them, every refinement only splits classes toward the concrete state space, and a real counterexample ends the procedure; so the loop revises which of the concrete system’s distinctions the model draws, never the property or the space of possible answers | Clarke, Grumberg, Jha, Lu and Veith (2000), LIT-1652, read in NOTE-1231. A textual fact about the algorithm as specified in §§3–4, and the exact form of the limit the exchange of 2026-10-27 (A16 §6) gave for CEGAR as the formal neighbour of inquiry as specification (CLAIM-470). It does not say that a property cannot be revised by anyone, only that no step of this procedure does it. | Active |
| THEORY-1096 | Blackwell’s comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces | Split from THEORY-156 by the status review of 2026-10-27 (ADR-057). THEORY-156’s equivalence needs a second proof of the general converse and stays Proposed. Ten of its twelve dependents used only its definitions (an experiment, a garbling, the informativeness order), or only the easy direction, that a garbling never helps a decision-maker free to choose any rule. The easy direction holds on any outcome spaces, and its proof is two lines, re-derived below under ADR-047 v2’s ruling, which ADR-057 keeps. It is Theorem 3 of Blackwell (1953), LIT-781, read in NOTE-595, and Blackwell’s 1951 result. The finite converse is THEORY-717, Active. | Active |
| THEORY-1098 v2 | On Karnkowska and colleagues’ genome analysis, the oxymonad Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed one by one, Fe–S cluster assembly to a bacterial-type SUF system shared with mitochondriate relatives and then the glycine cleavage system by loss, and the loss left the rest of the cell canonical, with no compensating changes found | Karnkowska et al. (2019), LIT-1574, read in NOTE-1243. The SUF replacement of the mitochondrial Fe–S pathway is shared by all examined Preaxostyla, including relatives that keep a mitochondrion-related organelle, and those relatives’ organelle still houses the glycine cleavage system that M. exilis lacks. So the organelle lost its jobs in at least two steps before it was lost. Apart from the mitochondrion, peroxisomes and mitophagy, the cell’s systems are those of an ordinary eukaryote. The order is inferred, not dated, and absence is limited by annotation. | Active |
| THEORY-1099 v2 | On Kay and colleagues’ relaxed-clock dating of pre-LECA gene duplications, most duplications in eukaryotic gene families of archaeal origin, including actin, tubulin, vesicle-trafficking and nuclear-targeted families, predate the onset of duplications in genes of alphaproteobacterial origin at about 2.2 billion years ago, which the authors take as the start of mitochondrial endosymbiosis, so the host was elaborating its cytoskeleton, endomembranes and nuclear machinery before the mitochondrion; each date is a maximum age for its paralogues’ functions, many intervals reach below 2.2 billion years, and the order of phagocytosis and the mitochondrion is left open | Kay et al. (2026), LIT-1566, read in NOTE-1234. A dated species tree places nFECA at 3.05–2.79 Ga, mFECA at 2.37–2.13 Ga and LECA at 1.80–1.67 Ga. Against an endosymbiosis marker at about 2.2 Ga, 85% of archaeal-origin duplications are older, and most non-alphaproteobacterial bacterial duplications are younger. The authors reject mitochondria-early scenarios and argue against serial endosymbiosis. The sharp edges are that duplication dates bound functions from above, that the priors shape which family can be older, and that the authors themselves cannot order phagocytosis and the mitochondrion. | Active |
| THEORY-1100 v2 | On Sørensen and colleagues’ review, a bacterial endosymbiont becomes genetically integrated, the authors’ mark of an organelle, when nuclear genes whose products are imported into it functionally replace essential genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, many of which fill gaps left by lost genes, the imported proteins come mostly from the host’s own genes or from other bacteria, and in the nitroplast no gene transfer from the endosymbiont has been found | Sørensen, Stiller, Kröninger & Nowack (2025), LIT-1560, read in NOTE-1241. Protein mass spectrometry of isolated endosymbionts found hundreds of host proteins imported into Paulinella’s chromatophore and more than 350 into the nitroplast, where they fill gaps left by lost genes. In Paulinella the imported proteins come mostly from the host’s own genes, with some from gene transfer from the chromatophore and from other bacteria. In the nitroplast no gene transfer from the symbiont has been found at all. So the route to an organelle is import, and the endosymbiont’s genes need not move. The origins are stated in summary, not counted, and the import mechanisms are unknown. | Active |
| THEORY-1101 v3 | Tobiasson, Luo, Wolf and Koonin find, on 5,850 eukaryotic gene stems, that normalized stem length does not behave as a clock of acquisition time: alphaproteobacterial stems are longer than Asgard ones for the 70% of stems below 0.3 and shorter only above 0.35, oxidative-phosphorylation genes presumed acquired together vary by more than two orders of magnitude, and information-processing genes have short stems; they conclude that even normalized stem lengths are set primarily by factors other than acquisition time, at least over the long timescales analysed, and suggest that adaptation after acquisition lengthens stems | Tobiasson, Luo, Wolf & Koonin (2026), LIT-1558, read in NOTE-1242. They test the stem-length method of Pittis and Gabaldón on their own trees and reproduce its mitochondrion-late signal only for long stems. Below 0.3, where 70% of stems lie, the order reverses, and genes presumably acquired together scatter widely. They conclude that “even normalized stem lengths are determined primarily by factors other than the gene acquisition time”. The paper still favours a late mitochondrion, on other grounds. | Active |
| THEORY-1102 v2 | On Pittis and Gabaldón’s stem-length analysis of the LECA proteome, families of alphaproteobacterial origin and mitochondrial location sit closer to their prokaryotic relatives, after correction for each family’s rate, than families of other bacterial origin (P = 0.044), which sit closer than archaeal families, and the authors read this as late acquisition of the mitochondrion by a host whose proteome was already chimaeric, under their stated assumption that rates before and after LECA are correlated | Pittis & Gabaldón (2016), LIT-1595, read in NOTE-1240. Of 1,078 LECA families, bacterial ones have shorter normalized stems than archaeal ones (P = 1.38e-25), and alphaproteobacterial ones shorter than other bacterial ones (P = 0.044). Origin tracks location: alphaproteobacterial with the mitochondrion, other bacterial with endomembranes, archaeal with the nucleus. A plastid control runs the right way. The sharp edges are that stem length measures distance to the closest sampled relative, so sampling can move it, and that the chimaeric host could have come from partners or from transfer, which the paper cannot tell apart. | Active |
| THEORY-1103 v2 | On Shigenobu and colleagues’ genome of Buchnera sp. APS, the aphid symbiont keeps the genes for the amino acids its host cannot make and has lost almost all of those its host can, uses host-made glutamate and aspartate as precursors, and lacks phospholipid synthesis and nearly all its own transcriptional regulators, so that its metabolism interlocks with the host’s and it presumably lives only inside the bacteriocyte, a state reached by gene loss from an E. coli-like ancestor | Shigenobu et al. (2000), LIT-1596, read in NOTE-1236. The first complete genome of an obligate insect endosymbiont: 640,681 bp, a subset of E. coli’s. What is shown is gene content. The interlocking of host and symbiont metabolism is inferred from it and from earlier physiology. That Buchnera cannot live outside the bacteriocyte, and that the host supplies its membrane lipids, are the authors’ inferences. The paper names no mechanism of loss beyond a homeostatic niche: it says nothing of drift. | Active |
| THEORY-1104 v3 | In Tobiasson, Luo, Wolf and Koonin’s constrained-tree survey of widespread eukaryotic protein families, Asgard archaea are the largest single prokaryotic source, with about half the likelihood weight in the best-resolved families and the lead in nearly every functional system, while Alphaproteobacteria contribute a small, coherent set centred on the mitochondrion and other bacteria about a third of families, scattered across functions with no pathway led by one donor except a narrow Myxococcota signal in nucleotide-sugar and nicotinate metabolism; the authors read this as one Asgard host lineage, one endosymbiont and piecemeal gene transfer rather than further symbioses | Tobiasson, Luo, Wolf & Koonin (2026), LIT-1558, read in NOTE-1242. In 4,290 well-resolved eukaryotic–prokaryotic clusters, Asgard takes 50% of likelihood weight and leads in almost every functional class. Alphaproteobacteria account for mitochondrial translation, oxidative phosphorylation and 4Fe-4S assembly. About a third of clusters point elsewhere, with no functional pattern except a narrow Myxococcota signal. The authors infer a complex Asgard ancestor, a late mitochondrion and “no indication of another symbiotic event”. The composition is shown; the scenario is read off it. | Active |
| THEORY-1105 v3 | In Bernabeu, Manzano-Morales, Marcet-Houben and Gabaldón’s reconstruction of the last eukaryotic common ancestor’s proteome, two bacterial phyla besides the mitochondrial ancestor, Myxococcota and Planctomycetota, each account for a share of the proteome comparable to that of Alphaproteobacteria and Asgard archaea under thresholds calibrated on those accepted partners, and stem-length distributions order the acquisitions as waves, Asgard first, then Planctomycetota, then the mitochondrial ancestor and Myxococcota together; the authors read this as serial associations with several partners, while conceding that their data cannot say whether any donor was a symbiont | Bernabeu, Manzano-Morales, Marcet-Houben & Gabaldón (2026), LIT-1561, read in NOTE-1237. Under thresholds chosen to maximize assignments to Alphaproteobacteria and Asgard, Myxococcota (3.92% of the proteome) and Planctomycetota (2.24%) pass in every dataset, beside Alphaproteobacteria (3.99%) and Asgard (3.28%). A control removing Alphaproteobacteria from the database does not reproduce their pattern. Stem lengths put Planctomycetota before the mitochondrion. The authors “challenge models depicting eukaryogenesis as a one-off event resulting from a binary symbiotic interaction”, and also say their data “do not provide information on the host or endosymbiotic nature of any partner”. | Active |
| THEORY-1106 v2 | Campbell and colleagues report, from four cicada species with five or six eggs each, that Hodgkinia cells per egg range from about 4,900 in a species with one Hodgkinia lineage to about 30,000, some six times as many, in Magicicada septendecim, whose Hodgkinia has split into many interdependent lineages and the only species that differs significantly, while Sulcia numbers are not raised, the transmission route is the same, and every egg sampled carries every lineage tracked by marker genes; the authors take the raised titre as a host adaptation that compensates for the symbiont’s fragmentation, and a cost of it | Campbell, Łukasik, Meyer, Buckner, Simon, Veloso, Michalik & McCutcheon (2018), LIT-1590, read in NOTE-1235. Measured: Hodgkinia cells per egg from 4,889 in a single-lineage cicada to 30,154 in M. septendecim, with Sulcia at 2,572 to 5,643; marker genotypes present in every egg sampled; the same route under the light microscope. Not shown: a trend across complexity beyond M. septendecim, or that splitting came first. The authors also speculate that a larger transmitted population weakens bottlenecks and could feed further splitting. | Active |
| THEORY-1107 v2 | On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it | Sørensen, Stiller, Kröninger & Nowack (2025), LIT-1560, read in NOTE-1241. Their closing scenario orders the steps from endosymbiont to organelle: effectors first, then dependence through one redundant imported protein, then many imported proteins once a targeting signal evolves. Rhizobia, targeted by about 700 host peptides yet not vertically transmitted, are their case of control without integration. It is a scenario built from systems at different stages, not a sequence observed in any one. | Active |
| THEORY-1108 v2 | On Imachi and colleagues’ cultivation of Candidatus Prometheoarchaeum syntrophicum MK-D1, the first Asgard archaeon grown in the laboratory is a small, internally simple coccus with long branching protrusions that degrades amino acids only in syntrophy, handing hydrogen and formate to partners that can be exchanged, and from its genome and other Asgard genomes the authors infer that the Asgard ancestor was likewise dependent on metabolic partners; the isolate is a modern relative separated from the host lineage by about two billion years, grown only in co-culture, and the entangle–engulf–endogenize model built on it is a hypothesis the authors call not definitive | Imachi et al. (2020), LIT-1591, read in NOTE-1238. Partner inhibitors impair MK-D1’s growth, added hydrogen or formate stops it, and it grows with either of two methanogens, so its dependence is on a service that several partners can supply. Cryo-EM and TEM show no internal compartments. The step from this one organism to the Asgard ancestor is a genomic inference, which the authors hedge. The sharp edge for the record is that the most famous transition in individuality may have begun from a host already dependent on others, with no partner specificity, cohesion or control. | Active |
| THEORY-1109 v3 | Santana-Molina, Williams, Snel and Spang find that the last eukaryotic common ancestor’s central carbon metabolism was chimeric: a few Asgard-archaeal enzymes in cytosolic glycolysis and the pentose phosphate pathway, four placed with Asgard archaea at bootstrap 39–97% and two ambiguously, most alphaproteobacterial ones in the mitochondrial TCA cycle and pyruvate conversion, and others from various bacteria with no third dominant donor, since remodelled by loss, horizontal replacement and retargeting so that compartment no longer follows origin; the authors take the cytosol–mitochondrion split as support for syntrophy | Santana-Molina, Williams, Snel & Spang (2025), LIT-1564, read in NOTE-1239. LECA had a complete central carbon metabolism. Six enzymes are placed with Asgard archaea, mostly cytosolic (bootstrap 39–97%, two ambiguous). Seventeen are alphaproteobacterial, ten of them in the TCA cycle, mostly mitochondrial. Others come from several bacteria, none dominant. Since LECA, Asgard enzymes have been replaced in many lineages and enzymes retargeted regardless of origin. The authors read the split as support for syntrophy; it does not discriminate syntrophy from other symbiogenetic models. | Active |
| THEORY-1110 | Shenhav, Botvinick and Cohen propose that control is allocated top-down by a dedicated specifying system: dACC computes the expected value of a control signal, its probability-weighted payoff less an intrinsic cost rising with intensity, and specifies the signal that lateral prefrontal and subcortical structures implement | What the expected-value-of-control paper proposes, split from THEORY-070 by the sixth status review (ADR-058’s route) so that a claim needing only the proposal can stand on it. The paper was read in full (NOTE-446), so the reading of what it proposes is held. Whether the proposal is true, a fit of the computation and a dissociation of specifying from implementing, stays THEORY-070’s question, Proposed. | Active |
| THEORY-1117 | Ainslie proposes that self-control over time needs no faculty of will: hyperbolic discounting makes preferences that dominate at different delays, these grow into interests that bargain with no governor above them, and resolve is the recursive staking of expected future choices on the present one | What Ainslie’s précis of Breakdown of Will proposes, split from THEORY-067 by the ninth status review (ADR-058’s route, on THEORY-1110’s pattern) so that a claim needing only the proposal can stand on it. The précis was read in full (NOTE-443), so the reading of what it proposes is held. Whether the proposal is true, which turns on bundling by self-prediction, the credibility of precedents and the order of lapse districts, stays THEORY-067’s question, Proposed. | Active |
| THEORY-1118 | McKenna and Coates argue, with Dryden and with Buss and Westlund, that a condition on free agency stated only in the agent’s present psychological structure is met by an agent whose structure a manipulator installed, so a mesh account must count that agent free or add a historical condition; Frankfurt, read first-hand, takes the first option | The manipulation argument as its sources make it, split from THEORY-040 by the tenth status review on THEORY-1110’s pattern, so that a claim needing only the argument’s first half can stand on it. All five sources were read in full (NOTE-270, NOTE-266, NOTE-267, NOTE-545, NOTE-264). Whether any historical condition escapes the argument stays THEORY-040’s question, Proposed. | Active |
| THEORY-1119 | Barandiaran, Di Paolo and Rohde propose that minimal agency requires individuality, interactional asymmetry and normativity set by the system’s own organisation, each necessary and together, they say, sufficient; that individuality without the other two is not agency; and that a system optimizing an externally fixed function is not a model of agency | What the paper proposes, split from THEORY-071 by the tenth status review on THEORY-1110’s pattern, so that claims needing only the proposal can stand on it. The paper was read in full (NOTE-458), so the reading of what it proposes is held. Whether the definition is right, which turns on an asymmetry criterion its authors call incomplete and on applying the conditions to explicit models, stays THEORY-071’s question, Proposed. | Active |
| THEORY-1120 v2 | On 416 translations into English by four LLMs of 104 culturally loaded utterances in Hindi, Punjabi, Tamil and Malayalam, each scored for pragmatic adequacy by one native rater per language, the reference-based COMET metric correlates with the ratings at Pearson r = 0.813 and Spearman ρ = 0.860, against r 0.80 and ρ 0.79 for XCOMET and r 0.73 and ρ 0.79 for MetricX, with no test of the differences; four outputs picked after the fact that invert or misplace the source’s meaning get COMET scores of 0.65–0.86 against the rater’s 1 or 2 | Singh et al. (2026), LIT-1817, read in NOTE-1345. On 416 outputs (104 items × 4 models), wmt22 COMET correlates with 1–5 pragmatic-adequacy scores at Pearson 0.813, Spearman 0.860; XCOMET ρ 0.792, MetricX ρ 0.788 (Table 3). Four hand-picked outputs scored 0.65–0.86 by COMET and 1–2 by the rater. Each language had one rater, so there is no reliability against which to read 0.81, and the claim that metrics fail “especially for implicature and deixis” rests on per-phenomenon differences of 0.1 or less. | Active |
| THEORY-1121 | With the couplings fixed, the square-loss Gromov–Wasserstein barycenter of several similarity matrices is their realigned average weighted by the barycenter weights, and the KL barycenter its weighted geometric average, so PSD and infinitely divisible inputs give barycenters of the same kind; but the barycenter’s size, histogram and weights are chosen beforehand, the couplings are only stationary points of a non-convex problem, and the iteration’s convergence at the step size used is not proved, so a structure carried by one input is shrunk by its weight at best and can be lost or redistributed by those choices | Peyré, Cuturi and Solomon (2016), LIT-1797, Propositions 3–5 and Remark 3, read in NOTE-1343 from the PMLR PDF. The closed forms and the cone preservation are proved and the PSD case was checked; the lack of guarantees is the authors’ own statement. Active as a statement of what is and is not guaranteed. It does not say a barycenter erases minority structure; a toy run showed seed-dependent redistribution, not erasure. | Active |
| THEORY-1122 | Weiss’s mark of a system, that the whole varies far less than the summed variances of its parts, is met by any aggregate of independent parts when the whole is measured as their average; when it is measured as their sum, it holds only if the parts’ fluctuations compensate one another, with ∑ᵢ<ⱼ Cov(xᵢ, xⱼ) close to −½ ∑ᵢ Var(xᵢ) | Nicholson (2026), LIT-1821, read in NOTE-1340, reports Weiss’s one formula, VS ≪ ∑ (va + … + vn), as a property “we would invariably find, if we studied any system”, and as a way of identifying systems empirically. The variance of a sum settles what it can show. Active: an identity, checked numerically. It does not say Weiss’s systems lack compensating covariance, only that the inequality as stated does not test for it. | Active |
| THEORY-1123 | In COTA’s formulation of causal abstraction learning as multi-marginal optimal transport, the term meant to impose do-calculus relations between the transport plans of comparable interventions depends only on the plans’ marginals and on normalisers built from them, which the transport constraints fix to the empirical distributions, so it is constant on the feasible set; its weight changes the solution only by rescaling the cost and entropy weights, and the paper’s evidence that the constraints help, a sweep over that weight, cannot separate the two | Felekis, Zennaro, Branchini and Damoulas (CLeaR 2024), LIT-1806, eqs. (6), (9), (10), (12) and Appendices A–B, read in NOTE-1342 from the arXiv preprint v1. A derivation from the paper’s own equations, which state the term as a divergence between column (or row) sums of two plans; Theorem 1’s proof writes it the same way. Active as mathematics about the problem as written. The released code, read but not run, builds a different problem for posets with several chains, in which the term acts on copies of plans that carry no marginal constraints. | Active |
| THEORY-1125 | For discrete X and Y, with an intervention on a variable abstraction T of X defined as the mixture of interventions on X weighted by Bayes inversion of the encoder under a protocol prior, no abstraction gains causal information about Y over X itself, and at full weight on causal control the causal information bottleneck is optimised by the partition of X by p(y | do(x)), mapped one class to one value of T, uniquely up to relabelling when T has exactly as many values as there are classes; these are Chalupka et al.’s causal macrovariables | Simoes, Dastani and van Ommen (UAI 2025), LIT-1802, Proposition 20 with its proof (pp. 14–15), read in NOTE-1347 from the arXiv preprint v5. Active: the proof is a strict-concavity argument, checked here on the paper’s Genetic Mutations SCM. It is the interventional counterpart of THEORY-743, and unlike the bottleneck there it reaches the partition exactly, because γ = 1 drops the compression term. It holds for the paper’s stipulated intervention on T and a fixed protocol; it says nothing about intermediate γ. | Active |
| THEORY-1128 | If a sequence is generated by a random walk over points of a compact Riemannian manifold whose step probabilities depend only on (locally scaled) distance, then negative log co-occurrence over a window that grows suitably with the number of points, less a term for each point, approaches the squared geodesic distance, whatever the link from distance to one-step probability; so log co-occurrence statistics identify the metric, and coordinates only up to isometry | Hashimoto, Alvarez-Melis and Jaakkola (2016), LIT-1801, Lemma 1 and Corollary 6, read in NOTE-1329. Proved from the convergence of graph random walks to a diffusion and Varadhan’s small-time formula, with an open conjecture needed for directed graphs; checked numerically on an unweighted ε-ball walk. Active as a theorem about the toy model. The authors say the model is not a model of language, and nothing here shows real corpora satisfy it. | Active |
| THEORY-1129 | Anchors entered as node features in fused Gromov–Wasserstein break a structural tie only when they move under it: with α strictly between 0 and 1, the zero-cost matchings of a structure with itself are exactly its automorphisms that fix every anchor, so a 6-cycle has 12 with no anchor, 2 with one anchor, 1 with two adjacent anchors and still 2 with two antipodal ones, and whether a set of anchors identifies the correspondence can be decided from the structure before any fit | Derived from Vayer et al. (2019), LIT-1813, Theorem 3.2’s zero case, read in NOTE-1328; the paper does not state it. Checked by enumerating all 720 matchings of a 6-cycle with four anchor patterns. Active: an elementary consequence of a proved theorem, checked. It is about exact ties; with noisy data, symmetries become near-ties, and the theorem says nothing about how near. | Active |
| THEORY-1130 | The Gromov–Wasserstein distance, half the least Lp distortion of a coupling, is a pseudometric on all networks with a probability measure on the nodes and any bounded edge weights, symmetric or not, and it is zero exactly when the two networks are weakly isomorphic, images of one network under maps that preserve weights and measures; so it compares only what the weight function and node measure record | Chowdhury and Mémoli (2019), LIT-1822, read in NOTE-1330 from arXiv 1808.04337v2. Theorems 16 and 18, with full proofs in the main text (pp. 8–9). Active: proved. The sharp edge is the zero set. A distance of zero says the recorded weights and measures agree up to merging and splitting nodes with identical profiles; it says nothing about relations the record leaves out, and the measure is chosen by the user. | Active |
| THEORY-1132 | For K networks compared under square loss, the multi-marginal fused Gromov–Wasserstein objective of a joint coupling tensor is exactly the sum of the pairwise fused GW objectives of its two-way marginals, so the joint optimum is at least the sum of the pairwise optima, with equality exactly when some choice of pairwise optimal couplings is jointly realizable as the marginals of one tensor; a joint alignment is consistent in that sense by construction, which for deterministic couplings means the pairwise maps compose and for soft ones does not | Zeng et al. (2024), LIT-1814, Theorem 1 and its proof (pp. 5, 12), read in NOTE-1334. The identity was checked numerically; the equality condition and the remark on composition are the record’s, short consequences of it. Active. The edge: “consistent by construction” means jointly realizable, not composable, once couplings are soft; and the paper never reports the gap that measures how far independent pairwise optima are from joint realizability. | Active |
| THEORY-1133 | The Gromov–Wasserstein objective between two spaces depends on each only through its within-space similarity matrix and weights, so every transformation of a space that preserves them, a single orthogonal map or per-vector rescaling under cosine similarity, leaves every coupling’s cost unchanged, and any coupling composed with a symmetry of either structure costs exactly what it did; GW can therefore identify a correspondence only up to the automorphisms of the two similarity structures, and on symmetric data it cannot choose between branches or categories that the symmetry exchanges | Alvarez-Melis and Jaakkola (2018), LIT-1818, use the GW objective (their eq. 8) for word translation and do not state its ambiguity; read in NOTE-1336. The invariance follows in two lines from the objective’s form and was checked numerically, including a symmetric tree on which a branch-swapping coupling costs exactly what the true one does. Active: elementary and checked. It does not say how large the symmetry group of real embedding spaces is, and exact symmetries are rare in noisy data; near-symmetries give near-ties instead. | Active |
| THEORY-1134 v2 | A translation metric that predicts a sentence score and token-level error spans from one shared encoder reaches segment-level agreement with expert MQM annotation of Kendall τ about 0.4 and span F1 under 0.33, the best among open metrics and GPT-4 prompting at the time, while the score computed from its own spans correlates with its regression scores at Pearson 0.73 to 0.82, and at 0.90 to 0.92 with its final score, which includes the span score by construction | Guerreiro, Rei, van Stigt, Coheur, Colombo and Martins (TACL 2024), LIT-1816, Tables 2, 4 and 5, read in NOTE-1341. The numbers are the paper’s, on WMT22 News MQM (zh-en, en-de, en-ru). The paper reads the heads’ agreement as support for the spans as explanation; the record reads it as coherence. The top of the agreement, 0.90–0.92, is partly arithmetic, since the final score includes the span score with weight 2/9. Active as a report of one test set (ADR-065); the record’s reading of the gap, across different statistics, is CLAIM-507. | Active |
| THEORY-1135 | Optimal transport between two point sets related by one unknown global linear map from a Schatten ℓp ball reduces, when every admissible map preserves angles or the target is whitened, to choosing the coupling Γ that maximises the dual Schatten-q norm of the transported cross-covariance XΓYᵀ: the nuclear norm when the maps are orthogonal, and the Frobenius norm when they lie on a Frobenius sphere, which is exactly the Gromov–Wasserstein problem with cosine similarity and squared loss; in every case one transformation acts on the whole space | Alvarez-Melis, Jegelka and Jaakkola (2019), LIT-1805, Lemmas 4.1–4.3 with proofs in Appendix D; read in NOTE-1333. Each step was checked numerically. Active: proved, and checked. The conditions matter. The p = 2 link to GW holds for the original transport problem only when the target is whitened and all vectors have equal norm, which the paper states in two separate places. Nothing here makes the chosen class the right one for given data, and nothing in it is local. | Active |
| THEORY-1138 | Optimal transport whose cost is the Lovász extension of a submodular set function on source–target assignments stays a convex problem, a zero-sum game in which an adversary picks the cost matrix from the function’s base polytope, and reduces to ordinary optimal transport exactly when the set function is modular; the convexity needs submodularity, so the paper’s own threshold cost, which is not concave at its threshold, falls outside it, and the induced discrepancy is symmetric only when the set function is | Alvarez-Melis, Jaakkola and Jegelka (2018), LIT-1815, §3, read in NOTE-1324 from arXiv v1. The convexity and game form are standard submodular analysis; the failure of eq. (12) and the asymmetry were checked numerically. Active. It does not say structured couplings predict better; the experiments are single runs on 100 samples. | Active |
| THEORY-1139 v2 | A graph network that learns its sheaf restriction maps from node features at each layer reaches the top, mostly within one standard deviation, of the standard small heterophilic node-classification benchmarks and stays within about one point of the best on homophilic ones; on a synthetic balanced bipartite graph it learns the negative transport the theory prescribes, but whether the maps it learns on real graphs have the geometry the theory credits for its success has not been examined | Bodnar et al. (NeurIPS 2022), LIT-1811, §§5–6 and Appendix F, read in NOTE-1332. The results are empirical: Table 1 on nine datasets of 183 to 18,717 nodes with ten fixed splits, baselines copied from earlier papers, spreads of one to nine points; and one synthetic bipartite experiment in which the learned maps are inspected. Active as a report (ADR-065); the paper’s explanation of its accuracy, that the model learns the right geometry, is tested only on the synthetic case and is THEORY-1220, Proposed. | Active |
| THEORY-1140 | If every entity’s O(d) frame may change independently and relation transports change as Tᵢⱼ ↦ gᵢTᵢⱼgⱼ⁻¹, then functions of the transports alone carry no invariant information except around cycles, a triangle’s invariants are exactly the class functions of its holonomy H = TᵢⱼTⱼₖTₖᵢ, and an equivariant update of an edge from one cycle through it is Ψ(H)Tᵢⱼ with Ψ conjugation-equivariant; the symmetry fixes a family of updates, not one mechanism, and says nothing about whether a task has the symmetry | Sarkar, Mukherjee and Sato (2026), LIT-1820, §2 and App. B, read in NOTE-1331. The gauge-fixing argument is standard in lattice gauge theory and group synchronization, as the paper says, and its covariance was checked numerically here. Active: a conditional theorem. What it does not support is the paper’s framing: “unique” holds for a family, and the antecedent is an inductive bias by the companion paper’s own statement (LIT-1803). | Active |
| THEORY-1141 | In the time limit, sheaf diffusion on a graph projects node features onto the harmonic space of its sheaf Laplacian, and which labellings that space can linearly separate depends on the class of restriction maps: a weighted graph Laplacian cannot separate the sides of a balanced bipartite graph, signed one-dimensional maps separate any two classes and no one-dimensional sheaf separates three, while diagonal maps with d ≥ C and orthogonal bundles with C ≤ 2d (d = 2, 4 and, against the paper, 8) separate any labelling; every separating sheaf is built from the labels | Bodnar, Di Giovanni, Chamberlain, Liò and Bronstein (NeurIPS 2022), LIT-1811, §3.2 and Appendix B, read in NOTE-1332 from the arXiv preprint v4. Active as mathematics: each separation result has a proof, and Propositions 9, 10 and 13 were checked numerically. The paper says the orthogonal construction exists only at d = 2 and 4; it exists at d = 8 as well, by the Hurwitz–Radon theorem, and was built. The results are existential: the separating sheaf is defined from the class labels, so they say what a sheaf can do, not that one can be found without the labels. | Active |
| THEORY-1142 | A translation between two logics presented as institutions is correct, in the Satisfaction Condition’s sense, when a translated sentence holds in a model exactly when the original holds in the model’s translation; given the model translation this fixes the sentence translation up to semantic equivalence, and when every model of the target is reached it preserves and reflects entailment, so one logic’s prover can decide the other’s; but the condition alone admits collapsing translations and is relative to a model translation it does not supply | Goguen and Burstall (1992), LIT-1661, read in NOTE-1348 from the published JACM article: Definition 32 (p. 125), Proposition 33 and Proposition 35 (p. 126), Theorem 39 (pp. 127–129). Active: proved, and checked on toy morphisms between propositional institutions. The collapsing morphism and the atom-negating morphism are my constructions, not the paper’s. It is a criterion relative to a chosen model translation, and says nothing about translation beyond truth conditions. | Active |
| THEORY-1146 v2 | Under confounding the partition of a variable X by what it predicts about a target Y can differ from its partition by what setting it does to Y, so the information bottleneck at full sufficiency can learn distinctions no intervention could use and miss ones only intervention reveals, as it did in two confounded toy models, while the causal information bottleneck, computed from observational data through a backdoor set, recovered the causal partition in all three toy models, projected gradient search finding it in 95 to 100 percent of runs | Simoes, Dastani and van Ommen (UAI 2025), LIT-1802, §8 and Appendices E–F, read in NOTE-1347. Three SCMs with 7, 4 and 16 values of X; ensembles of 4 to 12 optimisers; a penalty against unused values of T was needed at full control. The difference between the partitions follows from the definitions; what the experiments add is that the search finds the causal one, in toy problems only. Active as a report (ADR-065); whether search finds it at scale is QUESTION-346. | Active |
| THEORY-1147 | For entropy-regularized optimal transport with a fixed cost, the gradient of any differentiable function of the optimal plan with respect to the cost is obtained exactly from one linear system of size n + m whose matrix [[diag µ, T], [Tᵀ, diag ν]] is positive semidefinite and singular only along the gauge direction (1, −1), which does not affect the gradient; so the first-order effect of supervising any candidate pair on a utility of the whole alignment can be computed without forming the (nm)² Jacobian | Yu et al. (2026), LIT-1807, Lemmas 4.2 and 4.3 with Appendix A (pp. 4–5, 14–17), read in NOTE-1327. Proved by implicit differentiation of the optimality conditions; checked by finite differences. Active. It is a derivative: a local, first-order estimate of a finite change, exact only for plain entropic OT whose cost does not depend on the plan. | Active |
| THEORY-1149 | For symmetric matrices compared under squared loss, co-optimal transport with independent row and column couplings is a lower bound on the Gromov–Wasserstein value, and for squared Euclidean distance matrices the two are equal and share optimal couplings, because GW is then a concave quadratic program whose bilinear relaxation is tight; for other symmetric matrices the bound can be strict | Redko, Vayer, Flamary and Courty (2020), LIT-1804, Propositions 2–3, read in NOTE-1335 from arXiv v3. Proved via Konno’s theorem on concave quadratic programs and checked numerically, including strict inequality for non-Euclidean symmetric matrices. Active. It does not say COOT’s two couplings coincide at every optimum, nor that either is unique. | Active |
| THEORY-1150 | In any logical system presented as an institution, theories can be glued wherever their vocabularies can: if the category of signatures has colimits, so does the category of theories, the glued theory being the semantic closure of the union of the parts’ translated sentences; and because constraints requiring free or initial interpretation also obey the Satisfaction Condition, theories carrying them glue the same way | Goguen and Burstall (1992), LIT-1661, read in NOTE-1348 from the published JACM article. Theorem 11 (p. 108) proves that the forgetful functor from theories to signatures reflects colimits, using only the Satisfaction Condition and the Closure Lemma; Proposition 23 and Theorem 24 (p. 119) extend it to theories with data constraints. Active: proved, and checked on a toy propositional institution. It does not say that signatures have colimits in a given logic, that the glued theory is consistent, or anything about proofs: closure is semantic. | Active |
| THEORY-1151 v2 | A learned translation metric trained with synthetic hallucinations and diagnostic perturbations alongside human MQM flags most injected localized critical errors (all but 0 to 17 percent, by type and model size) and ranks fully detached hallucinations below other translations with AUROC up to 0.96, and without a reference it ranks hallucinations better than with one; how far this reflects the synthetic training data has not been separated from how far it generalises | Guerreiro et al. (TACL 2024), LIT-1816, §7 (Tables 6 and 8, Fig. 4), read in NOTE-1341. Localized errors are SMAUG perturbations of English references; hallucinations are one de-en benchmark of real system outputs. The training data includes synthetic detached and oscillatory hallucinations and DEMETR perturbations. The localized-error table’s “All” row is inconsistent with its rows. The larger model is worse on oscillatory hallucinations. Active as a report (ADR-065); whether detection generalises is QUESTION-345. | Active |
| THEORY-1152 | Between two weighted directed networks with node measures, the Gromov–Wasserstein distance is bounded below by a hierarchy of invariants, each in an in-going and an out-going version, and the strongest of them, a transport problem whose costs are themselves transport problems, equals a transport problem whose costs are one-dimensional Wasserstein distances between the nodes’ weight profiles, so it is computed in closed form; the bounds are one-sided, and two networks can agree on all of them while being at positive distance | Chowdhury and Mémoli (2019), LIT-1822, Theorem 24 and Lemma 28 (pp. 11–13), read in NOTE-1330. Proved; the equalities were already known for discrete spaces from Schmitzer and Schnörr 2013, as the paper says. TLB = R-TLB and the order of the bounds were checked numerically on 25 random asymmetric pairs. Active. The edge the paper leaves implicit: a zero lower bound is no evidence of sameness. A 3-node pair with every bound zero is at dN,2 ≥ 0.215 (the record’s construction). | Active |
| THEORY-1153 v2 | In the PETra corpus of English-source TED and Europarl translations, the additions for the target audience that a named-entity and null-alignment filter, and a classifier trained on its finds, turned up, as labelled by one annotator with no agreement reported, are mostly system transfers, chiefly unit conversions, and entity changes, and their mix differs by target language | Osmelak et al. (2026), LIT-1798, read in NOTE-1344 from the LREC 2026 PDF. Of 3,754 type instances in Table 2, 1,722 are system transfers (mostly measurement conversion) and 992 entity changes; the mix varies by language. It is a count over a pool selected by a filter that requires a named entity, mostly found by a classifier, labelled with no reported agreement. It is not a base rate of explicitation in translation, and it says nothing about how readers take the additions. | Active |
| THEORY-1155 | Co-optimal transport between two data matrices of equal shape with uniform weights vanishes exactly when one is the other with its rows and its columns separately permuted, and its objective is unchanged when a pair of optimal plans is composed with any joint relabelling of rows and columns that leaves a matrix unchanged; its sample and feature couplings are therefore identified at best up to that joint symmetry group, which can exchange a block of samples together with its block of features though neither exchange alone is a symmetry, and the paper proves no uniqueness beyond it | Redko, Vayer, Flamary and Courty (2020), LIT-1804, Proposition 1, read in NOTE-1335; the composition step is elementary and was checked by enumeration on small matrices, including a two-community block matrix with 32 tied optimal plan pairs. Active. It does not say real response matrices have exact symmetries; noisy ones give near-ties, which is the practical form of the threat. | Active |
| THEORY-1156 | The fused Gromov–Wasserstein discrepancy between two attributed graphs, a single coupling that pays (1 − α) for moving node features and α for distorting pairwise structure, tends to the Wasserstein distance between the features as α → 0 and to the Gromov–Wasserstein distance between the structures as α → 1, lies above their α-weighted mixture, and for q = 1, α strictly between 0 and 1 and distance-matrix structures is a metric that vanishes exactly when a bijection of the nodes preserves weights, features and structure together | Vayer, Chapel, Flamary, Tavenard and Courty (2019), LIT-1813, Theorems 3.1–3.2 with the supplement’s proofs, read in NOTE-1328 from arXiv v3. Proved; the lower bound and the concavity in α of the exact value were checked on enumerated permutations. Active. It says nothing about the uniqueness of the optimal coupling or about the solver, which reaches only stationary points. | Active |
| THEORY-1158 v2 | On synchronization of S4 relations with outliers, an untrained mechanism that composes relations through third vertices gains 26–45 points over the single direct path, and weighting the candidate paths by the Wilson loops of squares that avoid the missing edge adds 0.6–7.8 points, most where clean two-hop paths become a minority; it stays below spectral synchronization with one reweighting pass on every setting and within 2.2 points of plain spectral synchronization either way; its 0.99 AUC for corrupted triangles is the ceiling that exactly flat clean data set, not a measure of detection power | Sarkar, Mukherjee and Sato (2026), LIT-1820, §3 and Table 1, read in NOTE-1331. The accuracies are the paper’s, on 1000 instances per cell, and were not rerun. The AUC ceiling was recomputed here: 1.42% of corrupted triangles at p = 0.3 have cancelling corruptions, so the ceiling is 0.9929. What the experiment does not show is anything about learned operators, language, or relations not given as group elements. | Active |
| THEORY-1159 v2 | Aligning three networks jointly, through one multi-marginal fused Gromov–Wasserstein coupling with an anchor-based random-walk cost and a cluster decomposition, finds held-out correspondences better than pairwise and existing multi-network methods in most cases, as the paper’s plots show, in both pairwise and whole-tuple hits, on benchmarks whose true correspondences are consistent across the three networks by construction; whether the joint coupling itself is the cause is not separated from the new cost and the clustering | Zeng et al. (2024), LIT-1814, §5 and Appendix C, read in NOTE-1334. Reported as plots against 13 baselines on ER, Douban, DBLP and ACM triples, with 10% anchors and 90% held out. Active as a report (ADR-065). Its ground truth is a consistent set of node triples, and no ablation holds the cost and the clustering fixed while changing joint for pairwise fitting, so the paper’s attribution of the gain to joint fitting is THEORY-1210, Proposed. | Active |
| THEORY-1161 v2 | In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two | Friedkin (2011/2012), LIT-1404, read in NOTE-1374. Under reflected appraisal, the next self-weight set to the member’s share of control over the last issue, with each issue settled by Friedkin–Johnsen anchoring, his simulations give a single leader in 99.0–100% of 100,000 random networks. Re-runs on sparser networks gave 93% and 96%. The contrast with the DeGroot step, and which step real groups follow, are not this reading’s: the first is THEORY-1232. | Active |
| THEORY-1162 | In a consumer-resource model whose species-resource benefits and impacts have correlation rho, the replica-symmetric cavity fixed point loses stability where rho squared equals surviving species over non-depleted resources | Blumenthal, Rocks and Mehta (2024), LIT-1236: in MacArthur’s consumer–resource model with benefits c and impacts e correlated by ρ, the replica-symmetric steady state is unstable when ρ < ρ*, ρ² = S/M* at the boundary. With ρ = 1 competitive exclusion keeps S*/M* ≤ 1, so the symmetric model never crosses it. What this does not say: that non-reciprocity by itself breaks the optimization picture; it does so only past a boundary set by species packing, and that the other side is chaotic is shown numerically, not derived. | Active |
| THEORY-1163 | Krakauer et al.’s organismal and colonial individuality, A* = I(S_{n+1}; S_n) and A = I(S_{n+1}; S_n | E_n), cannot decrease when variables are moved from the environment into the candidate system, so maximizing either over candidate boundaries selects the largest candidate, and a boundary found this way needs a size cost the measures do not contain | Krakauer, Bertschinger, Olbrich, Flack and Ay (2020), LIT-1137, read in NOTE-1400. Mutual information cannot fall when either argument is enlarged, and A′ ≥ A follows by the chain rule with the remaining environment as condition. It is a fact about the measures, not about any system. It does not say how to price size, and nC, the third measure, can move either way. It is the formal content of A11’s remark recorded in QUESTION-115. | Active |
| THEORY-1164 | Propp proposes, from his analysis of Afanasyev’s tales 50–151, that the Russian wondertale is built from 31 character functions, each defined by its consequence for the action, occurring in one order and falling to seven roles, so that wondertales share one composition and differ in who performs the functions and how | What Propp’s Morphology proposes, split from THEORY-167 by the twelfth status review (ADR-058’s route, on THEORY-1110’s pattern) so that claims needing only the proposal can stand on it. The book was read in full in Russian (NOTE-615, with appendices II–III in a second text), so the reading of what it proposes is held. Whether the proposal is true of the class, which an independent coding would test, stays THEORY-167’s question, Proposed. The order is shown on 45 schemes Propp selected, partly for keeping it. | Active |
| THEORY-1165 | Kahl proposes that agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, which is absent where revision is imposed from outside, escaped by rollback or chosen by a fixed higher-order rule; that allocating authority among a system’s own components coordinates execution without authoring its governing frame; and that planning, autonomy, memory and coordination do not supply that standpoint | What the paper proposes, split from THEORY-219 by the twelfth status review on THEORY-1110’s pattern, so that a claim needing only the proposal can stand on it. The paper was read in full (NOTE-723), so the reading of what it proposes is held. Whether the condition is right, which turns on a line between settling a frame and selecting under an antecedent rule that the paper does not draw precisely, and on a positive case it does not give, stays THEORY-219’s question, Proposed. | Active |
| THEORY-1166 | Di Paolo argues that autopoiesis as originally defined entails conservation and not active homeostasis, so it gives a system only an all-or-nothing norm, to go on or to die, which an observer and not the system can use; that a graded norm the system itself has needs adaptivity, a capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them; and that sense-making and agency need adaptivity | What the paper argues, split from THEORY-628 by the twelfth status review on THEORY-1110’s pattern, so that claims needing only the argument can stand on it. The paper was read in full (NOTE-719), so the reading of what it argues is held. Whether the argument is right stays THEORY-628’s question, Proposed. It turns on three things: a reading of autopoietic theory that Di Paolo grants later formulations may not share (note 1), cases chosen to illustrate it, and its rival THEORY-666. | Active |
| THEORY-1167 v2 | When one intervention distribution over micro states is used at both levels, the effective information of a coarse-graining of a discrete mechanism, with each macro intervention realized by that distribution within its block, cannot exceed the micro effective information; so a positive causal emergence of effective information, scored with a uniform distribution at each level, equals the micro gain from changing the distribution minus a non-negative loss, and is a gain from the change of distribution | The test QUESTION-204 asks for has an answer that needs no system: with one distribution over micro states at both levels, EI(macro) = I(f(X); f(Y)) ≤ I(X; Y) = EI(micro) by data processing. Hoel’s causal emergence uses the uniform distribution at each level, which is a non-uniform micro distribution whenever blocks differ in size, so its positive values come from the micro chain scored under that other distribution. Comolatti and Hoel (LIT-1843) concede the stationary case; Eberhardt and Lee (LIT-1839) name the distribution as the source of the artefact. It says nothing about measures that data processing does not bound, nor about whether EI’s level agrees with an interventionist test. | Active |
| THEORY-1168 | In a Lotka–Volterra community with a symmetric interaction matrix, the growth-weighted biomass rᵀx strictly increases along every edge of the invasion graph, so the invasion graph has no cycles | Re-derived from Serván and Allesina (2021), LIT-1824, read in NOTE-1422, and applied to Hofbauer and Schreiber’s invasion graph (LIT-1233). At an equilibrium V = 2rᵀx − xᵀAx equals rᵀx, and for two equilibria the difference is a sum of invasion growth rates weighted by abundances, which an edge of the invasion graph makes positive. It needs symmetry and one equilibrium per vertex, not positive definiteness. It does not say assembly is order-independent: an indefinite symmetric matrix has several sinks. | Active |
| THEORY-1169 | For a discrete mechanism scored with one full-support intervention distribution over micro causes at both levels, a coarse-graining of cause and effect keeps all of the micro effective information if and only if it merges only causes with identical rows P(Y | do(x)) and only effects with proportional columns, whatever the distribution; so the coarsest coarse-graining that ties the micro is the causal partition of causal feature learning, with effects grouped by normalized likelihood, and any merge of causes with different effects loses information under every such distribution | Causal feature learning (LIT-1833, LIT-1834) defines macro causes as classes of micro states with the same interventional effect, with no distribution over interventions. Under one full-support distribution w at both levels, the loss of effective information from a coarse-graining (f, g) is I(X; Y | g(Y)) + I(X; g(Y) | f(X)), and both terms vanish exactly when f merges only equal rows and g only proportional columns, for every w. So the ties that THEORY-1167 allows are the causal-feature-learning partitions and their refinements: on which coarse-graining loses nothing, EI under a common distribution and the interventionist test agree, and the agreement is distribution-free. How much a merge loses still depends on w. Hoel’s EI, uniform at each level, parts from it on Eberhardt and Lee’s (13), (14) and note 6. | Active |
| THEORY-1194 | Clark and Chalmers argue that if belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so that the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and on a parity principle they state rather than argue | What the extended-mind paper argues, split from THEORY-579 by the sixteenth status review on THEORY-1110’s pattern (ADR-058’s route), so that CLAIM-175, which argues “on the extended-mind thesis’s own terms”, can stand on the statement of those terms. Both papers were read in full (NOTE-180, NOTE-363), and Clark (2005) says in his own words that the parity principle was not argued. Whether the conditional holds, that coupling of this kind makes a store part of a mind, stays THEORY-579’s question, Proposed. | Active |
| THEORY-1227 | Haken, Kelso and Bunz model the switch of rhythmic bimanual coordination from anti-phase to in-phase at a critical frequency as the loss of one stable state of an order parameter, relative phase, derived from two coupled oscillators; Kelso, Scholz and Schöner report the variability of relative phase rising before the switch in the anti-phase pattern only, in a few subjects and with no statistical test | What the Haken–Kelso–Bunz paper proposes and what the first report of critical fluctuations measured, split from THEORY-069 by the twenty-third status review on THEORY-1110’s pattern (ADR-058, ADR-065), so that a claim needing only the model and that report can stand on it. Both papers are read in full (NOTE-1424, NOTE-1433). Whether the switch is a nonequilibrium phase transition with critical slowing and hysteresis as its warning signs, which needs relaxation-time and hysteresis data neither paper shows, stays THEORY-069’s question, Proposed. | Active |
| THEORY-1247 | Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone | What Griesemer and Shavit’s sketch proposes. The twenty-fourth status review split it from THEORY-418 (ADR-065 §3), so that the record holds the reading of the proposal apart from the proposal. The paper was read in full (NOTE-173), so the reading is held. Whether the proposal is right is THEORY-418, Proposed, which extends this entry. That needs criteria that sort real groups into the three grades, and worked cases. The paper’s two headline claims, non-circularity and individuality as a special case of collectivity, are in its abstract only and are not part of this entry. | Active |
| THEORY-1248 | Catalysts cancel out of the stoichiometric matrix, so a cycle of mutual catalytic production among species is not a stoichiometric cycle: the cycles that carry a reaction network’s steady currents and affinities are fuel-to-waste pathways through it, and who catalyses whom enters only the rates, never the thermodynamic driving | From the definitions of Dal Cengio, Lecomte and Polettini (2023), LIT-1897 (NOTE-1476), and Rao and Esposito (2016), LIT-1889 (NOTE-1464), neither of which says it in these words. It makes exact A13’s “a productive cycle need not be a stoichiometric cycle” from the exchange “Analyze nucleation concepts”. Checked on a reversible rewriting of CASE-189’s first construction. Active: definitional and checked. It does not say that catalytic cycles are thermodynamically idle. They decide which of the states the driving allows is reached, and bistability needs them; but the budget they work within is set elsewhere. | Active |
| THEORY-1249 | In a fully observed, finite, communicating, stationary controlled Markov process, a deterministic goal-conditioned policy that achieves every composite goal of depth up to n with probability at least 1 − δ times the optimum determines every transition probability to within √(2P(1−P)/((n−1)(1−δ))), read from its first action on either-or goals; for a policy competent only at one-step goals no bound follows | Richens, Abel, Bellot and Everitt (2025), LIT-1891, Theorems 1 and 2, read in NOTE-1469. The reduction offers the agent a choice between “at most k successes of a transition in n attempts” and “more than k”; its switch point locates the binomial median. The final bound was re-derived and checked numerically against an adversarial δ-bounded agent. Active: a theorem, checked. It says the transition function is recoverable from the input-output map, not that any component of the agent represents it, and it needs full observation, determinism and the regret bound on every listed goal. | Active |
| THEORY-1250 | A chemostat is an emergent structure of a closed reaction network: when exchange reactions are fast and the reservoir species are abundant by a factor of order Omega, the closed network behaves to leading order as the open one, with its local detailed balance and dissipation, over a window that opens once exchange equilibrates and closes on a time of order Omega, after which a driven network relaxes to equilibrium; neither separation alone suffices | Remlein, Esposito and Avanzini (2026), LIT-1887, read in NOTE-1468: a multiscale expansion in the rate ratio ε and a WKB argument in the abundance ratio Ω, for arbitrary stoichiometry. The lifetime scaling was checked by integrating the paper’s §IX network, T_δ = 0.00483 Ω against its fit 0.0048 Ω + 0.33. Active. It does not say that every driven closed network departs from open behaviour at T ~ Ω, only that it can from then on, nor anything about systems that are not chemical. | Active |
| THEORY-1252 | In ℓ2-regularized regression held to a fixed training-error tolerance, refitting to one’s own predictions on the same inputs shrinks each eigencomponent of the labels by a factor that falls with its Gram eigenvalue, while the regularization weight needed to meet the tolerance rises as the labels shrink, so repeated rounds amplify regularization and reach the zero function after finitely many; the loss comes from the regularizer acting again, not from information the student lacks | Mobahi, Farajtabar and Bartlett (2020), LIT-1895, eqs. 24–37 and §§3.2–4; read in NOTE-1478, with the paper’s illustrative example reproduced from its own sample values. Active for the recurrence, the rising multiplier and collapse, which are elementary and reproduced; the sparsity rate of Theorem 5 is the paper’s, proof not checked. It does not show that deep networks behave so, and the amplification depends on holding the error tolerance fixed. | Active |
| THEORY-1253 | Treating the Bayesian evidence as a partition function with sample size as inverse temperature, the analogue of heat capacity, the rate at which leave-one-out predictive loss falls with sample size, equals half the parameter count in a regular model, as equipartition, and the real log canonical threshold in a singular one, falling below these where prior or discreteness fixes parameters; with multiplicity above one the approach to the singular value is only logarithmic, and the correspondence has no counterpart of physical current or dissipation | LaMont & Wiggins (2019), LIT-1882, read in NOTE-1481: the learning capacity C = −N²∂_N U and its equipartition values (Table II, eqs. 16–18, 24, 28, 30). Active: the regular entries were checked by differentiation, and the singular case exactly in K = a²b² (CASE-152’s model), where C approaches λ = ½ as ½ − 1/log n, not with the O(1/N) error the paper states. The statistical “temperature” is a sample size, so nothing in the dictionary is a physical heat flow. | Active |
| THEORY-1254 | A controller that must stay safe under every model it has not ruled out cannot use the evidence its committing action produces: if the information it can gather before its first action that forecloses a safe continuation under an alternative is bounded by β, every such controller forfeits at least about ½e^{−β} of the value reachable only by committing, so a linear oracle gap gives linear regret, and acting on an exclusion at risk α needs about log(1/(2α)) nats gathered beforehand | Saligrama (2026), LIT-1880, Theorem 1 and its examples, read in NOTE-1466. The proof is data processing on the stopped record plus Bretagnolle–Huber, and the explicit relaxation and one example’s optimization were checked numerically. Active: a theorem with a short proof, checked. It is a lower bound only; bounded information is sufficient for an unavoidable gap, and unbounded information does not by itself give recovery. | Active |
| THEORY-1255 | Near a non-degenerate minimum, constant-rate SGD’s stationary distribution is a Gaussian set by learning rate, batch size and preconditioner, which can be tuned to match the posterior at a given sample size; training time enters only as the wait for stationarity | Mandt, Hoffman and Blei (2017, LIT-1898; first in 2016, LIT-1899), read in NOTE-1475 and NOTE-1473. Under Gaussian, constant gradient noise BBᵀ, the continuous-time limit and a quadratic loss with positive-definite Hessian A, the stationary covariance solves AΣ + ΣA = (ε/S)BBᵀ; the preconditioner (2S/εN)(BBᵀ)⁻¹ makes it the posterior’s A⁻¹/N exactly. Checked numerically. Active as a result about that model. It does not cover singular posteriors, several basins or the search phase, and it relates the optimiser’s noise scale to sample size, not training time to sample size. | Active |
| THEORY-1256 | When a conservative experiment is safe under every plausible model, can be repeated from a common state and keeps informing, a controller that acts on the lower end of an anytime confidence sequence stays safe with probability 1 − δ and approaches the safe oracle, at O(√(T log T)) regret in a braking example; a certainty-equivalent controller in the same example is unsafe with probability ½ in every episode however much data it has | Saligrama (2026), LIT-1880, §5 (Lemma 5, Theorem 2, Corollaries 5–7, §5.6.1), read in NOTE-1466 with the proofs of Lemma 5 and Theorem 2. Active: a short proof by a Gaussian mixture martingale and Ville’s inequality, checked by hand, and an elementary calculation for certainty equivalence. Narrow: a scalar parameter with a known conservative direction, an experiment that returns to a common state, and Gaussian observations. | Active |
| THEORY-1257 | Averaging constant-rate SGD iterates at stationarity gives a covariance that shrinks with the number of examples processed, and equals the Gaussian posterior’s for that many observations when the noise covariance equals the Hessian | Mandt, Hoffman and Blei (2017), LIT-1898, §6 and App. G, read in NOTE-1475. In their OU model, the mean of T stationary iterates at batch S has covariance A⁻¹BBᵀA⁻¹/(TS) to leading order in 1/(Tελ), whatever the step size or constant preconditioner; under a well-specified regular model (BBᵀ = A) that is A⁻¹/(TS), the Gaussian posterior’s covariance for TS observations. Checked against the exact discrete covariance. It is the one place the paper ties examples processed to a sample size, and it is about spread in one basin after equilibration, not about which region dominates. | Active |
| THEORY-1258 | The steady state of Hennick and De Baerdemacker’s fractional Fokker–Planck model of SGD is a tempered posterior only within a region of constant effective diffusion; the weights between regions, where accessibility would act, are not derived | Hennick and De Baerdemacker (2025; ICLR 2026), LIT-1879, read in NOTE-1467. Their Lemma 3.3 and Corollary 3.2 show that where a scalar effective diffusion D_ξ is constant, the steady state is exp(−L_m/D_ξ), the posterior at effective sample size 1/D_ξ. That is a temperature, and inside such a region it leaves no room for an accessibility correction. Across regions their own equation’s zero-flux state is ∝ exp(−∫∇L/D), set by D along the paths between regions; checked on a double well. Active: the algebra and the counterexample are elementary and checked. It does not say SGD is far from Bayesian across regions, only that the paper does not show it is close. | Active |
| THEORY-1259 | When a teacher’s next-token distribution depends on a context the student cannot observe, the forward-KL-optimal student is the teacher averaged over the contexts compatible with the student’s observation, the loss it cannot remove is the conditional mutual information between the teacher’s output and the hidden context, and the optimal student is at least as uncertain as the teacher is on average; the self-distillation objectives actually trained may not be forward KL at all | Hossain, Kousar and Tourni (2026), LIT-1883, Proposition 1; read in NOTE-1479. Active: elementary and checked numerically. This is the published statement of the limit QUESTION-361 starts from. It holds for the unclipped forward KL; the OPSD objective the paper trains clips each vocabulary entry, which the paper shows reverses the gradient on the teacher’s most preferred tokens. | Active |
| THEORY-1260 | Richens et al.’s sense of “an agent contains a world model” is that the transition function can be computed from the agent’s goal-conditioned input-output map, uniquely to within the theorem’s error, so the attribution is not a stance an observer may withhold, unlike a representation-theorem fit; but it holds of every implementation of the same map, a lookup table included, and so cannot show that a predictive component is realized inside the agent | Richens, Abel, Bellot and Everitt (2025), LIT-1891, read in NOTE-1469. The paper says its recovery map is “from the agent’s policy rather than its activations”, that this is “strictly weaker”, and that it shows “a world model encoded in the agent’s policy, not its specific use”; it contrasts itself with Savage, whose fit exists even for a random policy. That the result applies to Block’s lookup table (LIT-642) is immediate from Def. 5, which constrains only the map. Active: the paper’s own statements plus a one-line corollary. It does not say whether a compact implementation must have a separable predictive part. | Active |
| THEORY-1261 | At stationarity a driven reaction network is confined to a thermodynamic space: each reaction’s affinity lies between zero and the largest cycle affinity per use of that reaction among the elementary flux modes containing it, and each product of concentrations lies between the effective equilibrium constants of the extremal pathways; with no driving the space collapses to the equilibrium point, so multistability, symmetry breaking and amplification above equilibrium need driving and are bounded by it | Liang, De Los Rios and Busiello (2024; PRX 2026), LIT-1886, read in NOTE-1463: eqs. 20, 24 and 29, proved for arbitrary stoichiometry from local detailed balance and the conformal decomposition of steady fluxes into elementary flux modes. Checked on the paper’s Schlögl parameters and on a reversible rewriting of CASE-189. Active. It bounds where a stationary state can be, for every choice of kinetics; it does not say which state is reached, whether it is stable, or how long the driving lasts. | Active |
| THEORY-1264 | Far from equilibrium, the weak-noise escape rate from an attractor goes as the exponential of minus system size times a barrier in the quasi-potential, not in any free energy, and that barrier is bracketed by thermodynamics: it is at most the entropy produced in relaxing from the saddle into the attractor and at least minus the entropy produced along the most likely escape path; the two bounds coincide at detailed balance and separate by twice the dissipation of the probability circulation, so far from equilibrium they can be very loose | Santolin, Freitas, Esposito & Falasco (2025), LIT-1881, read in NOTE-1465, for overdamped diffusions; Falasco & Esposito (2025), LIT-1892, §IV.H and appendix A, read in NOTE-1472, for jump processes with a macroscopic limit. Active: the derivation is two lines from the orthogonal decomposition of the drift and was checked on a linear nonequilibrium diffusion. It does not say that the more dissipative state lives longer, and in the one worked far-from-equilibrium chemical example the bracket is very loose. | Active |
| THEORY-1265 | Approximating a chemical master equation by a Fokker–Planck or nonlinear Langevin equation gets the quasi-potential right only at the fixed points: between them it mis-estimates the barriers, so escape rates come out wrong by a factor exponential in system size and the approximation can pick the wrong one of two states as the long-run favourite, and its apparent entropy production vanishes at every steady state, missing the dissipation that sustains it | Falasco & Esposito (2025), LIT-1892, §IV.D–E and §VII.A.1, read in NOTE-1472; Vellela & Qian (2009), LIT-1890, appendix A, read in NOTE-1474. Active: both formulas were checked on the Schlögl model, including Vellela & Qian’s counterexample, where the exact and diffusion quasi-potentials favour different states. It does not say diffusion approximations fail near fixed points or for Gaussian fluctuations; there they are exact to leading order. | Active |
| THEORY-1266 | In an open reaction network of reversible elementary mass-action reactions, sustained currents need an emergent cycle with nonzero affinity: each chemostatted species either breaks a conservation law or opens an emergent cycle, so one chemostat never suffices; with every emergent affinity zero the network relaxes to a unique equilibrium fixed by its unbroken conserved quantities; and a nonequilibrium steady state dissipates exactly the sum over emergent cycles of current times affinity | Rao and Esposito (2016), LIT-1889, read in NOTE-1464, with the one-chemostat proof of Dal Cengio, Lecomte and Polettini (2023), LIT-1897 (NOTE-1476). Active: rank–nullity, Wegscheider’s condition and two lines of algebra, checked numerically on a reversible rewriting of CASE-189’s first construction, which has a single equilibrium at zero affinity and is bistable only above 3.52 RT. It does not say that a driven network is far from equilibrium, multistable or organized; only that without an emergent affinity it cannot be, and what it costs when it is. | Active |
| THEORY-1267 | A safe set that a nominal barrier-function controller keeps invariant can be left under an arbitrarily small input disturbance, but if the barrier inequality holds with a disturbance penalty, ḣ ≥ −α(h) − ι(‖d‖), the state stays in a larger set {h + γ(‖d‖∞) ≥ 0} with γ class-K, and adding a damping term along the barrier’s input direction to any barrier controller gives this with an enlargement of ‖d‖²/(4ε) in units of h | Kolathaya and Ames (2019), LIT-1893, Theorems 1–2 and Examples 1–2, read in NOTE-1477. Short proofs, checked by hand, and both examples checked by simulation. Active. Matched input disturbances only, local, and the enlargement is in barrier units and quadratic in the universal construction, not proportional to the disturbance. | Active |
| THEORY-1268 | In a nonequilibrium steady state of a thermodynamically consistent Markov jump network, the entropy produced while a current is counted, times the squared relative uncertainty of that count, is at least 2k_B, independent of the counting time, so a current precise to ε costs at least 2k_BT/ε² of dissipation; the 2015 letter proves this within linear response and for single-cycle networks at any affinity, and for many-cycle networks far from equilibrium shows it numerically and states it as a conjecture | Barato & Seifert (2015), LIT-1885, read in full in NOTE-1480. Active for the scope stated in the title: the biased walk, the unicyclic formula (A/N) coth(A/2N) and the bound on random four-state networks were checked numerically. It bounds the precision of currents, not of occupations, so it does not price how long a maintained state lasts, and the general multicyclic proof is not in this paper. | Active |
| THEORY-1269 | In overparameterized linear regression, iterated self-training with fresh covariates, noiseless pseudo-labels and minimum-norm students applies a fresh random orthogonal projection to the estimate each round, so noise from the first fit decays geometrically while signal along a direction of variance s decays at the slower rate s/(s + τ); risk is U-shaped under anisotropy and monotone under isotropy, tends to the risk of predicting zero, and every student fits its teacher exactly on its own sample | Wu, Yang and Sun (2026), LIT-1884, Theorem 3.2, Corollaries 3.3–3.4 and B.1, and the projection identity of §3.2; read in NOTE-1471 and checked by Monte Carlo in the spiked model. Active for the spiked closed form; the general concentration result (Theorem 4.2) is the paper’s, proof not checked. It does not cover reused data, noise injected every round, or nonlinear students. | Active |
| THEORY-1270 | Entropy production does not select which of several maintained steady states is the stable one: in the bistable Schlögl model the state the system occupies in the long run can carry either the higher or the lower deterministic entropy production rate, and the mean stochastic entropy production follows whichever state is more probable; in general the most dissipative attractor is the longest-lived only under special conditions, and lifetime tracks relaxation entropy only near detailed balance | Vellela & Qian (2009), LIT-1890 (§3.3, Fig. 6), read in NOTE-1474, show the counterexample; Falasco & Esposito (2025), LIT-1892 (§§III.F, IV.H, VIII.B.6), read in NOTE-1472, give the general account: minimum entropy production holds only near detailed balance with linear dynamics, and the two maximum principles hold only where relaxation and escape dissipations nearly coincide. Active as a negative result. It does not refute every extremum principle, only those that rank steady states by dissipation alone. | Active |
| THEORY-1273 | In a driven chemical system with two stable states and a finite number of molecules, each maintained state is metastable: it is left by fluctuation after a time exponential in system size, set by a barrier in a nonequilibrium quasi-potential that closes at the saddle-node, and the long-run share of time in each state is fixed by the parameters and system size, not by the initial condition; the deterministic rate equation, where the initial condition decides, holds only on times short of that escape time | Vellela & Qian (2009), LIT-1890, read in NOTE-1474, on the Schlögl model, with Falasco & Esposito’s general frame (LIT-1892, §§III.C, IV.H, VI; NOTE-1472). Active: the slow eigenvalue, the barriers, the initial-condition independence and the closing of the barrier at the fold were checked numerically on the paper’s model. It does not say which state is the more stable without computing the quasi-potential, and the paper’s own large-volume rate formula carries a spurious 1/V factor. | Active |
| THEORY-1274 | Under a shared linear output head, the cross-entropy gradient of a multi-token-prediction head at its hidden state is exactly the gradient of a forward KL toward the model’s own position-aligned next-token prediction, held fixed, plus the next-token cross-entropy gradient at the aligned state, which alone carries the sampled target; the identity is about gradients and computations, not information, and in heads fed the intervening tokens the student and its teacher condition on the same evidence | Teoh (2026), LIT-1896, eqs. 1–3 and Appendix B; read in NOTE-1470 and checked by finite differences. Active: elementary and checked. It is a gradient identity, not a decomposition of the loss’s value, and it does not by itself make the self-distillation a projection onto less information: in the DeepSeek-style heads of the paper’s main results the student sees the same tokens as its teacher. | Active |
| THEORY-1287 | The experiments mechanists use to find a mechanism’s components (correlation, lesion, stimulation, and intervening on the whole while recording the parts) run in both directions between part and whole and, by Craver and Bechtel’s own account, show only prima facie that an item is a component, because an item not interactively integrated into the system could be a background condition; they give no test of integration, so the experiments do not by themselves fix where a mechanism ends | Craver and Bechtel (2007), LIT-1928, pp. 553–554, read in NOTE-1491. The four kinds of interlevel experiment “share a form with Mill’s methods” and are symmetrical. A lesion deficit is “only prima facie” evidence of componency, since a part “not interactively integrated” may be a background condition (n. 10). “Integrated” is never defined, and nothing in the four experiments tests where the system’s edge is. That leaves the mechanism’s boundary to be fixed some other way. | Active |
| THEORY-1288 | If a mechanism’s behaviour supervenes on its constituents’ behaviours and constitution is not causal, every cause of that behaviour is also a cause of some constituent along another path, so no intervention can change the whole without separately changing a part; mutual manipulability by ideal interventions is then unsatisfiable, and once interventions are relaxed to admit these common causes, every model with constitutive dependence has a model without it that implies the same correlations under every manipulation, even in ideal experimental conditions | Baumgartner & Gebharter (2016), LIT-1934, pp. 10–13, 18–20, and Baumgartner & Casini (2017), LIT-1939, pp. 6–11; restated by Baumgartner, Casini & Krickel (2020) as Systematic Fat-Handedness. Read in NOTE-1493 and NOTE-1490. Active as a conditional: valid on supervenience and non-causal constitution, and checked step by step. It does not say that constitution cannot be known, only that no finite set of manipulations of whole and part shows it; and it is about simultaneous whole–part dependence, not about a diachronic cause inside a system, which can be intervened on surgically. | Active |
| THEORY-1289 | In a dynamical model anchored to an identified mechanism, holding one part’s variable fixed and finding that the phenomenon persists indicates that the part is not a component of the mechanism responsible for that phenomenon, as when Smolen et al. fixed CLOCK and circadian oscillation continued; the verdict is relative to the phenomenon and only as good as the model, and it decides which pathways the phenomenon needs, not which lie inside the system | Bechtel and Abrahamsen (2010), LIT-1936, contribution C3 (p. 326), read in NOTE-1492. Smolen et al. (2001) fixed the concentration of CLOCK in a reduced model of the Drosophila clock, and oscillation continued. They concluded that the positive CLOCK loop “was not critical”. That is a part that belongs to the system and not to “the responsible mechanism for that particular phenomenon”. The clamp is an ideal intervention, cheap in a model and hard in vivo. It is not a test of where the system’s boundary lies. | Active |
| THEORY-1290 | A dynamical model whose variables are properties of parts and operations already identified by experiment explores the dynamics of that mechanism, while one that posits its parts de novo and is checked only by fit to behaviour shows at most how a mechanism of that kind could produce the behaviour; fit can rule such models out but usually cannot decide between competing architectures that each fit, and what decides is independent evidence about the parts | Bechtel and Abrahamsen (2010), LIT-1936, read in NOTE-1492. Circadian models came after the clock’s parts were found in the laboratory, and they explore those parts’ dynamics. Cognitive models posit their parts. “Empirical evidence can rule out specific models, but usually cannot decide between competing architectures” (p. 332). It does not say de novo models are worthless: they make the same six kinds of contribution, at weaker standing. Nor does it say that anchoring makes a model right. | Active |
| THEORY-1293 | Given that every change in a phenomenon is simultaneously realized by some constituent on every lower level, that simultaneous changes are not causally related, that co-occurrence is transitive, and that the parts and their levels are already known, a single intervention that changes a part and the phenomenon simultaneously while directly changing at most one behaviour on each lower level shows that the part is a constituent; the condition presupposes the boundary and the level hierarchy it is applied to and cannot supply them | Baumgartner, Casini & Krickel (2020), LIT-1924, §§3–4, read in NOTE-1494. Active as a conditional: the eight-line reductio is valid on Universal Constitution, Simultaneity and transitivity of co-occurrence, and checked. It does not say such interventions exist in any real system, that the condition is necessary, or where the boundary lies; and its line between causation and constitution is simultaneity, which is sharp only at a fixed time resolution. | Active |
| THEORY-1295 | Metzinger 2024 calls consciousness without an egoic self or first-person perspective “empirically established” on the strength of retrospective, self-selected reports from meditators, while its own methods chapter grants that a remembered report of a selfless state is “highly dubious” evidence that the state was selfless when it occurred, and leaves that problem open | Metzinger (2024), LIT-1943, read in part in NOTE-1499. The summary chapter says that “full-absorption episodes, nonegoic self-awareness, and nondual modes of consciousness are empirically established phenomena” (p. 457), and that the strong reading of subjectivity “is false” (p. 459). The methods chapter grants self-selection, theory contamination, unreliable recall and, at p. 19, that reports of having been selfless are “highly dubious from a methodological perspective” because autobiographical memory represents events as a self’s. It calls the question “open”. Active: the two passages are read and quoted. Whether selfless consciousness occurs is not settled either way here. | Active |
| THEORY-1301 | In the Haken–Kelso–Bunz potential V(φ) = −a cos φ − b cos 2φ, switching the parameters in time from no landscape (a = b = 0) to two wells (a = b = 1) to one well (a = 1, b = 0) makes the spread of the relative phase across trials rise while both wells are open and fall after one is removed, the pattern Sánchez-Cañizares compares qualitatively with post-stimulus quenching of trial-to-trial variability in decision tasks; under Gaussian white noise of the stated strength q = 1 the spread of cos φ settles near 0.40, not at 0 | Sánchez-Cañizares (2026), LIT-1992, read in full in NOTE-1506 from the publisher’s open-access PDF. Active for the model’s behaviour, which this reading checked: the bistability condition |a| < 4b analytically, and the rise and fall by re-simulation (20,000 paths). The paper’s statement that the spread decreases “to 0” holds without noise and fails at q = 1. It does not say that the model fits the cited EEG data, which the author calls “only qualitative”, or that the parameter changes are the mind’s (THEORY-1302). | Active |
| THEORY-1303 | For an uncertain linear program, a decision fixed before the data are revealed must satisfy every realization at once (the robust counterpart, ∃ solution ∀ data), while a decision that may adjust after them need only have a feasible adjustment for each (the adjustable robust counterpart, ∀ data ∃ adjustment); the second is never more conservative and can be feasible where the first is not, the two coincide when each constraint’s uncertainty is its own, and decisions may be restricted between them by what part of the data they read or by the form of the rule | Ben-Tal, Goryashko, Guslitzer and Nemirovski (2004), LIT-1975, read in NOTE-1503. Active: the definitions, the two worked examples (checked by hand), Theorem 2.1 and its proof, and the reported inventory results with their sizes. It does not say the information-basis case is solved in general (the paper asserts it extends), nor that affine rules lose nothing. | Active |
| THEORY-1304 | In latent Dirichlet allocation each document draws its own topic mixture from a Dirichlet shared across the corpus and draws a topic afresh for every word, a three-level model of conditionally independent assignments; the mixture of unigrams, which draws one topic per document, is its two-level comparison, and the Dirichlet is a distribution over mixtures, not over topics | Blei, Ng and Jordan (2003), LIT-1948, read in full in NOTE-1519. Active as a statement of the model’s structure, which is definitional and is stated in the text (pp. 996–1001); the variational updates were checked against Appendix A. It does not say LDA fits text well, which rests on perplexity on two corpora, or that its topics are anything beyond devices for representing word distributions, which the authors disclaim. | Active |
| THEORY-1305 | In Kasser and Ryan’s two cross-sectional samples, holding constant how much a person aspires to overall, the relative importance of extrinsic aspirations (money, fame, image) goes with less self-actualization and vitality and more physical symptoms, and of intrinsic aspirations (growth, relationships, community, health) with the reverse; the paper does not test whether this survives the reasons for pursuing the goals | Kasser and Ryan (1996), LIT-1977, read in NOTE-1525 from the scan on the SDT site. Active as a careful reading of reported associations with their sizes and samples: 100 community adults and 192 students, all self-report. The intrinsic and extrinsic results are nearly one contrast, since both enter after an overall score averaged across all seven domains. Not shown: causation, need satisfaction as mediator, or that the pattern holds once the autonomy of one’s reasons is controlled, which the paper names as an open question. | Active |
| THEORY-1307 | In Slemp et al.’s meta-analysis of 881 mostly cross-sectional samples, perceived support for autonomy, competence and relatedness each correlated strongly with satisfaction of all three needs (ρ = .44–.62) and negatively with their frustration, but the three supports intercorrelated .67–.71, and their need-specificity appears only in joint models: with all three entered, relatedness support carried relatedness satisfaction (β = .56) and autonomy support added nothing to it (β = .007) | Slemp et al. (2024), LIT-1969, read in NOTE-1507 from the CC BY Online First PDF. Active as a statement of what the meta-analysis reports, its Table 2 reproduced from its Table 1 matrix. It is a statement about correlations, which the authors interpret as effects on theoretical grounds only; it does not say support causes need satisfaction, or that the needs are distinct in the world. | Active |
| THEORY-1309 | For a small-noise diffusion the Freidlin–Wentzell quasi-potential equals 2 inf ∫ |b| sin²(θ/2) ds over curves, θ the angle between curve and drift, and the minimizing curve is the path a noise-driven transition follows with probability tending to one; with gradient drift and identity noise that curve runs against the drift, the time-reversed relaxation path, while a far-from-gradient drift, as in the Maier–Stein model above β = 4, makes it depart from the heteroclinic orbit | Heymann and Vanden-Eijnden (2008), LIT-1981, read in part in NOTE-1504 (§§1, 2.1–2.2, 4.1, 5). Active: the geometric formula and the gradient case were re-derived for this reading; the path concentration is Proposition 2.3, whose proof was not read but whose statement is standard Freidlin–Wentzell theory; the Maier–Stein threshold is the paper’s computation agreeing with Maier and Stein’s. It does not say anything at finite noise, or for degenerate noise, which the paper’s ellipticity assumption excludes. | Active |
| THEORY-1310 | In Sheldon, Elliot, Kim and Kasser’s three studies of most satisfying events, self-esteem was as salient as autonomy, competence and relatedness in every sample and the largest unique predictor of event affect balance among ten candidate needs in two of the three US samples (β = .28, .38); the authors, SDT researchers, concluded that there may be four fundamental needs and judged it imprudent to treat self-esteem as an outcome | Sheldon et al. (2001), LIT-1966, read in NOTE-1502 from the scan on selfdeterminationtheory.org. Active as a statement of what the paper reports and concludes, checked against Tables 2–7 and p. 336. It does not say self-esteem is a need: the criteria are self-reports of one remembered event, and the self-esteem items read like well-being, which is QUESTION-369’s problem in miniature. | Active |
| THEORY-1312 | A phase-amplitude coupling measure can be significant when only one non-sinusoidal rhythm is present, because the rhythm’s harmonics carry amplitude that depends on its own phase; such coupling is confined to bands spanning the rhythm’s harmonics, so a measured coupling is evidence of one rhythm modulating another only after waveform shape is controlled | Donoghue, Schaworonkow and Voytek (2021), LIT-1999, §5 and Fig. 5D–E, read in NOTE-1529 from the PsyArXiv preprint, standing on Kramer et al. 2008 and Cole et al. 2017. Active: checked by a short simulation in which a sharp-peaked 10 Hz wave gave a modulation index of 0.021 between its phase and 25–60 Hz amplitude against 0.0001 for a cosine, and none in a single-harmonic or an 80–250 Hz band. It does not say any particular published PAC is an artefact, nor that broadband coupling far above the harmonics is. | Active |
| THEORY-1313 | In two 7–8B chat models, the shift that finetuning induces along a difference-of-means persona vector correlates with how strongly the finetuned model then expresses that trait, r = 0.76–0.97 across synthetic datasets for evil, sycophancy and hallucination, above cross-trait baselines of 0.34–0.86; the last-prompt-token projection predicts a response’s trait at r = 0.75–0.83 overall but mainly by telling prompt types apart | Chen et al. (2025), LIT-1998, read in NOTE-1524 (main text in full, appendices C.2, G.2 and M). Active as the reported measurements, with their sizes and scope: two models, three main traits, synthetic datasets and an LLM judge. It does not say that persona vectors are the mechanism of persona change, which the authors leave open, or that the method finds personas it was not told to look for. | Active |
| THEORY-1315 | As a stochastically forced system approaches a fold bifurcation its dominant eigenvalue goes to zero, so it recovers ever more slowly from small perturbations, and under ongoing perturbation this critical slowing down shows as rising lag-1 autocorrelation and rising variance; skewness and flickering are further warnings not caused by slowing; but the same signals, less pronounced, also precede thresholds that are not bifurcations, so they mark proximity to a broad class of thresholds and not which kind, nor when the shift will come | Scheffer et al. (2009), LIT-1956, read in NOTE-1536 (Skimmed) from the version reprinted as chapter 2 of Dakos’s 2011 thesis. Active for the slowing-down result, which is standard (Wissel 1984), checked in the chapter’s Box 2.2 example, and re-derived and simulated here for variance σ²/(2|λ|) and autocorrelation e^{λΔ} since the chapter’s Box 2.3 was not extractable. The scope and timing caveats are the authors’ (p. 31). It does not say the signals have been shown in chaotic systems, which the review calls “less well studied”, or that any of the reviewed empirical cases was checked here. | Active |
| THEORY-1316 | In eight samples pooled to 3,692 respondents, the covariance of the BPNSFS items is organized more by item keying than by need: factors sharing a keying direction correlate .64 to .75 while satisfaction and frustration of the same need correlate .17 to .38, a higher-order model grouping the six scales by keying fits better than one grouping them by need, and the frustration items carry almost no thwarting content | Murphy et al. (2023), LIT-1978, read in NOTE-1530 from the PMC author manuscript. Active as a statement of what their analyses show, checked against Tables 1–3 and the reported fits: the factor intercorrelations, the two higher-order models and their indices, the external correlates sorted by keying, and the items. It does not say that the frustration scales measure nothing but keying; the authors hold that this cannot be settled with existing data, and the inference is THEORY-1311’s. | Active |
| THEORY-1317 | During held covert spatial attention in human fronto-parietal and visual cortex, the amplitude of broadband high gamma (80–250 Hz) is coupled to the phase of an endogenous 2–5 Hz rhythm, more strongly for the contralateral field, and the trial-by-trial strength of that coupling correlates with faster detection; the evidence is correlational, so it shows that the coupling tracks attention, not that it improves it | Szczepanski et al. (2014), LIT-1982, read in NOTE-1528 from the publisher’s open-access PDF. Active for what it states: the reported counts (123 of 273 electrodes coupled; 55 stronger contralaterally; 28 correlating with RT, r(41) = −0.51 and r(47) = −0.37 in the examples) were checked against the text, and the ERP and power controls are in it. The sharp edge is the causal one: no manipulation of the coupling was made, and the paper’s “facilitate” and Jennings’s “improve” outrun it. | Active |
| THEORY-1318 | Among 31 original self-report scales of psychological need fulfillment found by a 2023 systematic search, 21 measure satisfaction only, one measures frustration only and one thwarting only; all are self-reports of experience or of the perceived environment, the field uses satisfaction, frustration, support and thwarting interchangeably in places, and several frustration scales are negatively worded satisfaction items | Kermavnar, Avsec, Huang and Desmet (2024), LIT-1988, read in NOTE-1509 from the version of record. Active as a statement of what the review counted and found, checked against its §3 and Tables 2–4, within the scope it set (English, healthy general populations, title terms). It does not rank the scales, test their validity, offer a criterion for need-hood, or take a side on the two-dimensional model. | Active |
| THEORY-1320 | In Ntoumanis et al.’s meta-analysis of 73 SDT-informed health interventions, mostly randomized, the interventions raised perceived need support (g = 0.64), autonomy and competence satisfaction (0.37, 0.31), autonomous motivation (0.30) and health behaviour (0.45) by the end of the intervention, but not relatedness, controlled motivation or amotivation, and their small effect on psychological health (0.29) was gone at follow-up (0.14, ns); need frustration was not an outcome and mediation was not tested | Ntoumanis et al. (2020; issue 2021), LIT-1962, read in NOTE-1526 from the open-access PDF. Active as a statement of what the meta-analysis reports (Table 1, p. 222), with its sizes, intervals and k. It is the record’s experimental evidence that manipulating need support moves SDT’s proximal constructs; it is not evidence that need frustration is a process in psychopathology (THEORY-052). | Active |
| THEORY-1322 | In Belgian adolescents, need frustration measured by the twelve BPNSFS frustration items statistically absorbs most of the association between depressive and eating disorder symptoms: cross-sectionally their latent correlation falls from .50 to a non-significant .18, and in a two-wave latent change model their correlated change falls from .43 to .24; its mediating role is shown in correlated change over the same six months, not in temporal order | Campbell, Boone, Vansteenkiste and Soenens (2018), LIT-1964, read in NOTE-1533 from the early-view PDF on selfdeterminationtheory.org. Active as a statement of the reported models, checked against the text and Tables 1–2. It is the study THEORY-052 cites through LIT-561 p. 12 as the one longitudinal evidence for need frustration as transdiagnostic. It does not show that frustration precedes the symptoms, does not reach externalizing problems, and its measure is the one THEORY-1311 doubts. | Active |
| THEORY-1323 | Under weak local noise a many-body operator decays at a rate proportional to its size, so a Hamiltonian’s small-sized integrals of motion, which chaotic dynamics does not spread, decay slowly while generic operators, which it spreads, decay fast; in exact diagonalization of Ising and Heisenberg chains of up to nine sites at weak depolarizing noise, the slowest Lindbladian eigenoperators are linear combinations of the known exact and approximate small integrals of motion, and which approximate ones rank among the slowest moves with the noise strength | Yang and Abanin (2025), LIT-1958, read in NOTE-1534. Active as a statement of what the paper derived exactly and what its exact diagonalization showed at the sizes it reached. It does not say that the correspondence holds at large system size (THEORY-1343), that every integral of motion is found (those above size 3–4 are not), or that slowness ranks conserved quantities independently of the environment. | Active |
| THEORY-1327 | As Ryan and colleagues’ 2022 meta-review maps it, the meta-analytic evidence for self-determination theory is concentrated in cognitive evaluation, organismic integration and basic needs theory, is mostly cross-sectional and heterogeneous, and has gaps: no meta-analysis of relationships motivation theory or of need frustration, one or two of causality orientations and goal contents, and none that separates integrated from identified regulation | Ryan, Duineveld, Di Domenico, Ryan, Steward and Bradshaw (2022), LIT-1967, read in NOTE-1500 from the accepted manuscript on OSF. Active as a statement of what the review reports about the 60 meta-analyses it found to July 2021, which is checkable against its Tables 1 and 2 and its own summaries. It does not say the meta-analyses are sound, since the review grades none of them, and it is written by the theory’s co-founder, with several included meta-analyses by its authors. | Active |
| THEORY-1330 | In Gemma 2 27B, Qwen 3 32B and Llama 3.3 70B, activation vectors for 275 prompted character roles span a low-dimensional space whose first principal component runs from fantastical to Assistant-like roles, with role loadings correlated above 0.92 between models and the default Assistant at its extreme; the contrast vector from the default Assistant to the role mean aligns with it and is largely present before post-training | Lu et al. (2026), LIT-1991, main text read in NOTE-1505. Active as the reported geometry, with its sizes: three open-weight models, judge-filtered role vectors at one layer, base-model evidence from one model’s PCs and two models’ steering. It does not say the Assistant is a single direction (the authors call the linear assumption “likely flawed”), nor that the axis’s meaning beyond its ends is settled. | Active |
| THEORY-1331 | Deci and Ryan’s General Causality Orientations Scale measures autonomy, control and impersonal orientations as three separately scored and largely independent dispositions, reliable and stable over two months, whose validity rests on correlations with self-report measures in small samples, with behaviour its weakest support | Deci and Ryan (1985), LIT-1955, read in NOTE-1531 from the scan on the SDT site. Active as a statement of what the scale is and what its founding paper showed: α .69–.74 and near-independent subscales in 636 students, retest .71–.78 in 51, and a nomological table in samples of 42 to 87. It does not say that orientations cause behaviour, or that the orientations parallel the three kinds of event of cognitive evaluation theory, which the paper asserts and its data only suggest. | Active |
| THEORY-1332 | In Bishop’s account, chaos is a property of deterministic dynamics, so sensitive dependence is not indeterminism; under determinism it may model deliberation but is not sufficient for incompatibilist freedom, and under indeterminism the incompatibilist (Kane) must still show that agents can harness amplified indeterminism; and amplification itself may fail, since damping, the sign of the local Lyapunov exponents when a perturbation arrives, and the brain’s parallel redundancy limit it | Bishop, SEP “Chaos” (2008; revision of 2024-10-11), LIT-1961, §§4, 6.1 and 6.4, read in NOTE-1535. Active as a statement of the entry’s position, checked verbatim, and its §4 dynamics is elementary. The sharp edge: the brain-redundancy point is asserted without evidence, and the entry does not argue that harnessing is impossible, only that it has not been shown. | Active |
| THEORY-1333 | Bishop’s contextual emergence, as stated in his 2024 SEP revision, is an ontological account: stability conditions not given at smaller and faster scales (parameter values, dimensionality, large-scale flows) constrain or enable what is possible and actual at those scales, as convection rolls constrain the motions of fluid molecules, without the brute new causal powers of strong emergence; laws bound possibility and contexts fix actualization | Bishop, SEP “Chaos” (revision of 2024-10-11), LIT-1961, §§6.2–6.3, read in NOTE-1535, summarizing Bishop, Silberstein and Pexton 2022 (LIT-1993, not reached). Active as a statement of the view, checked verbatim. It corrects the record’s gloss (THEORY-1314, NOTE-1508) that Bishop’s contextual emergence concerns only the well-definedness of a higher-level description; it is not evidence that the view is true, and it sets the view against causal powers, not for them. | Active |
| THEORY-1335 | In Juarrero’s constraint account, context-independent constraints take conditions away from equiprobability and context-dependent ones away from independence (after Gatlin); enabling constraints couple separate processes into a coherent dynamic whose constitutive constraints then act top down as second-order governing constraints that confine its components to a subregion of state space; a constraint is defined as what raises or lowers barriers to energy flow without transferring kinetic energy, carries no time-scale condition, and the governing regime is held not to be other than the components | Juarrero (2023), LIT-1995, read in NOTE-1532 (Skimmed; chapters 2, 3, 6, 7, 11, 12, 15 and 16 in full). Active as a statement of what the account is: its kinds of constraint, its definition, and its top-down step, all stated in the text (pp. 25, 40, 67, 71, 79–84). It does not say that the account is true, that constraints transfer no energy, or that her constraint is the record’s (TERM-070’s). | Active |
| THEORY-1337 | Neural field recordings carry aperiodic 1/f-like activity with power at every frequency, whose exponent varies with age, state and task; so power in a band, correlated power changes across bands, and ratios of band power can arise from the aperiodic component with no oscillation changing, and 1/f structure is a ubiquitous background of field data rather than a mark of any particular organization | Donoghue, Schaworonkow and Voytek (2021), LIT-1999, §§1 and 3, read in NOTE-1529. Active: the spectral argument is elementary (an aperiodic signal has power at all frequencies, and a change in its exponent moves every band), and the simulations and cited cases (the ADHD theta/beta ratio) are in the text. It does not say aperiodic activity is noise without physiological meaning; the paper calls it “a physiologically informative feature”. | Active |
| THEORY-1338 | In the Basic Psychological Need Satisfaction and Frustration Scale as Chen et al. built it, the distinction between satisfaction and frustration coincides with item direction: every satisfaction item states a positive experience and every frustration item an aversive one, so the better fit of six factors over three, which is the scale’s evidence that frustration is a separate dimension, is also what a keying or valence factor would produce | Chen et al. (2015), LIT-2000, read in NOTE-1527 from the publisher’s typeset copy in the ULB repository. Active as a statement about the instrument, checked against its Table 3 item by item and its model comparison (ΔSBSχ²(12) = 450.79, p. 226). It does not say the frustration dimension is an artefact, only that this evidence cannot tell it from one; that test is Murphy et al.’s (LIT-1978), unread. | Active |
| THEORY-1342 | In Transformer attention the weights each query puts on the positions depend only on query–key compatibility, softmax(QKᵀ/√d_k), and the output is those weights applied to the values, so for a fixed input what a head selects and what it transports are separate factors; because queries, keys and values are all projections of the same input, the separation holds only with the input, and so the pattern, held fixed | Vaswani et al. (2017), LIT-1959, read in full in NOTE-1501, eq. 1 and §3.2.2; the circuit vocabulary is Elhage et al.’s (LIT-1534, THEORY-988). Active: it is the definition of the mechanism, checked against the equations. It does not say that heads are interpretable observers, which the paper supports only with visualizations, or that the separation survives composition across layers. | Active |
| THEORY-1344 | Empowerment is the channel capacity from an agent’s next n actions to its sensor n steps later, an open-loop, interventional measure of how many perceptibly different futures the agent can bring about, which counts no control the agent cannot perceive; given side information about the channel’s state it is never lower, so it is ordered by what the agent can read, from full state through its own sensor history to none | Klyubin, Polani and Nehaniv (2005, 2008), LIT-1965 and LIT-1951, read in full in NOTE-1511 and NOTE-1514. Active as definitions and one inequality, re-derived. It does not say empowerment is what organisms maximize, which the authors offer as a hypothesis in a weak form only, nor that it prices control cost or viability. | Active |
| THEORY-1345 | In three cross-sectional studies of undergraduates, most of the variance in adult attachment security lay within persons, across relational partners (56–79%, error included), and relationship-specific satisfaction of the needs for autonomy, competence and relatedness predicted relationship-specific security in every relationship tested, also with the relatedness items removed; relatedness was the strongest unique predictor, then autonomy, then competence | La Guardia, Ryan, Couchman and Deci (2000), LIT-1979, read in NOTE-1523. Active as a statement of the associations the three studies report, with their sizes. It does not say that need satisfaction causes security, which the authors say their data cannot show and believe runs both ways (p. 381). | Active |
| THEORY-1346 | In rat cortical slices with synapses blocked, extracellular fields of the local field potential’s size, applied by an electrode, shift the membrane potential of layer 5 pyramidal neurons by less than 0.5 mV at frequencies up to 100 Hz and, with spiking driven by current, phase-lock spike times without changing spike counts, most effectively for slow fields, and synchronize nearby neurons; the authors explain the effect by Kirchhoff’s circuit laws and a fixed firing threshold | Anastassiou et al. (2011), LIT-1953, read in NOTE-1516 from the gwern.net scan at the owner’s direction. Active as a statement of what the experiments measured, which the paper reports with sizes, samples and corrected tests. It does not say that the fields neurons generate do this in vivo, that ephaptic coupling bears on attention or experience, or that it is evidence against micro-level description, which is the use Jennings makes of it (LIT-1963, p. 35). | Active |
| THEORY-1347 | In Patrick, Knee, Canevello and Lonsbary’s three correlational studies, perceiving that a romantic partner fulfils one’s needs for autonomy, competence and relatedness goes with better relationship quality, with relatedness the strongest unique predictor and the partner’s fulfilment adding to one’s own; but the measure does not separate the needs, its autonomy and competence subscales correlating .96 once corrected for unreliability | Patrick et al. (2007), LIT-1971, read in NOTE-1515 from the article on the SDT site. Active as a careful reading of reported results with their sizes and samples: eight pooled samples (N = 1,918), 66 couples, and 120 diarists with 908 disagreements, all self-report, mostly women students. It does not say need fulfilment causes relationship quality, that autonomous motivation mediates it (the mediation is cross-sectional), or that the three needs are distinct dimensions. | Active |
| THEORY-1348 | In Froese’s Irruption Theory an agent’s motivations make a difference to behaviour only as an increase in its material underdetermination, measurable at best indirectly as neural or behavioural entropy, and such irruptions are by the theory’s own terms undirected: scaled up by chaos, 1/f noise or criticality, they loosen the current organization but “cannot force the formation of the behavior that is to come”, so the theory places behaviour’s direction in the body’s attunement and not in the irruption | Froese (2023), LIT-1950, read in full in NOTE-1521 from PubMed Central’s copy. Active as a statement of what the theory says: its three axioms, its three theses, and the commitment that irruptions are undirected, all in the text (§§5.1–5.2.3). It does not say that the theory is true, that neural entropy measures motivation, or that attunement answers the luck objection (THEORY-1321, Proposed). | Active |
| THEORY-1349 | In two cross-lagged panel studies of university students, a negatively worded self-report of need frustration predicted later depressive symptoms (standardized β .13 to .18 over three lags) and later negative affect, net of earlier depression and negative affect, while negative affect did not predict later depression; depression also predicted later need frustration at one lag of two | Levine et al. (2021), LIT-1974, read in NOTE-1510 from the version of record in PMC. Active as a statement of the reported paths, checked against Tables 2, 4 and 5. It does not say the frustration causes the depression: the models are traditional cross-lagged panels without random intercepts, so their lagged paths mix stable between-person differences, the reverse path was as large as the forward one at the first lag, and every variable is a self-report. | Active |
| THEORY-1350 | The Persona Selection Model holds that pre-training gives a language model a distribution over personas, that post-training updates it into a posterior over Assistant personas which runtime context keeps conditioning, and that the Assistant’s traits are a key determiner of behaviour; it disowns a single coherent persona and exhaustiveness, and names as open a router that selects personas and could carry goals no persona has | Marks, Lindsey and Olah (2026), LIT-1952, read in full in NOTE-1512. Active as a statement of what the model says and does not say, all of it in the text. It does not say that the model is true: the authors are “overall unsure how complete” it is, its evidence is surveyed rather than new, and its updating is offered as “can be viewed as”, with the strict form said to be too strong. | Active |