Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Mathematics — mathematics read for itself — category theory, topoi and sheaves, spectral geometry, set and measure theory — including mathematical frameworks applied to other fields.
259 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-005 v4 | The positive-pair density ratio is the kernel of the conditional-expectation operator on L²(p), so spectral representations are that operator’s eigenfunctions, well defined when the positive-pair χ²-divergence is finite | An inference assembled on 2026-09-26 from Johnson, El Hanchi & Maddison (LIT-258), Tosh, Krishnamurthy & Hsu (LIT-251) and HaoChen et al. (LIT-249), with a two-line derivation of the record’s own. Johnson et al. write the operator formulation on a finite space (App. E.2); no source states it with its integrability condition. | Proposed |
| 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-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-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-022 v4 | The reverse-engineered grokking network computes modular addition by multiplying characters of ℤ/p at a few frequencies, the same real two-dimensional irreducibles later found as circles in language models; in both the group comes from the task, not the network | Nanda et al. (2023), LIT-345, show that a one-layer transformer trained on addition mod 113 embeds its inputs at five key frequencies. It forms cos/sin(w_k(a+b)) and reads out Σ_k α_k cos(w_k(a+b−c)). The reading in terms of characters and irreducibles is NOTE-283’s, joined here to Engels et al.’s circles (LIT-322). It holds for one weight-decay model and its variants: dropout-trained generalisers are not Fourier-sparse. It is not evidence that a network discovered the group, which was given, or that the frequencies could be read off a spectrum. | Proposed |
| 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-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-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-038 | Category theory gives radical ontic structural realism no formal support: generalized elements exist in every category, morphisms relate objects the category presupposes, and Bain’s physical cases eliminate spacetime points, not relata | Lam & Wüthrich (2015), LIT-222, against Bain (2013), LIT-223. Bain’s “structure as object in a category” relabels relata rather than removing them, and his GR and TQFT cases dispose of points only. That the Yoneda lemma (LIT-221) strengthens the rebuttal is the record’s inference: no paper in the debate names it. This does not refute radical OSR. It removes one argument for it, and Eva 2016’s reply is unread. | Proposed |
| 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-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-084 v3 | Because the Gauss–Newton Fisher JᵀJ and the NTK Gram matrix JJᵀ share their non-zero spectrum, the lazy-regime Fisher of a ReLU network on spherical data inherits the NTK’s harmonic-degree blocks, and a spectral threshold on it is well posed only at gaps between blocks | The filer’s synthesis, stated by no source. The identity is linear algebra: JᵀJ (parameters × parameters) and JJᵀ (samples × samples) have the same non-zero eigenvalues. Under squared loss JᵀJ is the Gauss–Newton matrix and the Fisher. JJᵀ is the empirical NTK Gram matrix, which in the kernel regime on uniform spherical data has the degree-block spectrum of Bietti & Bach (2021), LIT-608: µ_k with multiplicity N(d, k). The threshold claim draws on Cao et al. (2019), LIT-624. Their guarantees are stated only for whole eigenspaces, with sample costs that grow as the gap below them shrinks. It also draws on Basri et al. (2019), LIT-612, where a structural zero block shows up in a finite sample as small non-zero eigenvalues. No paper filed computes a Fisher spectrum in this regime. | Proposed |
| 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-106 | In random-features ridge regression the double-descent peak is a divergence of the test error at the interpolation threshold 2N = n as the ridge goes to zero, which a positive ridge keeps finite and a ridge of the tuned size removes; minimum-norm linear regression has the same peak at n = d in exact finite-sample form | Liao, Couillet & Mahoney (2020), LIT-679: for random Fourier features with n, p and N large together, the test MSE has a closed form through a deterministic equivalent of the resolvent. As λ → 0 its corrections jump from O(1) to order 1/λ across 2N = n, det(Ω⁻¹) scales as λ, and the test error diverges at 2N = n unless the test data lie within about λ of the training data. At λ = 10⁻⁷ and 10⁻³ the curve has a singular peak; at the tuned λ ≈ 0.2 and at 10 it has none. Dereziński, Liang & Mahoney (2019), LIT-683: exact MSE of the minimum-norm estimator under a surrogate design, peaking at n = d with value σ² tr(Σ⁻¹), and below the threshold equal in expectation to ridge regression at the λn that makes the ridge effective dimension n. Proportional-limit theorems for one shallow model each; nothing here reaches trained deep networks. | Proposed |
| 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-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-157 | The classical theorems on sufficient statistics need only the copy/discard structure of Markov kernels: Basu’s theorem holds in every Markov category, and Fisher–Neyman and Bahadur need only strict positivity, not the existence of conditional distributions | Fritz (2020), LIT-772, read in NOTE-588, Theorems 14.5, 15.8 and 16.3 with the definitions of Sections 13–15. Sufficiency, completeness and ancillarity are stated with copy, discard, composition and almost-sure equality alone. Basu’s theorem follows with no further axiom; Fisher–Neyman and Bahadur use strict positivity, which Stoch has although it lacks regular conditionals. The match with the classical theorems is shown in FinStoch (Fisher–Neyman) and through bounded completeness in Stoch (Basu), not for measure-theoretic Fisher–Neyman. | Proposed |
| 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-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-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-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-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-180 v2 | In a tied single-head attention layer trained by ridge-penalized empirical risk minimization on high-dimensional Gaussian sequences, weight decay is a nuclear-norm penalty on the query–key map, and the learned map is a soft-thresholded noisy copy of the target: its outliers are recovered target directions and its bulk is finite-sample noise | Boncoraglio, Erba, Troiani, Xu, Krzakala & Zdeborová (2025), LIT-857: with S = WWᵀ, λ‖W‖²_F is exactly λ‖S‖_* (Appendix A), and in the limit n/d², p/d fixed the trained WᵀW/√(pd) has the law η·ReLU(S0 + δZ − εI), Z ∼ GOE, with δ a noise level that falls with the data and ε a threshold set by λ (Claim 4.1, replica/AMP, checked against Adam at d ≤ 400). It does not say that real transformers’ spectra arise this way: the data are isotropic Gaussian, the target is inside the model class, and the agreement with measured spectra is qualitative. | Proposed |
| THEORY-183 v2 | When the co-occurrence of words on a latent continuum depends only on their separation, a spectral word embedding places them on the Fourier modes of that continuum, so the circles and rippled curves of months, years and places come from a symmetry of the corpus statistics | Karkada, Korchinski, Nava, Wyart and Bahri (2026), LIT-860: proved for an embedding whose Gram matrix equals the co-occurrence matrix (or its absolute value) at full rank, with the exponential kernel solved for both boundary conditions; the symmetry measured in Wikipedia; matching PCA pictures in Gemma 2. That trained networks get their geometry this way is inferred, not tested: nothing in the paper changes the statistics and watches the geometry follow. | Proposed |
| 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-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-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-196 v2 | A finite tree embeds in the hyperbolic plane with worst-case distortion 1 + ε, but because a hyperbolic distance d takes about d bits to write, its coordinates need Θ((ℓ/ε) log deg_max) bits: logarithmic in the branching, linear in the longest path ℓ, so hyperbolic space is economical for shallow bushy hierarchies and not for deep ones | Sala, De Sa, Gu and Ré (2018), LIT-874. The upper bound is Sarkar’s construction with the paper’s bit count. The lower bound is proved for H² and one family of trees, with an approximation, and dimension’s help is shown only as an upper bound. The cost is a cost of coordinates in a fixed-point model, not of the information in the tree, and it says nothing about Euclidean or linear-direction representations of the same hierarchy. | Proposed |
| 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-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-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-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-254 | Tangle-defined cohesion matches network communities only where community structure is strongly delineated, and tangles of one to three vertices recur beside the ones that do | Salter-Duke (2024), LIT-955, a PhD thesis read in full in NOTE-814. Edge-connectivity tangles, computed exactly at low order on eleven small real networks, give communities close to standard methods where structure is strong (yeast PPI) and none or almost none where it is weak (karate club, C. elegans, college football at the orders reached), and where tangles divide a network some hold only one to three vertices. The relation is read by eye from eight networks, and the vertex sets depend on the author’s assignment rule. | Proposed |
| THEORY-257 v3 | Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many | Barnett and Seth (2023), LIT-979, read in NOTE-791: definition (2.6), transitivity (App. A), the causal-graph form (2.12), the state-space closed form (3.18) and the genericity count (3.25)–(3.26). The target is the coarse variable’s own future, so no outside query is chosen. But the system/environment boundary, the scale and the class of maps are inputs, and the worked theory’s candidates are linear subspaces, not partitions. The cost is multimodal, with every local optimum kept. The criterion is closure, not locality or informativeness about a far environment. | Proposed |
| 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-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-277 v2 | For individual data, at every bound on model complexity, the model with the shortest two-part code is, to within O(log n), the best-fitting model in randomness deficiency, although the fit itself cannot be computed or approximated | Vereshchagin & Vitányi (2002/2004), LIT-239, read in NOTE-787, Theorem IV.8: β_x(α) + K(x) and λ_x(α) agree to within O(log n) for a string of length n. Every set minimising K(S) + log |S| at complexity α is within O(log n) of the best fit there. h_x and λ_x are upper semicomputable, while β_x is not semicomputable. Restated by Grünwald & Vitányi, LIT-224, §6.2. Finite-set models; computable distributions at O(log n) cost. It says nothing at the (k/2) log n scale of parametric model selection, and nothing about K(A) + K(x | A). | Proposed |
| 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-288 v2 | In small CIFAR and MNIST autoencoders whose only basis-privileging primitive is the activation function, elementwise tanh and Leaky-ReLU concentrate latent representations into narrow beams arranged by the activation’s discrete symmetry group, even in layers no activation precedes, while their orthogonally equivariant radial counterparts leave them spread, so axis-aligned discreteness there comes from the primitive, not the task | Bird (2025, v4 2026), LIT-976, read in NOTE-759. Autoencoders of width 12–32 and up to three hidden blocks on CIFAR and MNIST, with rotation-equivariant affine maps, initialisation and optimiser. Elementwise tanh (B_n) and Leaky-ReLU (S_n) give beams along or against the basis, or on the ±1 diagonal, and radial tanh and Leaky-ReLU (O(n)) do not. The evidence is density plots, five seeds a condition, with no statistic. The downstream claims, that this causes grandmother neurons, superposition and neural collapse, are untested. | Proposed |
| 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-303 | A measurement scenario whose compatibility hypergraph is acyclic admits no contextuality: locally consistent context distributions always glue to a global one, so a contextual scenario needs an induced cycle of length at least four | Vorob’ev (1962), as stated in Budroni, Cabello, Gühne, Kleinmann & Larsson (2022), LIT-263, §V.B.2, Eq. 79, and App. A.6, read in NOTE-844. Distributions that agree on overlaps and sit on a hypergraph with the running-intersection property glue, step by step, to one joint distribution. For sharp measurements this means a chordal compatibility graph. So CHSH’s square and KCBS’s pentagon are the smallest contextual scenarios. Stated and illustrated in the review, not proved there. | Proposed |
| 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-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-323 v2 | Under spatially uncorrelated white noise, a stable non-normal linear system holds more variance than the sum of its pole contributions, and equality holds only when the operator is normal, so its spectrum understates the standing variance | Farrell & Ioannou (1996), LIT-931, read in NOTE-831, with Trefethen et al. (1993), LIT-932, read in NOTE-839. Linear algebra, worked in examples: growth over time t is the top singular value of the propagator, which the eigenvalues bound from below, not above. It holds in a chosen norm, and a norm can always be found that makes the operator normal. It says nothing about whether the growth survives nonlinearity. | Proposed |
| THEORY-325 v2 | Random variation of a linear operator destabilizes it generically, needing less variation the more non-normal and less damped the mean operator | Farrell & Ioannou (1996), Part II, LIT-929, read in NOTE-842. Necessity is a one-line bound and the destabilization is shown in explicit examples. Genericity under noise is cited to stochastic stability theory, and the role of non-normality and damping is shown numerically. In the deterministic examples, variation much faster than the transient growth destabilizes nothing. Since 2026-10-19 the deterministic half is THEORY-724, and this THEORY holds the stochastic half. | Proposed |
| 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-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-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-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-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-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-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-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-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-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-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-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-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-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-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-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-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-486 | For exact Betti Lagrangians in the cotangent bundle of a punctured surface, a finite creative spectral network exists, a point lies on its active walls exactly when a pseudo-holomorphic strip survives the adiabatic limit, and the Family Floer functor equals Gaiotto–Moore–Neitzke non-abelianization there | Casals & Nho (2025), LIT-017, an unrefereed v1 preprint read in full in NOTE-022 by a reader who is not a Floer theorist. Theorems 1–3 are stated for any rank and for positive-braid asymptotics. The proofs cover exact Betti Lagrangians with finite creative networks; the meromorphic case adds hypotheses in prose, and the Family Floer functor is not built there. The analytic core is taken by citation from Ekholm 2007, which the authors patch, and from the second author’s own preprint. Theorem 5, on networks from Demazure weaves, has no written proof and is not part of this account. | Proposed |
| 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-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-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-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-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-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-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-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-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-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-565 v3 | From a symmetric, positivity-preserving, positive semi-definite affinity, the normalized Markov operator is conjugate to a symmetric one, so its leading eigenfunctions scaled by powers of their eigenvalues embed the data with Euclidean distance equal to diffusion distance, and a density-corrected normalization recovers the Laplace–Beltrami operator independently of how the data were sampled | Coifman et al. (2005), LIT-981, read in full in NOTE-1441. The finite construction and its isometry are THEORY-728; the paper’s own diffusion distance is the one ã^(m) induces, not the inverse-stationary-weighted one. Theorem 2.1 and the α-family are stated without proof; the companion (LIT-1246, NOTE-985) proves the α-family’s limit, Active without boundary as THEORY-744, and its heat-kernel step is QUESTION-231. Self-adjointness comes from the kernel’s symmetry, and “directed diffusion” is anisotropic, not a non-reversible chain. | Proposed |
| 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-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-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-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-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-604 v2 | Most mathematical dualities are adjunctions between one category and the opposite of another, which become dual equivalences on their fixed points; what makes them dualities rather than equivalences is the reversal of arrows, so many of the ‘dualities’ physicists name are, in the mathematician’s sense, equivalences or symmetries | Corfield (2017), LIT-115, read in NOTE-141. A short programmatic paper. That most dualities of interest are covered by constructions around adjunction is shown by worked exposition of standard cases, not proved. There is unlikely to be “a single monolithic account” (pp. 59–60). Reserving “duality” for structural reversal is a stated preference, and the verdicts on string dualities are a paragraph each. The proposal that philosophers of physics need homotopy type theory is an expectation. | Proposed |
| THEORY-618 | Neural circuits that must represent hierarchically organised states in few dimensions use a hyperbolic geometry, three-dimensional where it has been measured, because hyperbolic space holds exponentially many well-separated states within a small radius; the argument from Zipf’s law to this geometry does not hold, and the evidence is two Betti-curve studies tested only against uniform Euclidean nulls | Sharpee (2019), LIT-875, read in NOTE-670: a four-page programmatic review with no new result, whose reading filed no THEORY because its claims are conjectures. Filed here as Proposed because the record holds two tests of the conjecture, odour statistics (LIT-872, THEORY-191) and CA1 place-cell correlations (LIT-876, THEORY-192), and no general statement. The Zipf step shows a shared exponential count of states, which any system with extensive entropy has, not a shared metric, and the rigidity argument for three dimensions cites Mostow’s theorem, which concerns finite-volume manifolds and says nothing about noisy point clouds. | Proposed |
| 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-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-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-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-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-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-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-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-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-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-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-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-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-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-747 | In Fourier–Galerkin large-eddy simulation, a first-order finite-memory Mori–Zwanzig closure tracks filtered-DNS energy decay and dissipation on the viscous Burgers equation and on the incompressible Taylor–Green vortex, isotropic turbulence and channel flow about as well as dynamic Smagorinsky where that model is at home and better than Smagorinsky and the t-model on the transitional Taylor–Green vortex; the memory length that fits it is set by the cutoff, not by the flow alone, scaling in Burgers roughly as 0.2 over the spectral radius of the resolved variables’ Jacobian | Parish and Duraisamy (2017), LIT-1259, read in NOTE-973. The closure is THEORY-760’s finite Markovian augmentation. Here is how it did, as far as the paper’s plots show: close to filtered DNS in integral quantities, comparable to dynamic Smagorinsky in equilibrium turbulence, clearly better on a transitional flow, and with the subgrid transfer peaked at the cutoff where DNS puts it but about 40% too deep. The paper calls the predictions “excellent” with no numbers. Its fitted memory length shortens as the resolved dynamics speed up, so the memory belongs to the cutoff as much as to the flow. | Proposed |
| 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-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-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-763 v2 | For every α < 1/3 there is a nonzero weak solution of the 3D incompressible Euler equations on the periodic box, with velocity Hölder-continuous of exponent α in space and time, that vanishes outside a finite time interval; its energy is not conserved, so the energy flux across a mollification scale ε does not vanish as ε → 0, and it runs in both directions, out of the vanishing scales while the energy rises and into them while it falls | Isett (2018), LIT-1255, skimmed in NOTE-974: the negative half of Onsager’s conjecture. With THEORY-749 (Constantin, E and Titi, the positive half) it makes 1/3 the threshold. For the record, its point is that below 1/3 the energy flux into vanishing scales can survive the limit, which is the half of A3’s point that THEORY-749 could not give. It does not say that this flux is a dissipative cascade. Isett’s solutions start from rest, gain energy and lose it again, so the surviving flux takes both signs. Nor does it say that such solutions are limits of Navier–Stokes or describe turbulence. | Proposed |
| 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-772 v2 | For every β < 1/3 there is a weak solution of the 3D incompressible Euler equations on the periodic box, in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}), whose Duchon–Robert dissipation is the smooth nonnegative density E(t, x) = −e(t)e′(t)ρ₀(x₁, x₂)² fixed at the construction’s first step; for such a solution the local coarse-grained energy flux into scales below ℓ converges, as ℓ → 0 and as a distribution in space and time, to E, so the transfer into vanishing scales is one-way at every place and time in the limit, nonzero inside a periodic array of pipes and zero outside them; where and how fast energy is dissipated is chosen by the construction, not forced by the equations, and nothing is shown about the sign at any positive scale | Giri, Kwon and Novack (2023; Annals 2026), LIT-1289, skimmed in NOTE-1022: the L³-based strong Onsager theorem. Below 1/3, in Constantin, E and Titi’s own scale of spaces (THEORY-749), there are weak Euler solutions satisfying the local energy inequality. In the proof the Duchon–Robert measure is a prescribed smooth function, positive in a periodic array of pipes and zero elsewhere. So the coarse-grained flux into vanishing scales is one-signed locally in space and time, in the limit, and not only for the whole box as in THEORY-776. Its location and rate are inputs of the construction; it is a smooth density on a few pipes, not concentrated on a thin set; nothing is said about the sign at a positive scale or about vanishing viscosity. | Proposed |
| 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-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-776 v3 | For every β < 1/3 and every smooth, strictly positive and strictly decreasing energy profile e(t) on [0, T], there is a weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous of exponent β in space and time, with ∫|v(t)|² = e(t) at every time; for such a solution the energy flux across a mollification scale ε, integrated over any time interval, tends as ε → 0 to ½ times the drop in e, which is strictly negative, so energy leaves the resolved field for vanishing scales on every interval; this is one-way for the energy of the whole box, not at every point, and it is chosen by the prescribed profile, not forced by the equations | Buckmaster, De Lellis, Székelyhidi and Vicol (2017; CPAM 2019), LIT-1300, read in NOTE-1013: the admissible version of Isett’s negative half of Onsager’s conjecture (THEORY-763). Any smooth positive energy profile can be imposed on a C^β solution, β < 1/3, so a strictly decreasing one gives solutions whose total energy only falls. Through THEORY-749’s energy balance, the coarse-grained flux then runs into the vanishing scales on every time interval. That is the one-way transfer THEORY-763 could not give. It is one-way for the box as a whole, not at every point (no local energy inequality is proved); the same theorem gives increasing energy as readily, so the direction is a choice of the construction; and nothing ties these solutions to Navier–Stokes, to a constant-flux inertial range or to uniqueness. | Proposed |
| THEORY-811 | With a kernel normalised to unit diagonal, the kernel quantum relative entropy of two distributions is within O(σ²) of their Kullback–Leibler divergence as the bandwidth σ shrinks, for translation-invariant kernels on the torus generally and for kernels on other compact groups | The general reading that THEORY-313 held before the promotion audit narrowed it to the one kernel family on the torus that the rate is shown for. Bach (2022), LIT-255, read in NOTE-837, asserts that other translation-invariant kernels and kernels on compact groups behave alike, and sketches a leverage-score argument without completing it. Filed under ADR-047 v4 so the wide form stays a thesis evidence can confirm or reject. | Proposed |
| THEORY-812 | As dimension grows with the undersampling and sparsity ratios fixed, the ℓ₁ phase transition for recovering sparse vectors from random measurements sits at the same curve for a broad class of random-matrix ensembles, those with a kind of orthogonality or weak independence, as for Gaussian matrices, while differing at finite size | The general reading that THEORY-391 held before the promotion audit narrowed it to what 2,948,000 optimizations at three problem sizes showed. Donoho and Tanner (2009), LIT-1160, read in NOTE-933, state it as the hypothesis of “weak universality” and pose the class of ensembles it holds for as an open problem. Nothing about non-Gaussian ensembles is proved. Filed under ADR-047 v4 so the wide form stays a thesis evidence can confirm or reject. | Proposed |
| 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-821 | For every α < 1/15 there are weak solutions of the 3D incompressible Euler equations on the periodic box, Hölder-continuous of exponent α in space and time, that satisfy the local energy inequality; in the construction their dissipation measure is a nonnegative function of time alone, equal at each time to the rate at which the total energy falls, zero while the local energy equality holds and positive on all of space during a short final interval in which the solution’s energy is drained to zero; uncountably many such solutions share one initial datum and one dissipation measure; for them the local coarse-grained energy flux into vanishing scales converges to that measure, so the transfer is one-way at every place and time in the limit and spread uniformly over the box, with its timing chosen by the construction, not forced by the equations | Isett (2017; ARMA 2022), LIT-1305, skimmed in NOTE-1027: the first continuous, indeed C^α with α < 1/15, weak Euler solutions satisfying the local energy inequality with nonzero dissipation, and a Cantor family of them sharing one initial datum and one dissipation measure. In the construction the dissipation measure is a function of time alone, so in the limit of vanishing scale the energy flux runs one way at every place, as in THEORY-772, but in a Hölder class and spread uniformly over the torus rather than set in pipes. It is switched on only while the constructor drains the solution’s energy to zero at the end of its life. Where, when and how fast the energy goes are inputs; the exponent is far from 1/3; nothing ties these solutions to Navier–Stokes. | Proposed |
| 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-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-875 | Phenomenal binding of the parts of an experience can be modelled as agreement, on overlaps, of measured similarity structures assigned to the parts by a presheaf, and the broad quale is fixed by its parts exactly when that presheaf is a sheaf; a sheaf failure may mark disunity, as proposed for binocular rivalry, or context dependence within one unified experience, as in the Ebbinghaus illusion, and the framework does not yet say which | Lee-Youngzie, Tsuchiya, Robinson, Dietz & Monti (2026), LIT-1352, read in full in NOTE-1059. A hypothesis article with no data. It proposes the presheaf and two readings of its failure, disunity for rivalry and context dependence for the Ebbinghaus illusion, without a criterion between them. It does not identify unity with a global section, since it assumes unity always holds. Its global section is measured data, its worked topology example is malformed, and by its own definition of gluing the rivalry case is not a sheaf failure. It answers QUESTION-255 only by showing that a sheaf failure need not be disunity. | Proposed |
| 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-916 v2 | In the DeGroot–Friedkin model, where a group reaches DeGroot consensus on each issue and each member’s self-weight on the next issue equals their control over the last, with fixed relative interpersonal weights that form a strongly connected network, self-weights converge from every non-autocratic start to one equilibrium ordered by the eigenvector centralities of those weights: power accumulates above a centrality threshold, is autocratic exactly when the network is a star and democratic exactly when it is doubly stochastic; and in the networks that are not strongly connected that the authors treat, members to whom some other member does not listen, even indirectly, end with none | Jia, MirTabatabaei, Friedkin & Bullo (2015), LIT-1146, read in NOTE-1135 from the version of record posted on the author’s site; proofs skimmed. Theorems about a model, not findings about groups. The empirical companion, LIT-1482 (THEORY-902), uses anchored dynamics and a changing network, whose attractor is autocracy rather than this interior equilibrium. | Proposed |
| 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-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-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-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-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-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-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-1052 | A quasi-static attractor describes an open system only while the drift of that attractor in state space is slow against the system’s slowest relaxation, scaled by the distance to the tipping threshold; rate-induced tipping is the failure of that time-scale separation, and it can occur while the parameter itself moves slowly, if the attractor is sensitive to it, and with no slowing of any mode | Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, read in NOTE-1215. Shown for the linear tipping-radius model, where the sufficient condition for tracking is a rate comparison, and stated for a fast-slow example (“rate dependent tipping occurs when the slow and very slow timescales are no longer separable”). Proposed because for nonlinear systems the paper says “no exact analogy possible” and gives no general condition. It bears on CLAIM-478, whose constraints hold over a window between time scales. | Proposed |
| 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-1056 | In angular synchronization with outlier offsets, a polynomial-time spectral relaxation, the top eigenvector of the Hermitian matrix of measured phase offsets, recovers the angles once the share of good offsets exceeds about 1/√n on the complete graph, within a factor √((L − 1)/(2 log₂ L)) of the information bound that limits every algorithm, while exact maximum likelihood is a non-convex search costing O(mLⁿ) exhaustively; in this model the price of recovering the global configuration cheaply instead of optimally is a constant factor in evidence, which vanishes for L ≤ 6 | Singer (2011), LIT-1655, §§2, 4 and 5, read in NOTE-1224 from arXiv v2. The threshold p > 1/√n comes from a cited rank-one deformation theorem adopted to a non-Gaussian mixture and a heuristic comparison with random correlation, confirmed by simulation at n = 100 and 400; the near-optimality ratio compares it with an information bound for a different success criterion. Proposed for both steps. It does not say that recovery is cheap on every graph, under every noise model, or for fine angular resolution. | Proposed |
| 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-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-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-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-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-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-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-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-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-1092 | Read over the sensor domains, the discrepancies between each observation and the restrictions of others can single out a faulty sensor where the consistency radius alone does not: in Robinson’s simulated search, a 2° bearing bias was attributed to the right sensor, without prior knowledge of it, because every large discrepancy involved that sensor’s domains, while the radius was lower than in the unbiased case | Robinson (2017), LIT-1658, Example 24, Case 2 (pp. 26–28 of arXiv v3), read in NOTE-1221. One simulated case, one noise realization, read from a figure. Proposed until a stated procedure is tested over trials, including faults in the restriction maps, which enter the discrepancies in the same way as faults in the data. | Proposed |
| 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-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-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-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-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-1136 | Poincaré embeddings of one hierarchy from different random seeds differ by more than an isometry of the hyperbolic ball, with branches placed differently and pairwise distances changed, so aligners that allow only a global orthogonal or Möbius map recover almost none of the correspondence, while a non-linear registration by a hyperbolic neural network, trained through a Sinkhorn divergence and started from a pretraining step, recovers a substantial part of it but finds the exact match for under a quarter of items even when the two hierarchies are identical | Alvarez-Melis, Mroueh and Jaakkola (2020), LIT-1808, §§5–7, Figs. 1–2, Tables 1–2; read in NOTE-1323. Proposed: single runs, one embedding method for the matching experiments, and GW, the obvious isometry-free baseline, is not run. It does not say why seeds disagree (the paper conjectures negative sampling), nor that the matching preserves the hierarchy’s direction, which it does not enforce. | Proposed |
| 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-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-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-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-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-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-1160 | Choosing which source objects to have an oracle match, by the derivative of a proxy utility of the entropic transport plan (its nearness to a permutation, or its consistency with the graphs) with respect to supervising each candidate match, averaged over matches by the current plan, raises held-out alignment accuracy more than uncertainty, centrality and matching-sampling query rules at the same label budget, by up to about three points on network alignment and less on image–text tasks; the gain depends on weighting by the current plan, without which the rule does worse than random | Yu et al. (2026), LIT-1807, Tables 3–5 and 7, read in NOTE-1327. Eight datasets, four aligners, a budget of 20% of true pairs in ten rounds after a random 20%, five runs, no variances. Proposed: an empirical result with small margins, on benchmarks where the truth is near one-to-one, so a coherence proxy and accuracy agree. It chooses among fixed questions; it does not extend what can be asked. | Proposed |
| 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-1170 | A convex map between two scenarios’ model sets is induced by a classical adaptive procedure exactly when some non-contextual model of the corresponding adaptive hom scenario induces it | The adaptive analogue of Theorem 44 that Barbosa, Karvonen and Mansfield assert in Closing Bell (LIT-845, §6.2) without proof, split from THEORY-174 when the fifteenth status review narrowed that entry to the non-adaptive procedures its re-derivation covers and promoted it (ADR-047, kind 1: a dropped clause the source asserts). The non-adaptive route does not transfer as it stands, since Lemma 39 fails for adaptive procedures. Proposed until a proof is read or re-derived. | Proposed |
| THEORY-1177 | For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation | The wide form THEORY-156 held before the twelfth status review narrowed it, kept as a Proposed reading by the ADR-047 kind-1 sweep of 2026-10-30: the source asserts it, and THEORY-156, Active, is its evidence. Blackwell (1953), LIT-781, claims the equivalence for any outcome space, through Theorem 2, quoted from his 1951 paper. The record’s re-derivation reaches only outcome spaces that are standard Borel, since its last step needs a regular conditional distribution. | Proposed |
| THEORY-1210 | When several networks are aligned, fitting one joint multi-marginal coupling finds held-out correspondences better than fitting them in pairs, and the gain comes from the joint fit itself, not from the cost or the search space that come with it | Zeng et al.’s (2024) attribution of HOT’s gains to joint alignment (NOTE-1334’s C6, asserted without an ablation), kept here when the twenty-second status review narrowed and promoted THEORY-1159 (ADR-065; ADR-047 kind 1). No ablation in the record holds the RWR cost and the clustering fixed, and every benchmark’s truth is consistent by construction. | Proposed |
| 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-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-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-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-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-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 |