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.
60 of 371 QUESTION documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| QUESTION-006 | Which pragmatic observables are conserved, exactly or within a bounded defect, by actual reconstruction kernels, and how does that depend on frame, genre and adaptation? | The third of the five central questions (A110 §IX), with the crystallized argument’s boxed question “Which observables are invariant under which communicative transformations?” The manuscript keeps the criterion (§4) and one sub-question, at what stage an invariant stops being conserved. | Open |
| QUESTION-008 | Is pragmatic fidelity a form of approximate bisimulation between communicative systems, and which parts of their update structure must it preserve? | Opened at A35 §10, made a formal layer of the outline at A38 §VI, and demoted deliberately at A39–A40 to an “optional refinement” whose usefulness “must be justified”. Absent from the manuscript. Set aside, not answered. | Deferred |
| QUESTION-017 v2 | Where the additive bound fails, what bound on accumulated drift holds when distortion is directed and need not satisfy the triangle inequality? | A50 derived drift bounds assuming a metric, and in the same reply dropped symmetry and the triangle inequality for distortion. The tension was never resolved; the manuscript states its recursion “in the chosen metric”. | Open |
| QUESTION-022 v2 | Beyond restriction-compatible kernels and kernels induced by one global kernel, which stochastic transports carry compatible empirical models to compatible ones and preserve global extendability: which conditions are necessary, and what does a transport that changes the cover do? | The first of the crystallized argument’s five central mathematical questions (A110 §IX). Propositions 1 and 2 give sufficient conditions, and that part is QUESTION-240, Answered. Necessary conditions, and transports that change the cover, are open. | Open |
| QUESTION-025 | Do correlated or hierarchical attributes in co-occurrence statistics still give linear attribute directions in an embedding, and with them a concept lattice that is not Boolean? | THEORY-185 derives linear directions for binary attributes from co-occurrence, but only for independent attributes, which make every combination possible and the concept lattice Boolean. The Lattice Representation Hypothesis (LIT-267) and CLAIM-119 need the opposite: attributes that imply or exclude each other. Whether the linear directions survive that structure is not answered by anything the record holds. | Open |
| QUESTION-031 v5 | When does a chosen family of observational probes separate the communicative realizations a task needs told apart, and when does it collapse distinct ones, beyond the cases in which the task’s target and the probes’ outcome laws are linear in one state? | A173’s secondary question (“When do restricted probes separate communicative realizations?”), v7’s RQ3 (A178) and A214 §3. It is the open half of CLAIM-142: observational equivalence is relative to the probes, so how rich must the probes be? The discrete case has exact answers, and so does a metric-enriched one. The stochastic, decision-relative version is open. | Open |
| QUESTION-036 | Can the admissible symmetries of a communicative kind be reconstructed from a sufficiently rich family of its observable realizations and transformations? | A198’s question (U55), put with Tannaka reconstruction as its model and labelled “an inspiration rather than an applicable theorem”. It was recorded on the workbench and not coded, and the next turn’s crystallized argument (A203, U56) left it out with no decision either way. Filed on 2026-10-11, when it came up a second time. An answer says what “sufficiently rich” means and what plays the role of the fibre functor, or shows that no such reconstruction can hold. | Open |
| QUESTION-052 v2 | Beyond linear decisions at the population level, if a kind’s admissible transformations are given as a distribution over pairs of its realizations, is the top eigenspace of the pair operator, or the top left singular space when the transformations are directed, its stable, decision-sufficient signature, and what makes one span the signature when other spans tie? | Filed in the fourth synthesis pass to give readings no question asked for a place in the argument. It bears on QUESTION-027, CLAIM-098, CLAIM-193, CLAIM-164 and CLAIM-043. For linear decisions at the population level it is answered, for symmetric and directed pairs, and that part is QUESTION-245. Open here: nonlinear decisions, and what makes one span the signature when other spans tie. The body gives each reading’s limit. | Open |
| QUESTION-055 v4 | If communicative categories form a non-Boolean lattice, and the communicative object is a presheaf over contexts, over which base category is it one, and does some probe family split it into sectors that no probe connects? | Filed in the fourth synthesis pass to give readings no question asked for a place in the argument. It bears on CLAIM-119, CLAIM-038, CLAIM-044, QUESTION-025 and QUESTION-031. Given a presheaf over contexts, the negation is the Heyting pseudo-complement, and that part is QUESTION-247, Answered. That the communicative object is a presheaf over contexts is QUESTION-332, Answered by CLAIM-038. Over which base, and whether probes split it into sectors, is open. THEORY-041 and THEORY-042 bear on it; the body gives each one’s limit. | Open |
| QUESTION-058 | When a trained network’s representation carries a group’s structure, how much of it comes from the task, the architecture, the training set’s relations and the symmetry of the update, and how much can be read off the network without supplying the group? | Filed in the fourth synthesis pass to give readings no question asked for a place in the argument. It bears on QUESTION-036, CLAIM-137, CLAIM-132, CLAIM-148, CLAIM-135 and CLAIM-193. THEORY-022, THEORY-181, THEORY-021 and THEORY-184 answer it in part; the body gives each one’s limit. | Open |
| QUESTION-060 v2 | When a network is trained on data from a structured system, are the variables it comes to represent fixed by its training objective or by what its architecture can represent cheaply, and since an architecture that can represent the system’s coarse-graining need not learn it, when do they coincide with the system’s own relevant variables? | Filed in the fourth synthesis pass to give readings no question asked for a place in the argument. It bears on CLAIM-152 and QUESTION-025. THEORY-197 and THEORY-206 answer it in part; the body gives each one’s limit. | Open |
| QUESTION-065 | Can the target that a sufficiency-based boundary is relative to be supplied by the system’s own locality, as informativeness about its far environment across a buffer, so that the coarse variables kept no longer depend on the analyst, and is the partition that wins for a given target unique? | Filed in the fourth synthesis pass to give readings no question asked for a place in the argument. It bears on CLAIM-152, CLAIM-154, CLAIM-157, CLAIM-164 and QUESTION-035. THEORY-198, THEORY-205 and THEORY-157 answer it in part; the body gives each one’s limit. | Open |
| QUESTION-074 | Can the transport structure fitted to single transformations predict what receivers recover under a composition of transformations no data were fitted on, within the conditions under which approximate transports compose? | From the interlocutor’s reading of PR #121 and its reconciliation, relayed by the owner on 2026-10-12. It names the test the reading takes to separate a structural formalism from a richer feature set. Its scope is set by CLAIM-260: approximate transports compose only under conditions. | Open |
| QUESTION-102 | Can a family of interpretations each feasible in its own context fail to admit a feasible global integration when a compatible global interpretation exists? | A15 §6. A computational analogue of the gluing question QUESTION-001 asks, kept apart from it. Failure to compute a global section is not the absence of one. | Open |
| QUESTION-110 v2 | How should the cost of recovering a distinction from a representation be defined, relative to an encoding, a class of procedures and a success criterion? | A18’s sixth candidate, “recovery complexity”, filed as a question at A18’s own suggestion rather than as a term. The exchange used the idea under five names and never fixed one. | Open |
| QUESTION-120 v5 | Beyond distances and sparse recovery, whose thresholds share a sufficiency proof and not a necessity proof, when the record meets a random linear map that keeps what a later stage needs, in random features for kernels, the superposition toy model’s bound and phase changes, and the expand-and-threshold codes of sparse distributed memory and the fly, is it one concentration-of-measure fact applied to different families, and does each threshold count the same thing? | Filed when a survey of the record found random projections in three places and the Johnson–Lindenstrauss lemma in none. The three uses carry three thresholds: ε⁻² log n for distances, k log(n/k) for sparse recovery, and 2N = n for random-features regression. The question is whether these are one union bound over different families or different phenomena that share a tool. Six anchor works are now read. They show the distance and sparse-recovery thresholds share a sufficiency proof, but not a necessity proof, and that part is QUESTION-251, Answered. They give nothing that reaches random features or expand-and-threshold. For SDM and the fly, the record’s SDM recall account and its LSH account offer candidate thresholds of a different shape. | Open |
| QUESTION-123 | How should an organization’s demands be tracked across differentiation, branching, merging and changes of organizational identity, when no one-to-one correspondence between earlier and later demands exists? | A39’s, from the exchange of 2026-10-17: “How should demands be tracked across differentiation, branching, merging, and changes of organizational identity?” CLAIM-361 judges relief against a fixed reference set of demands under an explicit correspondence between earlier and later demands. When the processes that bear the demands split or merge, that correspondence may not be a bijection, and CLAIM-287 says the continuity it would follow can branch and is not an identity relation. An answer says what structure the correspondence has, and how relief is judged through it. | Open |
| QUESTION-130 v3 | Does transport of evaluative representations between contexts, with the evaluated situation held fixed, admit a connection fixed by independently established relations rather than chosen, and is its holonomy around a loop of contexts ever nontrivial? | The owner’s, U19 of the exchange of 2026-10-17, self-hedged: “this might be a stretch, but given our framework already leverages fields and flows, are we perhaps moving towards a gauge theoretic account of something here?” A19 separated representational gauge freedom, which the record holds (CLAIM-148), from a connection between contexts and from nontrivial curvature, and kept the last speculative. The first objection is the record’s own: it rejected LIT-038, whose path-dependent transport across a network of causal models comes from arbitrary maps (NOTE-050). The first target is flat holonomy. An answer derives transport from shared commitments, causal effects or roles in a finite model and reports whether a loop’s holonomy is trivial. | Open |
| QUESTION-136 | As the number of stimuli grows, do the task-averaged readout identities with white-noise tasks have a non-degenerate population form, with (1/M²)Tr(K̂_X K̂_Y) converging to the Hilbert–Schmidt inner product of the kernel operators, and at what rate when the regularizer is estimated? | The population clause of THEORY-008’s former promote_when. On 2026-10-19 THEORY-008 was confined to a finite stimulus set, where Harvey, Lipshutz & Williams prove its identities, and promoted. Their App. A.3 sketch does not reach the K_z = I case, where its limit is zero (NOTE-834). GULP’s population analysis may answer it, and the record does not hold it. | Open |
| QUESTION-138 | When does the truncated n × n matrix of a functional map stay nearly orthogonal and nearly block-diagonal by eigenspace under a near-isometry, and how does the bound depend on the spectral gap at n? | The truncation clause of THEORY-297’s former promote_when. THEORY-297 is Active for bijections, with orthogonality for volume-preserving maps and eigenspace preservation for exact isometries. For an exact isometry the truncated block is orthogonal when n ends at a spectral gap. What is open is the near-isometric case, which LIT-993 cites to Kato and does not show. | Open |
| QUESTION-139 | Does the one-hidden-layer lower bound of ∏(rᵢ+1) neurons for the monomial x₁^r₁⋯xₙ^rₙ hold for uniform approximation on a box, and not only for Taylor matching at the origin, and what replaces it when hidden units have biases? | The open half of THEORY-206’s former promotion condition. On 2026-10-19 THEORY-206 was narrowed to the Taylor-matching bound for bias-free networks, which Rolnick and Tegmark prove and the record re-derived, and promoted. Their uniform lower bound rests on a step that fails, and the biased case is untreated. Both are here. | Open |
| QUESTION-140 | In trained networks, how far do CKA and normalized Bures similarity disagree in ranking pairs of representations, and does the rank-dependent envelope between them matter in practice? | The empirical clause of THEORY-233’s former promote_when. On 2026-10-19 THEORY-233 was promoted: its envelope between CKA and NBS was checked by a direct derivation. Harvey, Larsen & Williams (LIT-250) illustrate the gap only on random Wishart matrices. Whether it matters for real representations needs a measurement on trained networks, and the record holds none. | Open |
| QUESTION-141 | For an asymmetric self-supervised objective, such as BYOL or SimSiam with a predictor and stop-gradient, does the eigenfunction characterization of spectral methods fail, or is it replaced by the singular functions of a non-self-adjoint operator? | The open clause of THEORY-009’s former promote_when. On 2026-10-19 THEORY-009 was narrowed to its reading of the positive-pair papers, where self-adjointness comes from the symmetry of the positive-pair distribution, and promoted. What replaces eigenfunctions when the objective is asymmetric was never part of that reading. No published analysis held by the record answers it. The near misses are LIT-228, LIT-242 and LIT-975. | Open |
| QUESTION-142 | For unital JB-algebras, is the existence of a dynamical correspondence exactly what makes observables generate symmetries of what they conserve, as Alfsen and Shultz’s theorem, quoted by Baez, states? | The Alfsen–Shultz clause of THEORY-160’s former promote_when, moved here on 2026-10-19, when THEORY-160 was narrowed to the bracket equivalence and promoted. Baez quotes the theorem (his Theorem 12) without proof, and the record has not read Alfsen and Shultz. | Open |
| QUESTION-143 | Is the presheaf of causal functions on a space of input histories, with the lowerset topology, a sheaf exactly when the space has no solipsistic contextuality witness, as Theorem 2.34 of The Topology of Causality states? | The characterisation clause of THEORY-171’s former promote_when, moved here unchanged on 2026-10-19, when THEORY-171 was narrowed to its existence claim and promoted. That claim is settled by LIT-808’s checked Θ₃ example. The iff is Gogioso and Pinzani’s Theorem 2.34, in an unrefereed preprint whose proof the record has not read. | Open |
| QUESTION-144 | Beyond a finite state space, does an observable commute with a Markov generator exactly when its mean and second moment are conserved in every state, and is there a component form of that condition on general state spaces? | The general-space clause of THEORY-158’s former promote_when, moved here on 2026-10-19, when THEORY-158 was narrowed to the finite state space and promoted. Baez and Fong state the moment form on σ-finite measure spaces (Theorems 2–3). The record has read Theorem 3, but has not re-derived it, and Theorem 2 is stated there as a consequence. The paper gives no component form beyond finite X. | Open |
| QUESTION-145 | In trained models, do the features’ dimensionality-weighted projectors sum to the identity on their span, as they do in the exact polytope and tegum configurations of uniform superposition? | The promotion condition formerly on THEORY-021, moved here unchanged when THEORY-021 was narrowed to the exact configurations and promoted on 2026-10-19. The algebra is settled for those configurations. Whether trained weights reach them is a measurement no one has made. | Open |
| QUESTION-146 | Is there a map from an empirical model’s signed global sections to a quasiprobability or frame representation of the corresponding GPT fragment, and if so does it preserve positivity both ways? | The promotion condition formerly on THEORY-014, moved here unchanged when THEORY-014 was promoted on 2026-10-19. THEORY-014 sets the sheaf, GPT and Contextuality-by-Default results side by side and claims no map between them. This question asks for the map, or a proof that none preserves positivity. | Open |
| QUESTION-148 | Do bilinear knowledge-graph scores (RESCAL, ComplEx, HolE, the Neural Tensor Network, QuatE) admit a representation as a coboundary of a sheaf of vector spaces, or is there a proof that they do not? | The open half of THEORY-299’s former promote_when. On 2026-10-19 THEORY-299 was narrowed and promoted: structured and translational embeddings are 0-cochains of a pulled-back sheaf scored by a norm of the coboundary. Gebhart, Hansen and Schrater (LIT-232) state that bilinear models fall outside the framework as given (App. A.1) and do not prove it. Whether any sheaf of vector spaces takes them in is open. | Open |
| QUESTION-149 | In released language models, do the relational linear properties reported so far lie in the subspaces that enter the logits, with Γ ⊆ N and Γ_q ⊆ M, so that they are properties of the distribution and not of one fitted model? | The trained-model half of the promotion condition formerly on THEORY-234, moved here unchanged when THEORY-234 was narrowed and promoted on 2026-10-19. Theorem 14 of Marconato et al. says a relational linear property is shared by every model inducing the same distribution when it lies in N with Γ_q ⊆ M. Whether the linearities reported in trained models meet that condition has not been measured. | Open |
| QUESTION-150 | In the Poincaré ball H^r for r > 2, does embedding a tree with worst-case distortion 1 + ε need Ω((ℓ/(εr)) log deg_max) bits per coordinate for r up to about log deg_max, and Ω(ℓ/ε) beyond, matching the code-based upper bound? | The higher-dimensional half of the promotion condition formerly on THEORY-196, moved here when THEORY-196’s planar lower bound was split off as THEORY-734 on 2026-10-19. LIT-874’s Proposition 3.1 gives the upper bound in H^r. No lower bound in H^r is held, so whether dimension helps exactly as much as the construction uses is open. | Open |
| QUESTION-153 | Do the hyperbolic Betti-curve matches in odour space and in CA1 separate a hyperbolic metric from a flat sphere or a flat multiscale code, and do they replicate on other data? | The former promote_when clauses of THEORY-191 and THEORY-192, moved here unchanged when both were narrowed to what was measured and promoted under ADR-047 v3. Each Betti-curve test rejected only uniform Euclidean cubes. On a thin shell, hyperbolic distance is nearly a monotone function of angle, and uniform hyperbolic sampling is nearly an exponential spread of scales, so flat rivals remain. Neither study has been repeated on other data or by another group. | Open |
| QUESTION-160 | Does the heat kernel of a metric tree satisfy the Hellinger Hölder and separation bounds of the multiscale hyperbolic diffusion theorem, with which Hölder exponent, and when does the diffusion operator of sampled data approximate a latent tree’s heat kernel? | The promotion condition formerly on THEORY-478, moved here unchanged when THEORY-478 was narrowed on 2026-10-19 to the conditional equivalence under two Hellinger bounds, with the admissible range α < θ/β. Whether a metric tree’s heat kernel meets those bounds, with which θ, and whether sampled data yield such a kernel, is not settled by anything the record holds. | Open |
| QUESTION-168 | Does the ℓ₁ recovery transition of some non-Gaussian measurement ensemble provably converge to the Gaussian curve ρ(δ) as N → ∞, or do simulations an order of magnitude beyond N = 1600 keep the gap shrinking as N^(−1/2) for an ensemble not used to choose that exponent? | Filed when the promotion audit (batch 025, ADR-047 v3) narrowed and promoted THEORY-391 to what Donoho and Tanner measured. In 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached the Gaussian curve, consistent with N^(−1/2) gaps. Whether that agreement is a limit theorem is THEORY-391’s former promote_when, moved here unchanged. | Open |
| QUESTION-197 | Is there a variant of the Döring–Isham rules, or of the context category, under which tensor-product composites of quantum systems satisfy the translation condition? | Filed when the promotion audit (batch 044, ADR-047 v2) proved, for finite dimensions, the failure that THEORY-585 had held as asserted: along a tensor product the pull-back of a non-scalar quantity disagrees with the composite’s quantity at some context, and no operator realises it. The former promote_when’s refutation route, a variant of the rules that repairs the composite, is held here. | Open |
| QUESTION-198 | Does the O(σ²) gap between the kernel and Shannon relative entropies hold for translation-invariant kernels on the torus beyond the exponential Fourier family, and on other compact spaces? | Filed when the promotion audit (batch 043, ADR-047 v2) narrowed and promoted THEORY-313 to the kernel family Bach (2022) works out. The former promote_when asked for a confirmation of the O(σ²) gap beyond the torus; this question holds it. | Open |
| QUESTION-218 v2 | Can a collective mode that is dynamically closed, but is not a weakly coupled block of the units carrying it, be one system, and if so what should a boundary test take as the unit: a subset of members or a function of the whole state? | A5 of the exchange of 2026-10-19 on coarse-to-fine processes, its candidate Q2: “Does organizational unity require a subsystem boundary, or can it arise from a dynamically closed collective mode distributed across multiple otherwise entangled subsystems?” CLAIM-388 shows the two come apart; CASE-120 is the construction. The question is which of them the record’s “one system” should track. | Open |
| QUESTION-221 v3 | Does near-decomposability bound dynamical dependence: when the coupling into a block from the rest of the system is at most ε, is the dynamical dependence of the block’s aggregate small to a stated order in ε? | A5 of the exchange of 2026-10-19 on coarse-to-fine processes, its candidate Q1 and its “most valuable near-term result”. THEORY-661 says near-decomposability gives approximate closure of block aggregates over the long run, by coupling strength, and THEORY-733 characterizes exact closure for linear Gaussian VARs. No entry bounds the one by the other. CLAIM-388 shows that the converse fails: a closed aggregate need not be a weakly coupled block. | Open |
| QUESTION-222 v4 | Beyond the scalar linear Gaussian case and the deterministic case of a fast, contracting eliminated variable, when a coarse variable’s reduced dynamics carry memory of the eliminated variables, does it admit a finite-dimensional Markovian augmentation whose prediction error is bounded in terms of the eliminated dynamics, and is a level whose history, not its present state, does the predicting still a level? | A3 and A5 of the exchange of 2026-10-19 on coarse-to-fine processes, A5’s candidate Q3. The Mori–Zwanzig decomposition writes any projected dynamics as a Markovian term, a memory term and noise. The record holds no reading of it. Its criteria of closure (THEORY-257) condition on the coarse variable’s own history, so memory is not by itself a failure of them, and CLAIM-327 already locates a collective in a variable members read from the past. | Open |
| QUESTION-231 | Does Coifman and Lafon’s α-normalized diffusion limit hold on a compact manifold with boundary, in L² on band-limited functions at a stated rate, and does the iterated operator P_{ε,α}^{t/ε} converge to the semigroup of the limit operator? | The boundary and semigroup halves of the promotion condition formerly on THEORY-744. THEORY-744 was narrowed to compact manifolds without boundary and promoted on 2026-10-20 (ADR-047 v2), with the source’s kernel expansion re-derived. What it does not settle: the limit near a boundary, at the rate the source prints, and the heat-kernel limit that the source’s Proposition 11 claims. | Open |
| QUESTION-240 | Does a transport whose local kernels commute with restriction preserve overlap consistency, and does a transport induced by one global stochastic kernel preserve global extendability? | The part of QUESTION-022 the manuscript’s Propositions 1 and 2 settle, split off under ADR-049. Yes to both, for finite scenarios: CLAIM-121 for overlap consistency and CLAIM-070 for global extendability. Both are sufficient conditions. Whether they are necessary, and what transports that change the cover do, stays with QUESTION-022. | Answered |
| QUESTION-242 | Does exact dynamical closure of an aggregate require a nearly decomposable block, or can a collective variable be exactly dynamically independent when no partition of the units is weakly coupled? | The converse half of QUESTION-221, split off under ADR-049 and answered by CLAIM-388: closure does not require a nearly decomposable block. In a linear Gaussian VAR(1) with uniform coupling, the sum of all units is exactly dynamically independent at every stable coupling strength, while every partition couples as strongly across its edges as within them. The forward bound stays with QUESTION-221. | Answered |
| QUESTION-245 | For linear decisions at the population level, if a kind’s admissible transformations are given as a distribution over pairs of its realizations, is the top eigenspace of the pair operator a stable, decision-sufficient span, and what span takes that role when the pairs are directed? | The linear part of QUESTION-052, split off under ADR-049. For symmetric pairs, yes: the top eigenfunctions are the most view-invariant directions, and their span is minimax-optimal for linear prediction of approximately view-invariant targets (THEORY-007) and bounds linear-probe error by leakage over conductance (THEORY-228). For directed pairs, the top left singular functions of the conditional-expectation operator take that role (THEORY-239), and a linear readout of the full density ratio suffices for redundant labels (THEORY-319). Other spans can tie. Nonlinear decisions, and what picks one signature out, stay with QUESTION-052. | Answered |
| QUESTION-247 v2 | If the communicative object is a presheaf of propositions over a category of contexts, is the negation its topos logic gives an orthocomplement or a Heyting pseudo-complement, and what fixes it? | The conditional part of QUESTION-055, split off under ADR-049 and answered by THEORY-516 and THEORY-037. Over a presheaf of contexts the truth values are sieves, and negation is the Heyting pseudo-complement that the base category fixes, not an orthocomplement. That the communicative object is such a presheaf is QUESTION-332, Answered by CLAIM-038. Over which base, and whether probes split it into sectors, stays with QUESTION-055. | Answered |
| QUESTION-248 | In a stable linear Gaussian VAR(1), does a scalar linear coarse variable whose reduced dynamics carry memory admit an exact finite-dimensional Markovian augmentation? | QUESTION-222 for the linear Gaussian class, split off under ADR-049 and answered by CLAIM-398. Yes: a scalar coarse variable Y = LX of a VAR(1) on ℝⁿ is an ARMA process of orders at most (n, n − 1). Its Mori–Zwanzig memory is non-zero at every lag in general, but finitely many coordinates, built from Y’s own past and innovations, carry it exactly. Other classes, error in terms of the eliminated spectrum, and the ruling on levels stay with QUESTION-222. | Answered |
| QUESTION-251 | Do the Johnson–Lindenstrauss threshold for distances and the k log(N/k) threshold for sparse recovery rest on one concentration-of-measure argument, a tail bound for each fixed vector’s norm with a union bound over a family, both for sufficiency and for necessity? | The part of QUESTION-120 its readings on distances and sparse recovery settle, split off under ADR-049. For sufficiency, yes: one per-vector tail and a union bound give both, over pairs for distances (THEORY-397) and over net points on every sparse support for recovery (THEORY-390). For necessity, no: the distance lower bounds are encoding and bit-counting arguments (THEORY-395, THEORY-399), recovery’s rests on Gelfand widths, and its sharp location on face counting (THEORY-391). Random features, superposition and expand-and-threshold stay with QUESTION-120. | Answered |
| QUESTION-252 v6 | For the conditions an effective description is asked to meet (predictive sufficiency for a declared target, dynamical independence of the omitted variables, memory closure under a declared projection, approximation fidelity in a declared metric and horizon, and tractability within a declared budget), which can fail while all the others hold, and which such configurations are impossible in which model classes? | The owner’s idea at U3 of the exchange of 2026-10-20: “what if we censor or negate each virtue one at a time? that reduces our combinations from n! to n.” A3 corrected the count (2ⁿ presence/absence configurations, not n!) and fixed five conditions, asking whether each leave-one-out case is realizable: “An impossible configuration is itself a substantive result.” The record holds pairwise non-implications only, none a leave-one-out witness: CLAIM-389 gives zero memory without independence in VAR(1), CLAIM-304 retention without feasible recovery. A4 and A5 then found that some conditions are parameters rather than conditions (CLAIM-401); that part is QUESTION-324, Answered. The same form as QUESTION-093 and QUESTION-115, on a new list. | Open |
| QUESTION-255 v6 | Can an empirical model that genuinely fails to glue, under a specified cover and correspondence between its contexts, take part in one phenomenally unified process, given that a unified experience can contain conflicting interpretations or demands and that such conflicts are not by themselves gluing obstructions? | A7 §10 of the exchange of 2026-10-21, after A6 found a sheaf-theoretic account of global phenomenal unity. The record’s self line already reads a coherent self as a global section (CLAIM-153), and conflict without an overlap as no disunity (CLAIM-198). A7 denied that experiential unity is “the existence of one globally consistent assignment of interpretations”: “the conflict itself can be a unified experience”, and a global assignment “could exist in a merely descriptive model without any corresponding unified process”. A review of 2026-10-22 corrected the example: a felt conflict need not be a gluing failure, since one global state can hold both sides. The question is whether a model that genuinely fails to glue, under a cover and correspondence fixed independently, can be part of one unified experience. | Open |
| QUESTION-272 v2 | When the task’s target and the probes’ outcome laws are both linear in one unknown state, as for the pairwise comparisons of actions under partial monitoring and for the authority shares of a mixture of known response profiles, when does a family of probes separate what the task needs told apart? | The part of QUESTION-031 that the readings of 2026-10-25 settle, split off under ADR-049. Both answers are linear: a probe family separates what the task needs exactly when the task’s target lies in the span of what the probes reveal. For pairwise comparisons of actions under partial monitoring, that is the loss difference lying in the probes’ combined signal row space (THEORY-962). For the authority shares of a mixture of known profiles, it is the profiles over the probe contexts being affinely independent (CLAIM-452). The general stochastic, decision-relative case, where the target or the outcome laws are not linear in a common state, stays with QUESTION-031. | Answered |
| QUESTION-282 | Under what conditions do interactions change the physically operative correspondence structure among systems (the compatibility operator itself, not only the states it relates), rather than merely reveal it to a measurement or redescribe it in a model? | Raised at the owner’s review of the pull request that filed the exchange “Field Existence Question”, which called it “one of the genuinely unfinished parts of the program”. The formal results the record now holds take the correspondence structure as input. A sheaf’s restriction maps, a gauge connection, a port condition and a Laplacian are given, and the mathematics says what follows. CLAIM-471’s strong thesis is that interactions can make or change that structure. Only one model the record has read changes it through interaction, and there agreement is reached by learning to lie. The question asks for conditions under which the operative structure changes, with a way to tell that change from discovery and from redescription. | Open |
| QUESTION-283 | Can composing systems create an effective structure whose adequate specification needs a richer relational vocabulary than the parts needed separately (relations of higher arity, history, or a new kind) and, if so, under what conditions, and is the new vocabulary forced by the composite or only by the restricted class of descriptions in use? | Raised at the owner’s review of the pull request, which called it the result most worth preserving and asked that it “become a major research question”. Hansen and Ghrist show that eliminating an interior vertex of a cellular sheaf can leave a three-way boundary relation, v₁ = v₂ + v₃, that no sheaf on the boundary’s pairwise edges expresses (THEORY-1090; CLAIM-480). For scalar networks reduction stays pairwise. The question asks when composition forces a change in the type of relational structure an adequate description needs. It also asks whether that is a fact about the composite or about the class of descriptions, and whether the same thing is what happens when a record’s relation scheme cannot express what a composed organization requires. | Open |
| QUESTION-287 | In a deterministic singularly perturbed system whose eliminated variable is fast and contracting and whose reduced system is contracting, does the memoryless quasi-steady-state reduction predict the coarse variable with an error bounded uniformly in time in terms of the eliminated dynamics? | The fast contracting case of QUESTION-222, split off by the status review of 2026-10-27 under ADR-049, because THEORY-773 (Active) settles it. No memory coordinate is needed: the reduction on the slow manifold stays within O(ε) of the true slow trajectory for all t ≥ 0, with every constant explicit, set by the fast contraction rate λ_z, the reduced contraction rate λ_x and the manifold’s drift. | Answered |
| QUESTION-293 | Is the distortion of a composite of approximate transports bounded by the sum of its parts’ distortions, and does that additive bound survive when the distortion lacks a triangle inequality or a later kernel stretches distances? | The additive part of QUESTION-017, split off by the status review of 2026-10-27 under ADR-049, because CLAIM-260 (Active) settles it in full. The composite of an ε₁-transport and an ε₂-transport is an (Lε₁ + ε₂)-transport when the distortion obeys a triangle inequality, with L the later kernel’s stretch, so additive when that kernel is nonexpansive, as every Markov kernel is in total variation. Without a triangle inequality, or with an expansive kernel, the additive bound fails. | Answered |
| QUESTION-294 | When a network’s architecture can represent a structured system’s renormalization step exactly, must training by distribution fitting make the variables it represents the system’s relevant ones? | The representability part of QUESTION-060, split off by the status review of 2026-10-27 under ADR-049, because THEORY-197 (Active) settles it, in the negative. Mehta and Schwab’s RBM–renormalization correspondence is an identity of parametrizations, and an exact RG step is a perfect fit of the data distribution. Away from that point a contrastive-divergence RBM spends its hidden units on noise, not on the relevant fields. The architecture’s match does not fix what training keeps. | Answered |
| QUESTION-319 | Where an organization’s parts are related by transports established independently of the analysis, and those transports have nontrivial holonomy, is the holonomy an error the organization tends to remove, or a persistent feature it maintains, and does its pattern predict the organization’s response to intervention beyond what its pairwise relations and its dynamics predict? | From A2 §5 of the exchange “Paper assessment” (2026-10-28 to 29), which inverted LIT-1820’s use of holonomy as an error signal: “What if it is a persistent feature of the organization itself?” It is QUESTION-130’s question asked of organizations, not of evaluative contexts, and its first objection is the same: the transports must not be chosen. An answer needs transports derived from measurement, their holonomy followed under perturbation, and a test against matched systems whose holonomy differs. | Open |
| QUESTION-321 | For finitely many states, does “at least as informative in every bounded decision problem” imply simulation by a Markov kernel when the simulated experiment’s outcome space is not standard Borel? | The part of the promotion condition formerly on THEORY-156 that the twelfth status review’s re-derivation does not reach. Blackwell (1953), LIT-781, claims the equivalence for arbitrary outcome spaces through its Theorem 2, quoted from Blackwell (1951), which the record has not read. The record’s proof needs a regular conditional distribution of the simulated experiment’s outcome given its posterior, so THEORY-156 was narrowed to standard Borel spaces and promoted. | Open |
| QUESTION-332 | Is the communicative object a presheaf over contexts? | The first remaining part of QUESTION-055, split off by the fourteenth status review under ADR-049 because CLAIM-038 (Active since 2026-10-29) settles it: a communicative object is a context-indexed system of empirical constraints over a cover of contexts, supports or distributions with their restrictions (TERM-042), which is a presheaf over contexts. Which base category it has, and whether some probe family splits it into sectors, stays QUESTION-055’s. | Answered |
| QUESTION-354 | Do Monderer and Shapley (1996) state, as their Lemma 2.5, that a finite strategic game has the finite improvement property exactly when it has a generalized ordinal potential, with the standard definitions of improvement path, finite improvement property and generalized ordinal potential? | THEORY-896’s former promote_when, moved here when the twenty-fourth status review promoted it on its re-derivation, second-checked. The mathematics is proved in the record and stated by Apt (LIT-1866). What is open is only whether the attribution to Monderer and Shapley (LIT-1462, unread) is exact: lemma number, page, wording and class. | Open |
| QUESTION-358 | Does an independent computation of the linearized plane Couette and Poiseuille operators in the energy norm reproduce transient growth of order R, resonance of order R² and a minimal destabilizing perturbation of order R⁻², with leading constants near those of Trefethen et al.’s Table 1? | The promotion condition formerly on THEORY-304, moved here when the status review of 2026-10-30 promoted it under ADR-065. The lower bounds are CLAIM-383’s derivation; the upper orders and the constants rest on one computation (LIT-932, read in NOTE-839). Reddy & Henningson (1993) and Schmid & Henningson’s monograph are the obvious independent sources, unread. | Open |
| QUESTION-359 | Is the revision closure of Aevyra’s The Ledger of Will, Proposition 4, the Moore–Nerode minimal-machine construction applied to the expanded target family {T_d ∘ J_u}, under the same hypotheses, as its authors say? | The part of THEORY-884’s former promote_when that asked for more than its title. THEORY-884 was promoted on its own re-derivation, second-checked at the twenty-fourth status review. Whether its criterion is the classical construction, which the authors say it is, needs a reading of Moore (1956), Nerode (1958) or a standard automata text. | Open |