Lines of inquiry — A named line of the record’s own argument, so that a line can be set aside as a whole and what depends on it can be found. Retracting a line is a status change on its members, never a deletion (ADR-032).
Formal apparatus — results that hold whatever any substantive line concludes — gluing and contextuality, comparison of experiments and probes, identifiability, conservation, and thresholds for naming a unit; held as a second line by the general results other lines borrow, so that setting a substantive line aside does not retire them (ADR-040).
35 of 371 QUESTION documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| QUESTION-103 | Is computational indistinguishability the bounded-observer form of signal detection’s chance ROC, and is ideal-observer efficiency its empirical measure, so that the record’s formal results on bounded observers can be tested on real ones? | The owner asked whether material downstream of computational indistinguishability, or the signal detection literature, would help. A test’s distinguishing advantage is its hit rate minus its false-alarm rate, a point on its ROC; the best over all tests is total variation, reached by the likelihood-ratio detector (THEORY-031). Computational indistinguishability says every efficient observer’s ROC stays near the chance diagonal. Ideal-observer analysis measures how far real observers fall short of the ideal. This asks whether the two are one framework. | 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-113 | When a system names a category, which criterion is it meeting: that the data supports the distinction, that a deadline forces it, or that it survives refinement and compression, and do the three ever pick the same categories? | The record now holds three criteria for when a distinction earns a name. CLAIM-154 says when the data supports it, and CLAIM-307 when a deadline forces it, the two moving opposite ways with copy count. The third is from Part III of the owner’s LEGO monograph: a category is worth keeping when it persists as refinement adds distinctions and compression prunes them. MOM-CLM-138 rejected the metaphysics built on it and kept the observation as measurable, and CLAIM-164 says what a persistent spectral mode picks out. This asks how the three relate. | 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-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-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-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-156 | Which published models of concept combination reproduce overextension by dropping conjunction-as-meet, and which by dropping an order-preserving membership weight? | What remains of THEORY-255’s former promote_when after its promotion. In any lattice with an order-preserving membership weight, conjunction as meet cannot overextend. So every model that reproduces Hampton’s guppy-type items drops one of the two. Aerts drops the meet with an emergent subspace. Which assumption the other models drop, Busemeyer and Bruza’s (LIT-316, unavailable) and Franco’s among them, has not been read. | 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-185 | When does the information bottleneck’s first global transition between a lattice block and its far environment coincide with the point where the uninformative encoder loses stability, and is the first feature the leading excited transfer-matrix mode beyond the Ising check? | The promotion condition formerly on THEORY-205, widened. THEORY-205 was narrowed to the point where the uninformative IB encoder loses stability, β = (λ0/λ1)^(2L_B), and promoted on 2026-10-19 (ADR-047 v2). The global threshold, where the uninformative encoder stops being optimal, is 1/s*. It can come earlier, as it does on a skewed 3-state chain. In the circumference-3 Ising check the two agree. | Open |
| QUESTION-191 | Does the per-step nostalgia of a driven system without feedback fail to bound its dissipation when the drive is not Markov or the kernel does not keep each equilibrium stationary, and does any physical system show dissipation tracking nonpredictive information? | The promotion condition formerly on THEORY-026, with the part met removed. THEORY-026 was promoted on 2026-10-19 (ADR-047 v2) on a derivation that the per-step nostalgia I[s_t;x_t] − I[s_t;x_{t+1}] is nonnegative under a Markov drive and, summed, bounds dissipation from below when the kernel keeps each equilibrium stationary. What is left is whether the bound fails when either condition is dropped, and a physical system. | 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-203 | How does the average degree of the S¹ hidden-metric model flow under geometric coarse-graining when the closure fails between nearby supernodes, and does an exact ν(γ, β, r) confirm or move the phase boundaries γ = 3 and β = 2? | The degree-flow part of the promotion condition formerly on THEORY-190. THEORY-190 was promoted on 2026-10-20 (ADR-047 v2) narrowed to what was proved: the coarse-grained link probability tends to the S¹ form as the supernodes’ separation grows, and exceeds it between nearby supernodes. The source’s flow of ⟨k⟩ as r^ν, and its phase boundaries, rest on the closure holding at every separation and on a power-law approximation. | Open |
| QUESTION-205 | Inside Kadanoff’s kernel families, where each hidden spin couples to a block through a kernel of fixed form, does the exact trace condition force the coarse Hamiltonian to keep the relevant operators, or does some further stated condition still have to choose them? | The promotion condition formerly on THEORY-202. THEORY-202 was narrowed and promoted on 2026-10-20 (ADR-047 v2): over general Boltzmann machines, neither a matched partition function nor Kadanoff’s exact trace condition fixes which variables a coarse-graining keeps. Its second counterexample needs visible–visible couplings, so it lies outside Kadanoff’s kernel families. Whether the trace condition suffices inside those families is open. | Open |
| QUESTION-208 | Does the expected structure function of a computable i.i.d. source match its distortion-rate function, with the right-hand step of Grünwald and Vitányi’s Theorem 6.27 filled in? | Filed when the promotion audit (batch 050, ADR-047 v2) narrowed and promoted THEORY-244 to the entropy and mutual-information bridges, which it re-derived. The distortion-rate bridge rests on a step in Theorem 6.27 that the reading found asserted, so it left the title. The former promote_when’s clause for it is held here. | Open |
| QUESTION-215 | Does Landauer’s bound hold without assuming reversible microdynamics, and does any logically reversible operation have a positive entropy floor? | The open part of the promotion condition formerly on THEORY-030. THEORY-030 was narrowed and promoted on 2026-10-20 (ADR-047 v2) on a derivation that assumes the memory and its environment evolve reversibly together and start uncorrelated. Whether the bound survives without that assumption (the Earman–Norton charge that it is circular), what Landauer 1961 itself states, and whether any logically reversible step has a positive floor are open. | Open |
| QUESTION-223 | Is Ince’s common change in surprisal a correct redundancy measure: is it continuous, does it agree with a redundancy fixed by mechanism where rivals with non-negative atoms do not, may unique information depend on the source-source marginal, and can a modified sign rule restore subset equality? | Filed when the promotion audit (batch 056, ADR-047 v2) corrected and promoted THEORY-714. What I_ccs is and which axioms it keeps is now settled, including that it fails subset equality. Whether it is the right redundancy measure is not: that was the old promote_when, and it moves here. | 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-232 | Does the coarse-graining that maximizes mutual information with the system beyond a buffer find the renormalization-group variables of a system where they were not known beforehand, and does it give short-ranged effective Hamiltonians in two or more dimensions? | The promotion condition formerly on THEORY-194, moved here when THEORY-194 was narrowed to what Koch-Janusz and Ringel measured on the 2D Ising and dimer models and promoted on 2026-10-20 (ADR-047 v3). Every 2D case in that study had a known answer, and the link to a short-ranged effective Hamiltonian is proved only for 1D and quasi-1D chains. | Open |
| 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-324 | Are the conditions an effective description is asked to meet (predictive sufficiency, dynamical independence, memory closure, approximation fidelity, tractability) of one logical kind, or are some of them parameters of the regime under which the others are assessed rather than conditions in their own right? | The last clause of QUESTION-252, split off by the thirteenth status review under ADR-049 because CLAIM-401 (Active) settles it: they are not of one kind. Predictive sufficiency and approximation fidelity are adequacy relative to a declared target, metric and horizon; tractability restricts the operations under which adequacy is assessed; dynamical independence and memory closure are relations of the representation to the system, memory relative to a chosen projection. Which conditions can fail while the others hold, and which configurations are impossible in which model classes, stays QUESTION-252’s. | Answered |
| QUESTION-335 | Does causal emergence of effective information in Hoel, Albantakis and Tononi’s temporal and spatio-temporal macros survive a common intervention distribution, or does it come from comparing a macro step that spans several micro steps with a single micro step? | THEORY-1167 bounds the effective information of a coarse-graining of one transition by the micro’s. LIT-1038’s temporal and spatio-temporal macros are not of that form: the macro effect lies several micro steps ahead, and the comparison is with one micro step. The question is whether their positive causal emergence survives a common distribution, and whether that comparison is a fair one. | Open |
| QUESTION-343 | Does a rerun of Sarkar, Mukherjee and Sato’s released harness, or an independent implementation, reproduce Table 1’s single-path, uniform and holonomy-weighted accuracies on S4 synchronization with outliers within their intervals, including the paired holonomy − uniform gains? | The promotion condition formerly on THEORY-1158, moved here when the status review of 2026-10-30 promoted it under ADR-065. The accuracies are the paper’s, on 1000 instances per cell, and were not rerun; the AUC ceiling, which is arithmetic, was recomputed and needs no rerun. | Open |
| QUESTION-346 | On confounded causal systems larger than toy models, with a known causal partition, does gradient search for the causal information bottleneck through an adjustment set find the partition by p(y | do(x)) reliably, and how often does it fail? | The promotion condition formerly on THEORY-1146, moved here when the twenty-second status review narrowed that reading to what three toy models showed and promoted it (ADR-065). That the causal partition is the optimum is THEORY-1125; whether search finds it at scale is this question. | Open |
| 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 |