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).
How training finds structure — why trained networks come to have the structure they have, which features, symmetries and connected solution sets training reaches and why — understanding, not instruction for practice, which belongs to the anthology.
15 of 547 CLAIM documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| CLAIM-263 v3 | Whether two trained networks compute the same way can be tested by quotienting the architecture’s full symmetry group and measuring what remains: a barrier that survives alignment marks a different mechanism, connection after alignment means shared feature structure, and the test cannot run where features have no counterpart | The general content of CLAIM-187, filed alone in learned-structure under ADR-040: a measured protocol for asking whether two trained solutions are the same solution. Quotient the full symmetry group, including the orthogonal map on a transformer’s residual stream (THEORY-112). Read a barrier that survives as a difference of mechanism (THEORY-116). Between linearly connected image classifiers, nearly every layer’s features interpolate (THEORY-111). The test cannot run where features have no counterpart (THEORY-113) or the objectives share no structure (THEORY-115). It is the first claim in the learned-structure line. | Proposed |
| CLAIM-290 | A positive-definite kernel’s minimal realizations are unique up to a unitary, and two finite representation matrices with the same centred Gram matrix differ by a translation and an orthogonal map | The record’s own proofs of two steps that THEORY-004 rests on and that its sources state without proof. (a) Jorgensen & Tian’s Cor. 1.35: minimal realizations of a positive-definite kernel are unique up to a unitary. The book leaves this “as an exercise”, and NOTE-829 supplied the book’s own proof of Thm 4.35, transferred. (b) The finite case that LIT-250 §5.3 states as known: XXᵀ = YYᵀ implies Y = XQ with Q orthogonal, with no assumption that the rows span R^d. With centring this becomes a translation and an orthogonal map. It does not say Q is unique, which fails when the rank is below d. | Active |
| CLAIM-291 | The adjoint of a square-integrable integral operator is the integral operator with the conjugate-transposed kernel | The record’s own step, written down in THEORY-009 and marked there as not the Riesz article’s, which covers matrices and bounded operators only. For a kernel k in L²(μ⊗ν), the operator (Tf)(x) = ∫ k(x,y) f(y) dν(y) from L²(ν) to L²(μ) has adjoint (T*g)(y) = ∫ k̄(x,y) g(x) dμ(x), by Cauchy–Schwarz and Fubini. So T is self-adjoint, when μ = ν, exactly when k(x,y) = k̄(y,x) almost everywhere. It does not say where the symmetry of k comes from, which is THEORY-009’s question, and it is not claimed for kernels that are not square-integrable. | Active |
| CLAIM-356 v4 | An objective that originates outside a learning system can become a goal of the training organization, through its causal role in selecting, stabilizing and reproducing the learned structure it rewards; exogenous origin does not disqualify a goal, but it does not show that the network governs or endorses the objective, and what is selected is performance on the updating distribution, not generalization | The owner’s, U35 of the exchange of 2026-10-17, against A34’s guardrail that “Loss minimization does not automatically mean the model has its own goals”: the externally assigned objective “can motivate identification and persistence of learned features”, so the training process can be described as “having endogenous goals”. A35 granted it: “Yes, in a qualified but important sense”, and weakened the generalization half, since selection is for performance on “the encounters and feedback distribution actually used for updating”. It objects in part to CLAIM-288: it grants that a modeller-fixed objective does not show self-constitution, and denies that exogenous origin bars a goal from becoming the training organization’s own regulation. The sharp edge: the goal belongs to the training organization (CLAIM-367), not to the network, and nothing here says the network endorses or governs it. | Active |
| CLAIM-360 v2 | The deep-versus-shallow learning results separate three effects that aggregate loss conflates: a larger realizable function class, a larger set reachable within a fixed data and compute budget, and a different internal organization with the same attainable outcomes; the deep linear results show the third, the random-hierarchy results the second, and the depth-separation result for products of inputs the first, at fixed size | The owner’s, U38 of the exchange of 2026-10-17, on A37’s proposed experiment: “I think we may already have an empirical demonstration that would be useful here in the wide vs deep literature.” A38 refined it into three effects, (A) enlarged realizable function class, (B) enlarged budgeted reachable set, (C) new internal organization without better attainable outcomes, filed Saxe et al. under C and the random hierarchy model under B, and said A “needs a separately established expressivity restriction”. The record has one: Rolnick and Tegmark (THEORY-206), a separation at fixed neuron count. The sharp edge: which effect a result shows depends on the budget, and A38 warned against “depth produces individuation while width merely provides redundant capacity”. | Active |
| CLAIM-366 v3 | Whether several objectives share a strict first-order descent direction, 0 ∉ conv{g_i}, is invariant under smooth invertible reparameterization, while the sign of a Euclidean gradient inner product is not, and a negative inner product does not exclude common descent; a budgeted version is invariant only with the metric pulled back | A35 §2 of the exchange of 2026-10-17, at the owner’s U35 “double click” on whether functionally equivalent parameterizations give equivalent frustration diagnoses. Gradients transform as g′ = Jᵀg and directions as v′ = J⁻¹v, so directional derivatives, and with them the common-descent test 0 ∉ conv{g_i}, do not depend on coordinates. A Euclidean inner product does: g₁ = (1, 1), g₂ = (1, −½) have product ½, and under θ = diag(½, 2)ψ their transforms have −7/4, while d = (−1, 0) lowers both throughout. The sharp edge: it covers coordinate changes of one parameter space at first order, not function-preserving symmetries or training trajectories (QUESTION-127). The practice corollary, not to diagnose task conflict by gradient cosine, is the anthology’s. | Active |
| CLAIM-367 v4 | Whether a training-time regulatory condition belongs to a network depends on which unit is individuated for the training dynamics and whose jurisdiction governs the updates; in conventional training most of that governance falls to the optimizer–model–data system, so a condition of training is not thereby a condition of the network | The owner’s correction, U31 of the exchange of 2026-10-17: “it might help clarify things if we apply our individuation and volition frameworks to this system”, the system being a deep network during training (U30). A31 granted it: “applying individuation first, then volition, then affect exposes a problem in our previous discussion”. Individuated for the training dynamics, the unit is at least the network–optimizer pair; mapped through the six relations of will-organization, most of the governance of updates is outside the network’s jurisdiction. So optimization difficulty in conventional training is real and belongs to the training system. The sharp edge: it does not say the governance is unreal, or that a network cannot come to hold some of it. | Proposed |
| CLAIM-382 | In an autoencoder initialised with i.i.d. Gaussian weights and zero biases and trained by SGD or momentum SGD on a loss of its output, the law of each hidden layer’s learned representation is invariant under every orthogonal map of that layer that commutes with its activation, so under a radial activation no basis is privileged in law, while elementwise tanh or Leaky-ReLU yields invariance only under the signed permutations or the permutations, which leaves the standard axes free to be privileged | THEORY-288’s symmetry argument, stated as the record’s own and proved. Transform the parameters by orthogonal maps Q_l of the hidden spaces that commute with the activation. The network then computes Q_l h_l and the same output, so the loss is invariant. Gradient steps commute with the transformation, and an i.i.d. Gaussian initialisation is invariant under it, so the law of the trained representations is too. For radial activations this is all of O(n), and no basis is privileged in law. For elementwise ones it is a finite group, so the axes may be privileged. Whether they are, and how strongly, is THEORY-288’s empirical content. The argument needs an orthogonally equivariant optimiser, and fails for Adam. | Active |
| CLAIM-387 | Optimal deterministic information-bottleneck partitions at two budgets need not be nested: with X uniform on four states and binary Y with P(Y=1|x) = (0, 0, 1/8, 3/8), the optimum is {0,1,2 | 3} for 6.91 < β < 9.26 and {0,1 | 2,3} for 9.26 < β < 16.06, two partitions neither of which refines the other | Found by brute force in the reading of Equitz and Cover (NOTE-988) and re-run here from scratch. It answers the deterministic half of QUESTION-220 in the negative: raising the budget can reorganize the optimal partition rather than split it. The two optima are not even laminar, so no fixed tree holds both as prunings, and NOTE-676’s nesting is bought by its tree restriction. It says nothing about stochastic encoders, where finite-β optima generally live (THEORY-743). | Active |
| CLAIM-392 v2 | Stochastic information-bottleneck optima at two budgets need not be nested: for X uniform on four states and binary Y with P(Y=1|x) = (0, 0, 1/8, 3/8), the unique optima at β = 8 and β = 12 admit no Markov chain X → T₂ → T₁, so raising the budget reorganizes the optimum even with noisy encoders | Computed in the reading of Charvin, Catenacci Volpi and Polani (2023), NOTE-981, on the source of CLAIM-387, and re-run here. With THEORY-753’s criterion (refinable iff some pair of optima forms the chain, iff the coarse decoder points lie in the fine decoder hull when that hull is non-degenerate), and optima certified unique, the hull test fails. It answers the stochastic half of QUESTION-220 in the negative for this source. The certificate is numerical, not interval arithmetic, and the loss is small: 0.49% of the larger budget, in unique information. | Active |
| CLAIM-498 | Averaging weights within a basin averages features: the network whose weights are the average of several networks in one low-loss basin has, in every layer, the average of their features | The averaging clause of THEORY-111’s old title, kept as a Proposed claim by the ADR-047 sweep of 2026-10-30 (kind 2) when the eleventh status review’s narrowing had dropped it. No source measured or asserts it: it is the record’s step from pairs to many-model averages, and its defeater is the averaging half of the old promote_when. | Proposed |
| CLAIM-499 | The linear barrier left after aligning independently trained networks over the architecture’s symmetries rises with depth | The depth clause of THEORY-112’s old title, kept as a Proposed claim by the ADR-047 sweep of 2026-10-30 (kind 2) when the eleventh status review’s narrowing had dropped it. No source measures or states it for barriers after alignment: it is the record’s composition of Entezari et al.’s unaligned depth curves with Ferbach et al.’s compounding width bound. Its defeater is the depth paragraph of the old promote_when. | Proposed |
| CLAIM-524 | Whether functionally equivalent realizations are connected depends on which paths are admissible: for f = u₁v₁ + u₂v₂ with nonnegative weights, (1, 1, 0, 0) and (0, 0, 1, 1) are joined by a curve along which f stays 1 though the straight line between them falls to ½, and under any resource budget below √2 times the endpoints’ cost no path with f = 1 joins them | From the owner’s remark at U17 of the exchange “Analyze nucleation concepts” (2026-10-31), that the reconstruction fibres “reminds me of loss landscape connected mode stuff (which might just be a consequence of high dimensional geometry? … probably already discussed on the nucleation record)”. It is: see below. A17 §9 gave the curve and A18 §7 the budget for a tied-weight family; the general bound here is the reader’s, rechecked. “Viable mode connectivity” is A18’s name and is not filed as a term. | Active |
| CLAIM-536 | A categorical model of perspectives and an LDA-style model differ in where the mixture lives: the first draws every act’s perspective from one fixed categorical distribution, the second gives each episode its own mixture, drawn from a Dirichlet, and draws each act from it; the Dirichlet is a distribution over mixtures, not the perspectives themselves, and with symmetric concentration a over K perspectives two acts in one episode share a perspective with probability (a + 1)/(Ka + 1) against 1/K across episodes, so the concentration runs from one perspective per episode, as in persona selection (a → 0), to a fresh independent draw for every act (a → ∞) | The owner’s U12 of the exchange “Assess SDT Coverage” (2026-11-02), stated precisely: “H1 isn’t LDA like though. that’s my point. it’s a categorical distribution. the topics in H@ are the dirichlet distribution which parameterizes the categorical distribution, which is what distinguishes H2 from H1 and why H2 is an LDA.” The structure is right; one word is not. The Dirichlet is a distribution over each episode’s mixture, and the topics, here the perspectives, are the categories that mixture weights. The agreement formula, derived here and checked by simulation, shows that both readings of H1 are limits of H2. | Active |
| CLAIM-543 | What separates a language model’s distribution over personas from a distribution over organizational perspectives is not whether the latent model is hierarchical, since a mixture or an LDA-style admixture already is, but whether some latent components condition which others are available or consequential: a model whose components are exchangeable given their weights is organizationally flat however many levels its prior has, and dependencies among components count as organization (organization as relational arrangement) only where interventions, not fit, support them | The owner’s U8 of the exchange “Assess SDT Coverage” (2026-11-02): “think of the relations as bayesian hierarchy, so H1 is a flat plate model and H2 has hierarchical/causal structure”, said of a language model read as a distribution over perspectives under the Persona Selection Model (U6). Stated here with A9’s correction: an LDA is itself a three-level hierarchical model drawn in plates, so “flat” has to mean organizationally flat, components exchangeable given their weights, not probabilistically flat. The precise form of the owner’s H1/H2 contrast is CLAIM-536. Proposed: the contrast is argued, and no study separates the two kinds of model on a language model. | Proposed |