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.
Learning theory — why learning generalizes and how it proceeds — compression and description length, Kolmogorov complexity and sufficient statistics, rate–distortion and PAC-style bounds, model selection; and training dynamics: solvable models of learning trajectories, stages, plateaus and phase transitions in training, and the implicit bias of optimisation. A work whose subject is how to train a model better is ML practice and goes to the anthology (ADR-036).
12 of 547 CLAIM documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| CLAIM-305 | Resolving a singularity by blow-up reparametrizes the states and adds no observation, so it cannot make indistinguishable states distinguishable: every point over a singular point gives the same distribution, and the Fisher information can stay degenerate there; what the resolution buys is a normal form for the divergence, and so the learning coefficient | The owner’s LEGO monograph says, in Part I (chapters 4, 6, 7, 9, the Overview and “Definitions and Theorems”), that blow-up “restores identifiability”, “introduces new observables” and makes the Fisher information non-degenerate, and maps it onto splitting a LEGO bin. The pulled-back model is p(x | π(u)), a function of u only through π(u), so every point of the exceptional divisor gives the distribution of the singular point. In p(x | a, b) = N(ab, 1), the chart a = u, b = uv gives mean u²v and a Fisher matrix of zero all along u = 0. Splitting a bin is refining the observable, which can separate states. MOM-CLM-099 records that the monograph uses “blow-up” in three senses; this records why the analogy cannot hold in any of them. | 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-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-369 | In symmetric resource-mediated competition with interaction matrix A = UᵀU and growth rates r = Uᵀb − c, Lotka–Volterra population dynamics are multiplicative gradient flow on the convex feature-selection objective ½‖b − Un‖² + cᵀn over the nonnegative orthant: the invasion fitness of an absent component is the negative partial derivative of that objective, and the uninvadable equilibria are its KKT points | From a synthesis the owner relayed on 2026-10-18 (workbench entry of that date), checked here and given a proof. When competition between components is mediated by shared resources with feature or niche vectors u_i, so that A = UᵀU and r = Uᵀb − c, the Lotka–Volterra dynamics are ṅ_i = −n_i ∂_i L with L(n) = ½‖b − Un‖² + cᵀn: entropic mirror descent on the positive orthant. A rare component’s invasion fitness is −∂_i L, the value of adding that feature, and uninvadable equilibria are the KKT points of the convex program. The sharp edge: the identity needs a shared scalar objective, and it fails for asymmetric interactions or processes with competing objectives, which is where an organization stops being an optimizer. Mehta et al. (2019) and Marsland, Cui and Mehta (2020) give the general duality for consumer–resource models. | Active |
| CLAIM-377 | For any differentiable model under squared loss with Gaussian noise of precision β, the model’s Fisher on n inputs is the Gauss–Newton matrix βJᵀJ, and its non-zero eigenvalues are exactly β times those of the empirical NTK Gram matrix JJᵀ, with the same multiplicities, its eigenvectors being Jᵀ applied to the Gram matrix’s, normalized | THEORY-084’s Step 1, stated as the record’s own and proved. Through the SVD J = UΣVᵀ, JᵀJ and JJᵀ share the non-zero eigenvalues σ_i² with equal multiplicities, and Jᵀu/σ is the eigenvector of JᵀJ for a unit eigenvector u of JJᵀ. Under y | x ~ N(f(x; θ), β⁻¹), the expected outer product of the score is β∇f∇fᵀ, which summed over inputs is βJᵀJ, the Gauss–Newton matrix. “Fisher” means the model’s Fisher, not the empirical Fisher of observed residuals. It holds at every width, in every regime and on any data. It says nothing about what the spectrum looks like, which is THEORY-084’s. | Active |
| CLAIM-384 | For a network linearised at its initial parameters, with an isotropic Gaussian prior of precision α on the parameter displacement and Gaussian noise of precision β, the log-determinant part of the evidence is −½ Σ log(1 + βλ/α) over the non-zero eigenvalues λ of the empirical neural tangent kernel’s Gram matrix on the training inputs, so at most n directions are priced however many parameters the network has | 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-391 | For any classes with weights π_c and per-class factor matrices H_c and Δ_c, the class-wise Kronecker product Σ_c π_c H_c ⊗ Δ_c exceeds the product of the weighted averages H̄ ⊗ Δ̄ by exactly the weighted between-class covariance Σ_c π_c (H_c − H̄) ⊗ (Δ_c − Δ̄), so the two agree when either factor is the same in every class | 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-444 v2 | A replicator dynamic on the distribution of authority among an organization’s processes is the special case of the meaning–authority–legibility model in which the organization’s interpretation is the authority-weighted mixture of its processes’ and evidence reaches scoring without a maintained regime of its own; the replicator tools (Bayesian reading, potential and harmonic games, stochastic stability) then apply, and the record’s separate interpretation and legibility states earn their place exactly where interpretation departs from that mixture or legibility keeps a memory of past authority | Filed at the owner’s direction, 2026-10-24. The owner asked whether the tools of Soma et al. (THEORY-899), “or from related work”, could formalize the record’s more abstract version of the dynamics. Their population–policy duality fits the record’s authority dynamics with one identification. Read as a reduction, it says what CASE-124’s extra states are for. | Proposed |
| CLAIM-467 v3 | Under full-information, independent audits, a learned allocation of influence among specialists is no better than a contextual router trained on the same audits, which is a mixture of experts whose experts earn responsibility by fitting better; whatever governance adds over routing has to be shown where evaluation fails, and the record has not yet found such a regime | The prediction CASE-158 states (“Under full information, governance should not beat it”) and held by no claim, found in the audit of 2026-10-27; the runs were reported in CASE-158 and reproduced in CASE-162, where the router had the lower error in all four cells, selective and captured audits included. It gives CLAIM-278’s objection a definite form for this model class. The edge: it bounds governance by routing under full information; it does not say governance never helps, and the regimes where it might (QUESTION-271’s four failures) are untested or, so far, lost. | Active |
| CLAIM-513 | Across a family of goals, a world model is not an optional explanation of competent behaviour: in a fully observed, finite, communicating, stationary environment, a goal-conditioned policy within a factor 1 − δ of optimal on every composite goal of depth n fixes every transition probability to within √(2P(1 − P)/((n − 1)(1 − δ))), so observers reading the same policy must ascribe the same world model; what it leaves open is whether a predictive component is realized inside the agent, since a lookup table over histories and goals has the same policy | From the reading of Richens, Abel, Bellot and Everitt (LIT-1891, NOTE-1469), registered as the objection to watch against the owner’s U20 thesis CLAIM-519. It defeats that thesis’s functional reading only. The thesis’s defeated_if is not met: the theorem holds “regardless of … architecture”, makes no matched-resource comparison and no intervention on internals, and its authors say the map runs “from the agent’s policy rather than its activations”. Within its scope, though, calling a world model an explanatory attribution is too weak. The bound was re-derived and checked against an adversarial agent. | Active |