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.
279 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-011 v2 | Spectral Sets | Badea & Beckermann (2013), ARXIV-1302.0546. A survey of when a set X ⊃ σ(A) satisfies ‖f(A)‖ ≤ K sup_X|f| for all rational f. Its headline entries are that the closure of the numerical range is completely K-spectral with a universal K_Crouzeix ∈ [2, 11.08] (Fact 107.7.2a; completely 2-spectral for 2×2, Fact 107.7.2b) and that σ(A) is K-spectral iff A is diagonalisable, with K the eigenvector condition number (Example 107.2.5). §107.8 turns these into Krylov error bounds such as Arnoldi ε_m ≤ 2K_Crouzeix ρ_{m−1}(f,E) for convex E ⊃ W(A). | Active — read in full 2026-09-25 (NOTE-009); worth keeping as a compact, correctly referenced index of what was known in 2013 about bounding ‖f(A)‖ for non-normal A by sup-norms over the numerical range, pseudospectra, disks, lenses and annuli. Consult the relevant Fact rather than read it through. The universal numerical-range constant has since been improved: Crouzeix–Palencia (SIAM J. Matrix Anal. Appl., 2017) give 1+√2. That comes from outside the chapter and was not verified in this session. Check it before quoting 11.08 as current. |
| LIT-016 v2 | The sheaf-theoretic structure of non-locality and contextuality | Abramsky & Brandenburger (2011), ARXIV-1102.0264. For finite measurement scenarios, an empirical model has a factorizable (local / non-contextual) hidden-variable realization iff its compatible family of distributions extends to a global section (Thm 8.1). Over the reals every no-signalling model has such a section (Thm 5.9: “negative-probability LHV” ⇔ no-signalling, because both span the same subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{|U|}, Thm 5.4). Contextuality comes in three strictly nested strengths (probabilistic ⊋ possibilistic ⊋ strong), with the Bell, Hardy and GHZ(n≥3) models as witnesses of each level. | Active — read in full 2026-09-25 (NOTE-016); the founding paper of the sheaf-theoretic approach to contextuality. It is short, fully proved and still the standard reference for its definitions (empirical model, strong contextuality, the global-section criterion). Later work (cohomology, the contextual fraction, the CbD comparison) extends it but does not replace it. |
| LIT-017 v2 | Spectral Networks and Betti Lagrangians | Casals & Nho (2025), ARXIV-2504.08144. For any rank n, and for Lagrangians with non-algebraic (positive-braid) asymptotics, spectral networks exist and are characterised by pseudo-holomorphic strips in the adiabatic limit (Theorems 1–2). For exact Betti Lagrangians, a Family Floer functor equals Gaiotto–Moore–Neitzke non-abelianization in the adiabatic limit (Theorem 3, i.e. 5.24) and is realised by A∞-operations in wrapped and partially wrapped Fukaya categories of T*S (Theorem 4). The analysis in §§4–5 leans on Ekholm 2007 and on the second author’s 2024 preprint, and Theorem 5 (spectral networks from Demazure weaves) is recorded without a written proof. | Proposed — read in full 2026-09-25 (NOTE-022); an unrefereed v1 preprint that is carefully motivated, but whose analytic core (§4 adiabatic convergence, §5 continuation strips) leans on Ekholm 2007 and on the second author’s 2024 preprint, and whose Theorem 5 has no written proof. What would settle it: a refereed version or independent expert check of §§4–5, a written proof of Proposition 7.5, and theorem statements narrowed to the hypotheses the proofs use (creative networks, exactness, trimmed limits). Only §1 is useful to a non-specialist here. |
| LIT-020 v2 | Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows | Weber et al. (2016), ARXIV-1607.08654. The paper specializes Forman’s (2003) Bochner–Weitzenböck curvature on weighted CW complexes to graphs, giving the closed-form edge curvature Ric_F(e) = ω(e)[ω(v1)/ω(e) + ω(v2)/ω(e) − Σ_{e_v1∼e, e_v2∼e}(ω(v1)/√(ω(e)ω(e_v1)) + ω(v2)/√(ω(e)ω(e_v2)))] (Eq. 11). It then proposes three uses: an EMD distance between curvature distributions, which puts Facebook, Google and BioGrid graphs at 0.050–0.060 from a Barabási–Albert model against 0.17–0.36 from Erdős–Rényi and Watts–Strogatz (Table 1); a discrete “2-D-form” Ricci flow γ̃(e) − γ(e) = −Ric_F(γ(e))·γ(e) (Eq. 36) for change detection; and a Laplacian flow ∂G/∂t = (□₁ − F)G (Eq. 45) for denoising. The authors say the flow applications are “only theoretically proposed and illustrated solely on a small example” (§5). | Active — read in full 2026-09-25 (NOTE-026); the standard early citation for Forman–Ricci curvature on networks and for the Forman–Ricci flow. Read it for §2.3 (Eqs. 11–17) and §4.1 (Eqs. 36–41), and cite it for definitions only. Its empirical and application claims are one-table or one-figure demonstrations. Several of its formulas contain sign and consistency slips, listed below, so a user should take Eq. 11 from here and check it against Sreejith et al. 2016. |
| LIT-030 v2 | A Counterexample to the Mizohata-Takeuchi Conjecture | Cairo (2025), ARXIV-2502.06137. For every C² hypersurface Σ ⊂ ℝᵈ not lying in a hyperplane (implicitly: one with a point of non-zero curvature; see corrections), and every large R, there are f ∈ L²(Σ) and a weight w = |ĥ|² with supp ĥ ⊂ B_{CR} such that ∫|Ef|²w ≳ log R · ‖f‖²₂ · sup_ℓ ∫_ℓ w (Thm 1.2 via Prop. 3.1). So the Mizohata–Takeuchi conjecture and Stein’s conjecture (Conj. 1.4) are false as stated, and the CHV route to loss-free endpoint multilinear restriction is closed. The mechanism is a lattice Q of middle-layer subset sums of N ≍ log R points on Σ, chosen so that no hyperplane meets more than 2^{d−1} of the R⁻¹-balls around Q (Lemma 3.5). | Active — read in full 2026-09-25 (NOTE-051); a short, readable disproof of a 40-year-old conjecture in Fourier restriction. The counting argument is checkable by a non-specialist; the incidence lemma (§4) is the part that needs an expert. It is worth reading as mathematics. The “local” replacement it proposes (Conj. 1.5) has since been refuted for a dense family of C^k hypersurfaces by Cairo–Zhang (arXiv:2512.08064), so it should be read together with that paper. |
| LIT-032 v2 | A type theory for synthetic ∞-categories | Riehl & Shulman (2017), ARXIV-1705.07442. A three-layer type theory adds a directed interval 2 to HoTT. Its layers are cubes, topes (the coherent theory of a strict interval) and types, and simplices become shapes that can be mapped out of by extension types. One contractibility condition, “Segal” (unique fillers for the 2-1-horn), then yields associativity (Prop. 5.9) and 3-horn fillers (Prop. 5.21). It also makes every function a functor (Prop. 6.1), gives a dependent Yoneda lemma for covariant families (Thm. 9.5), and makes having a left adjoint a proposition when the adjoint’s codomain is Rezk (Thm. 11.23). The bisimplicial-sets semantics is sketched. Segal and Rezk types are shown to match Segal and complete Segal spaces (Props. A.21, A.25), and soundness is not proved in full. | Active — read in full 2026-09-25 (NOTE-069); This is the founding paper of simplicial HoTT, where synthetic (∞,1)-category theory starts. It is worth reading as mathematics by anyone following directed type theory. Its semantics appendix is explicitly a sketch. |
| LIT-038 v2 | The Relativity of Causal Knowledge | D’Acunto & Battiloro (2025), ARXIV-2503.11718. The paper defines SCMs as functors I → Prob and interventions as natural transformations (which requires invertible mixing functions). It encodes each non-intervened SCM as the convex space Δ(V, Ω) of all probability measures on its endogenous space, and puts these spaces on a graph as stalks of a network sheaf and cosheaf. The restriction maps are interventionally consistent α-abstractions and the extension maps are unspecified affine maps. “Relative causal knowledge” is the image of alternating restriction and extension along a path (Def. 11). The two substantive-looking theorems are either unsound or vacuous. Theorem 1 (every convex combination of a model’s interventional measures is a soft-interventional measure of it) rests on a proof step that is false. Theorems 2 and 3 hold for any pushforward map, so interventional consistency does no work in them. There are no results about the sheaf. | Rejected — read in full 2026-09-25 (NOTE-050); Not worth a reader’s time as a source of results. Its one non-trivial theorem is unsound as proved and false in general, its encoding theorem is vacuous, and §4 contains definitions only. A reader wanting category-theoretic causal abstraction should go to Rischel’s thesis or Beckers–Halpern. A reader wanting sheaves on networks should go to Curry or Hansen–Ghrist. |
| LIT-039 v2 | Rigorous renormalization group | Kupiainen (2023), DOI-10.1038/s41567-023-02233-x. A Comment, read in the author’s accepted manuscript. Rigorous RG iterates Wilson’s block-spin map on lattice Hamiltonians. Each step is controlled by a polymer (cluster) expansion that converges where fields are small and treats large fields as improbable. It proved Gaussian long-distance behaviour for massless 4D lattice φ⁴ and constructed the 2D Gross–Neveu model. Aizenman and Duminil-Copin showed continuum 4D φ⁴ is always Gaussian. The method is limited to near-Gaussian regimes. | Active — read 2026-10-14 (NOTE-821) in full, from the author’s accepted manuscript in the University of Helsinki repository; worth reading as a three-page orientation, by one of its principals, to what rigorous renormalization group is: Wilson’s block-spin step iterated, with each step made rigorous by cluster expansions that separate small and large fields, and what it proved (Gaussian long-distance behaviour of massless 4D lattice φ⁴; continuum 4D φ⁴ trivial; Gross–Neveu constructed). Read it knowing that it is a Comment with no proofs, that it speaks for the block-spin school, that it says nothing about learning, and that its claim that a non-perturbative formalism is missing is the author’s judgement. |
| LIT-040 v2 | two-level type theory [descriptive title: nLab wiki entry “two-level type theory”] | nLab (2018), https://ncatlab.org/nlab/show/two-level+type+theory. Two-level type theory pairs a homotopy (univalent, fibrant) type theory with an outer level having UIP “exact equality” (the internalised metatheory). The nLab page is an accurate but thin gloss on two-level type theory. It says nothing about the result that makes 2LTT usable: basic 2LTT is conservative over HoTT (ACKS Prop. 2.18: an inner type provably inhabited in 2LTT is inhabited in HoTT), which answers much of the page’s own “open question” about the principle of equivalence. It also omits that a fibrant-replacement type former would force UIP on the inner level (ACKS Thm 2.20). The paper it cites, ACKS arXiv:1705.03307, is the proper source: definitions via models, conservativity, the axiom menu (A1)–(A6)/(T1)–(T3), and internal Reedy-fibrant diagrams over inverse categories with a definition of univalent (∞,1)-category. | Superseded — by LIT-095; read in full 2026-09-25 (NOTE-047); superseded by Annenkov–Capriotti–Kraus–Sattler, “Two-Level Type Theory and Applications” (arXiv:1705.03307; MSCS 2023, DOI 10.1017/S0960129523000130), which the page itself cites and which covers everything the page says, with proofs and the conservativity theorem the page lacks. That paper is now filed as LIT-095, and this page is superseded by it. |
| LIT-043 v2 | This Secret Math Equation let the US Government Spy on Anyone | Kibicho (2025), https://leetarxiv.substack.com/p/dual-ec-backdoor-coding-gui…. The post is a paywalled programming tutorial that implements the Dual_EC_DRBG random-number-generator backdoor in Python, as a chapter of a book on finite fields for programmers. Nothing beyond that framing was reachable. | Unavailable — seeded 2026-09-25 from the post’s visible front matter; on 2026-10-14 the body was still not reachable, so it is not read and no NOTE is filed. The Substack post API returned the post with audience ‘only_paid’, a word count of 1,364 and a body truncated after the introduction and chapter list (no free preview). The public page shows the same. A web search for the title, the author and LeetArxiv found no other copy of the chapter: the forthcoming book it belongs to is not published, and the search results were other authors’ accounts of Dual EC. A paid subscription is not a copy this client can use. It stays Unavailable until a copy is supplied. |
| LIT-044 v2 | The Epic Story of Maximum Likelihood | Stigler (2007), ARXIV-0804.2996. Stigler gives a history of maximum likelihood from Lagrange (1769) to Le Cam (1953). Fisher gave three optimality arguments. The 1922 one rests on the false belief that ML estimates are always sufficient. The 1925 “ANOVA” one derives the information inequality V ≥ 1/I(θ) from Var X = E Var(X|T) + Var E(X|T), treating approximate normality as exact. The third, sent privately to Hotelling in 1930, holds for multinomial models and smooth degree-0-homogeneous estimators, and was published only in disguise in 1935. Stigler’s verdict is that the Neyman–Scott growing-parameter example (Wald 1938, published 1948) is the practically important failure. He calls Hodges’s 1951 superefficiency example “an ingenious but minor technical achievement”, because the superefficiency set has Lebesgue measure zero (Le Cam 1953). | Active — read in full 2026-09-25 (NOTE-074); an authoritative, archive-based history by the leading historian of statistics, and worth reading for exactly which ML optimality claims were proved, under what conditions, and which counterexamples matter. It is a history, not a technical reference: for modern conditions it points to Bahadur (1964), Le Cam (1953, 1990), van der Vaart (1998) and Cox (2006). |
| LIT-050 v2 | Mean-Field-Type Game Theory with Rosenblatt Noise | Tembine et al. (2025), ARXIV-2506.08025. The paper’s control and game results use linear state feedback, and they depend on the Rosenblatt noise only through its covariance, which is exactly fBM’s with the same H. They are therefore identical to the fBM results. The ergodic LQ gain (Thm 4), K̂ = −[b₁ + √(b₁² + 4H(1−H)b₂²q/r)]/(2b₂(1−H)), is correct and reduces to the Brownian LQR gain at H = ½. But the paper’s headline “suboptimality of noise approximation” (§VI-A, repeated for the variance-aware control, zero-sum, non-zero-sum and MFTG settings) holds only against surrogates with a different covariance; against fBM with the same H it is false. The ML claim that Rosenblatt “super-diffusion transformers” “significantly enhance the performance of generative machine intelligence models” (§XIV-C) has no model, training, dataset or experiment behind it anywhere in the paper. | Rejected — read in full 2026-09-25 (NOTE-068); not worth a reader’s time. The control-theoretic results are earlier papers’ (its refs [43]–[46]) restated. Its central new claim, that Gaussian and fBM surrogates give suboptimal control, is false for fBM with matching H, since the costs depend only on covariance. Its empirical sections are unsourced vignettes: numbers without references, “Rosenblatt shape” fits without any fitting procedure, and data from the first author’s own startups. Its ML claim is an assertion with no model or experiment. A reader wanting Rosenblatt-noise LQ control should go to Čoupek–Duncan–Maslowski–Pasik-Duncan (CDC 2018, ref [43]) and Čoupek–Duncan–Pasik-Duncan (SPA 2022, ref [27]). |
| LIT-051 v2 | Lecture Notes on Control System Theory and Design | Başar et al. (2020), ARXIV-2007.01367. A conventional first-year graduate course in linear state-space control. It covers modelling and linearisation, LTI/LTV solutions, Lyapunov stability, controllability and observability with Kalman/Hautus tests and minimal realisations, then pole placement and observers, integral control, sensitivity, and finally HJB/LQR and a heuristic minimum principle. Its most useful non-standard content is the Pendubot thread (Chs 1, 9, 10): an observer-based design from one sensor is fragile (the compensator has an unstable pole at s ≈ 46.4), while full-state LQR has |S(jω)| ≤ 1, infinite gain margin and ±60° phase margin. Every worked example I recomputed is correct. | Active — read in full 2026-09-25 (NOTE-062); a sound, free, compact state-space control course, worth reading as background for LQR, Lyapunov and HJB when a reader needs one place for them. It is not a source for anything current, and the preface’s promise of a reinforcement-learning bridge is not delivered in the text. |
| LIT-053 v2 | The Homological Nature of Entropy | Baudot & Bennequin (2015), DOI-10.3390/e17053253. On a finite probability space, take an information structure S (a set of partitions closed under joins that have a common refinement in S and containing the trivial partition) and a probability family Q complete for it. The information cohomology H*(S;Q) is computed by a Hochschild-type bar complex whose action is Shannon’s averaged conditioning. If S is connected and “sufficiently rich” and cochains are measurable, then H¹ is one-dimensional and spanned by Shannon entropy (Thm 1). This makes entropy the unique measurable 1-cocycle, up to one multiplicative constant per connected component. In the unitary-invariant quantum case H¹_U = 0: Shannon entropy is the coboundary of minus the von Neumann entropy, which is the unique invariant 0-cochain with that property up to an additive constant (Thm 3). The tree (“arborescent”) version again gives H¹ spanned by entropy (Thm 4). | Active — read in full 2026-09-25 (NOTE-088); the founding paper of information cohomology, with complete proofs of the degree-one results (Thms 1, 3, 4). Read it for those. Its broader programme (higher mutual informations as cocycles, super/sub-harmonicity, max-entropy discrimination) is sketched or deferred here to unpublished companion papers. |
| LIT-054 v2 | A complete graphical calculus for Spekkens’ toy bit theory | Backens & Duman (2014), ARXIV-1411.1618. Take the maximal-knowledge fragment of Spekkens’ toy bit theory with post-selected measurements, viewed as relations over {1,2,3,4}ⁿ. It has a ZX-style red–green graphical calculus that is universal (§4.3), sound (§4.4) and complete (Thm 17). Completeness goes through a proof that the toy theory’s binary “check-matrix” formalism coincides with the qubit stabilizer one (valid states SᵀJS = 0, valid transformations QᵀJQ = J, Lemma 3). With it come graph-state/local-complementation results for toy states (Thms 4–5), map–state duality (§5.2) and an rGS-LO normal form (Thms 10–11). The rewrite rules differ from the stabilizer ZX-calculus only through the phase group: Z₂×Z₂ against Z₄. That difference appears as an “11-commutation” rule in place of π-commutation (§4.5). | Active — read in full 2026-09-25 (NOTE-066); the completeness result is proved and still the diagrammatic reference for the toy bit theory. Read §2.1, §4.2, §4.5 and §5.1 for the stabilizer comparison. The ψ-epistemic framing in the abstract and §1 is motivation only; the paper does no ψ-epistemic analysis. |
| LIT-064 v3 | Talagrand Meets Talagrand: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes with Finite Index Sets | Chu & Raginsky (2025), ARXIV-2502.06709. Take a centered Gaussian process on a finite set T with max variance σ² and minimum separation a. The expected softmax-weighted average g(β) = E[Σ_t X_t e^{βX_t}/Z] satisfies g(β) ≤ √(2σ²·E D(ν_β‖ν₀)) for every β (Thm 1, D = KL from uniform). For β ≥ β*, g(β) ≥ c·a·√(E D(ν_β‖ν₀)), with c the Sudakov constant (1/17 via Marcus–Rosen) (Thm 2). For i.i.d. N(0,σ²) logits there is an exact identity, g(β) = βσ²(1 − E‖ν_β‖²₂) (Lemma 2). The quenched free energy is bounded above by √(2σ²·E D_{1/2}(ν_β‖ν₀)), with an i.i.d. lower bound of (cσ/2)√(E D_{1/2}) (Thms 4–5). | Superseded — by ANTH-LIT-726; moved to the Anthology of the SOTA as ANTH-LIT-726 on 2026-09-26, at the owner’s request, with its Read NOTE; nothing is held in both records (ADR-001) |
| LIT-067 v2 | A Social Equilibrium Existence Theorem | Debreu (1952), DOI-10.1073/pnas.38.10.886. Debreu proves existence of an equilibrium for an “abstract economy” (generalized game), in which each agent’s feasible set A_ι(ā_ι) depends on the others’ actions. The conditions: each action set 𝔄_ι is a contractible polyhedron; A_ι has a closed graph with non-empty compact values; f_ι is continuous on the graph into the extended reals; the value function φ_ι(ā_ι) = max f_ι is continuous; and each best-response set M_ā_ι is contractible. The proof applies the Eilenberg–Montgomery/Begle fixed-point theorem to the product best-response map. A Remark replaces the joint continuity of φ_ι with continuity (lower hemicontinuity) of A_ι plus a compact graph. A Corollary gives a saddle-point theorem that contains Kakutani’s and von Neumann’s. | Active — read in full 2026-09-25 (NOTE-037); the original existence theorem for generalized (constrained) games, underlying Arrow–Debreu (1954). Short, fully self-contained, and still the canonical citation. Later results (Shafer–Sonnenschein 1975; the generalized-Nash literature) weaken its hypotheses but do not supersede it as the source. |
| LIT-068 v2 | How Local Separators Shape Community Structure in Large Networks | Frenkel & Carmesin (2025), ARXIV-2504.14501. This is a comparison of the authors’ local 1- and 2-separator decompositions against Infomap, Label Propagation, Leiden and Louvain (igraph) on five sparse graphs (62 to 9,133 nodes). Local separators score lower overlapping modularity than the modularity optimisers everywhere, and lower than Infomap on 3 of 5 graphs (Table 2). Their edge is confined to high density thresholds on the two road networks, where the best radius gives a handful more qualifying clusters: about 12 against about 5 at β ≥ 1.25 on Euroroads, and about 50 against about 40 for Infomap at β ≥ 1.5 on NRW (Figs. 3–4). On Netscience all methods are similar. At δ = 1 on both road networks, and at δ = 1.25 on NRW, Infomap yields more qualifying clusters than any local-separator setting. | Rejected — read in full 2026-09-25 (NOTE-085); not worth a reader’s time. Its critique of modularity (§2.4) contains two false statements. Its own “community detection problem” (§2.5) is defined but never scored. Read at the paper’s own thresholds, the figures do not support “consistently identify the densest communities”. The method itself is never defined here, so a reader wanting local separators should go to Carmesin & Frenkel, arXiv:2312.12354 (its ref. [9]). |
| LIT-073 v2 | General probabilistic theories: An introduction | Plávala (2021), ARXIV-2103.07469. A self-contained textbook on finite-dimensional GPTs whose central results are that state spaces and effects are dual (K = {φ ∈ A(K)*+ : ⟨φ, 1_K⟩ = 1}, Thm 3.19) and that optimal two-state discrimination succeeds with probability ½(1 + ‖λx₀ − (1−λ)x₁‖) in base norm (Thm 3.43). Every composite lies between the minimal and maximal tensor products (Prop. 5.6), and four statements are equivalent: id_K is measure-and-prepare, id_K is self-compatible, K admits a universal broadcasting channel, and K is a simplex (Thm 7.7); minimal = maximal iff one factor is a simplex is cited to [134], not proved. | Active — read in full 2026-09-25 (NOTE-035); the best single entry point to the GPT vocabulary this record’s Spekkens-programme readings presuppose (state space, effect algebra, GPT dimension, tomographic locality, minimal/maximal tensor products, measure-and-prepare). Read §2–3, §5.1, §5.5, §6.1–6.5 and §7.1. It is not the place for restricted theories like Spekkens’ toy theory, to which §3.7 gives a single paragraph. |
| LIT-081 v2 | Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories | Barnum et al. (2009), ARXIV-0912.5532. A bipartite state ω steers its B-marginal iff ω̂ : [0,u_A] → [0,ω_B] is a strong quotient map of ordered sets (Thm 5.4). When ω̂ is injective and the marginal is interior, a steering state is an order-isomorphism (Cor. 5.5). With Thm 4.1/Cor. 4.2, it follows that universal self-steering makes every irreducible state space homogeneous and weakly self-dual (Prop. 5.8). The step from there to C*-algebraic quantum theory needs strong self-duality, which the paper does not derive, and a Hanche-Olsen correspondence it defers (“as we will establish elsewhere”, §6). | Active — read in full 2026-09-25 (NOTE-082); a short, readable and mostly proved link from an operational feature (ensemble steering) to cone geometry (homogeneity, weak self-duality). It remains a standard reference for the steering route in GPT reconstructions. Read it with the open weak/strong self-duality gap and the thin argument for Prop. 5.7 in view. |
| LIT-093 v2 | Notes on the Multiplicative Ergodic Theorem | Filip (2017), ARXIV-1710.10694. Expository notes that prove the Oseledets theorem in four ways. §2 uses projective invariant measures and a splitting lemma, without Kingman. §3.5 checks Kaimanovich’s regularity criterion in the symmetric space GLₙ/Oₙ, using Kingman. §4 proves the Karlsson–Ledrappier horofunction theorem (for µ-a.e. ω there is a horofunction h_ω with −(1/N)h_ω(g_ω⋯g_{T^{N−1}ω}x₀) → l) for isometries of proper metric spaces. §5 gives an L² “mean” version via Monod’s direct integrals of CAT(0) spaces and Karlsson–Margulis tracking, with an L² mean Kingman theorem. Two of the notes’ own derivations do not work as written: Proposition 3.4.8(iii) is mis-stated, and the Lemma 5.1.5 step in the tracking proof does not apply as claimed. The main theorems are standard and not in doubt. | Active — read in full 2026-09-25 (NOTE-078); a good, short guide to the geometric view of Lyapunov exponents (symmetric spaces, horofunctions, CAT(0) direct integrals), worth reading for the §2 and §4 proofs. Read Kaimanovich’s theorem (proved here only for H²), Prop. 3.4.8 and the §5.3 tracking proof critically, and do not use it as a source of ML claims; it makes none. |
| LIT-095 | Two-Level Type Theory and Applications | Annenkov et al. (2017), ARXIV-1705.03307; Mathematical Structures in Computer Science 2023. Two-level type theory (2LTT) pairs a fibrant inner theory (HoTT) with an outer theory with strict equality, so that coherence towers such as semisimplicial types can be stated internally. Basic 2LTT is conservative over HoTT (Props 2.18–2.19): an inner type inhabited in 2LTT is already inhabited in HoTT. A fibrant-replacement type former would force UIP on the inner level (Thm 2.20). | Active — read in part on 2026-09-26, alongside the nLab page it supersedes (LIT-040) — §1–3, §4.7 and §5 in full, §4.3–4.6 at the level of statements; no NOTE yet, so a full reading is still owed |
| LIT-102 v2 | Pregeometry, Formal Language and Constructivist Foundations of Physics | Arsiwalla et al. (2023), ARXIV-2311.03973. A programmatic essay with no construction, theorem or model of its own. It argues that inferential justification in a materialist ontology regresses without end (§3), that a “coherentist” constructivism escapes the regress, and that Wheeler’s pregeometry is therefore best read as “structureless structure”: the types and compositional rules of a computational language such as homotopy type theory (§§4, 6). Every construction of space from such rules that it points to is in the authors’ own earlier papers ([13, 17, 18]) or is promised as “a forthcoming publication” (fn. 1, p. 3). | Rejected — read in full 2026-09-26 (NOTE-182); not worth a reader’s time as an argument for its thesis. The step from the regress problem to formal languages is asserted, not argued, and its one physics claim (that “widespread consensus” holds spacetime to be fundamentally discrete, p. 5) is overstated. A reader wanting pregeometry is better served by the sources it cites: Meschini, Lehto & Piilonen’s “Geometry, pregeometry and beyond” ([63]) for the history, and Döring & Isham’s topos programme ([36–40]) for a worked formal-language approach. |
| LIT-108 v3 | Philosophy of Mathematics and Natural Science | Weyl (1949), DOI-10.1515/9781400833337. Philosophy of mathematics and philosophy of physics are intertwined, with advances in philosophy led by scientific discovery. Weyl’s two-part treatment (mathematics, then natural science), drawing on Descartes, Galileo, Hume, Kant, Leibniz and Newton, is a classic mathematician-physicist’s statement of that view. | Unavailable — seeded from the abstract on 2026-09-26; a close reading was attempted the same day and found the book in copyright with no open copy: every Internet Archive scan (the 1949, 1959 and 1963 printings) is lending-only, and the De Gruyter e-book is paywalled. It stays Unavailable until a copy is supplied. |
| LIT-115 v5 | Duality as a category-theoretic concept | Corfield (2015), DOI-10.1016/j.shpsb.2015.07.004. A short programmatic paper. Its thesis is that category theory, not set theory, is the framework in which the scattered instances of mathematical duality can be drawn together. The constructions in §3 (dual adjunctions restricting to dual equivalences on fixed points, truth-value-enriched Galois connections, Isbell conjugation, dualising objects “keeping summer and winter homes”) all “revolve around” adjunction, though Corfield expects no “single monolithic account” (p. 59–60). On his usage, ‘duality’ is reserved for “involution with some form of structural reversal” (p. 57), so several physicists’ dualities (homological mirror symmetry as an A∞-equivalence, mirror symmetry’s Z2 on the Hodge diamond, S-duality as a vestige of SL(2,Z)) are better called equivalences or symmetries. He proposes, without argument beyond the examples, that philosophers of physics will need homotopy type theory rather than predicate logic for the sameness questions dualities raise. | Active — read in full 2026-09-26 (NOTE-141); worth reading, briefly, as the mathematician-philosopher’s counterweight to the physics-first accounts of duality this record holds: it is the clearest statement that what physicists call a duality is often what a mathematician calls an equivalence, and it is only seven pages. Its positive programme (HoTT for philosophy of physics) is announced, not carried out. |
| LIT-122 v2 | The Representation of Time and Change in Mechanics | Belot (2005), DOI-10.1016/B978-044451560-5/50005-1. For theories with ideal existence, uniqueness and a dynamical time-translation group, time is three intertwined ℝ-actions (on spacetime, on the symplectic space of solutions, and on the isomorphic space of initial data), and change is a slicing-indexed family of functions on those spaces. Singular dynamics, gauge freedom and time-dependence each damage this picture in a repairable way. Spatially compact vacuum GR breaks it, because the reduced Hamiltonian vanishes (h ≡ 0) and there is only one canonical isomorphism between the reduced spaces. That is the problem of time. It is not an artefact of the 3+1 split, it does not show change to be illusory, and it is absent from asymptotically flat GR and from artificially generally covariant theories. A parameterized “geometric time”, such as CMC, restores change and, in the CMC case, a time-dependent Hamiltonian flow. | Active — read in full 2026-09-26 (NOTE-118); worth reading as the careful technical-philosophical statement of what the problem of time is and is not in classical GR. It is long and assumes graduate differential geometry, and it offers a diagnosis, not a solution. Its quantum-gravity motivations date from 2005 and are not updated here. |
| LIT-124 v3 | Mereology | Varzi (2003), https://plato.stanford.edu/entries/mereology/. The entry argues that no mereological principle, not even the partial-order core M (Reflexivity, Transitivity, Antisymmetry), is merely a fixing of the meaning of “part”. It lays out the ladder M ⊂ MM (+Supplementation) ⊂ EM (+Strong Supplementation) ⊂ GEM (+Unrestricted Sum). GEM is “virtually” a complete Boolean algebra with zero removed, and it is reached from M by exactly three adequate pairings: Supplementation + Unrestricted Sum2, Strong Supplementation + Unrestricted Sum3, and Complementation + Unrestricted Sum1. The influential Simons (1987) pairing, Supplementation + Unrestricted Sum3, which the entry’s own 2003 edition also used, does not yield GEM. | Active — read in full 2026-09-26 (NOTE-163); the standard, current (revised June 2026) and formally exact map of the field, and the right background reference for the record’s mereology topic. Two caveats: by its note 1 it excludes the systems-theory and ecological senses of “mereology”, and by note 18 it excludes quantum part–whole formalisms. |
| LIT-143 v2 | What is a Mathematical Structure of Conscious Experience? | Kleiner & Ludwig (2023), ARXIV-2301.11812. The paper proposes a definition, (MSC). A mathematical structure S = ((A_i), (S_j)) is a structure of conscious experience iff (S1) its domains are sets of experiential aspects, and (S2) every relation or function S_j has an “S_j-aspect”. An S_j-aspect is an aspect that a variation of experience changes exactly when the variation fails to preserve S_j. The older condition (MDC), that a structure is “of” experience if its domains are aspects and its axioms hold, survives only as the necessary condition for describing experience, because as a sufficient condition it admits incompatible structures (discrete and non-discrete topologies on the same set), arbitrary re-definitions (every C·d for C > 0) and consciousness-indifferent structures. Worked examples: experienced relative similarity is a strict partial order that satisfies (MSC), given transitivity (assumed). Visual regions form a topology that satisfies (MSC), given Bayne and Chalmers’s subsumptive unity thesis. | Active — read in full 2026-09-26 (NOTE-135); worth reading as the most explicit criterion for when a formal model is of experience rather than a convenient description of it. Its two positive results are conditional: transitivity of relative similarity is assumed, and the topology result assumes the subsumptive unity thesis. Cite the Synthese version, which has dropped v1’s argument against metric structure. |
| LIT-145 v2 | Ultrafilters as Propositional Theories | Turner (2025), DOI-10.1111/phc3.70047. An expository piece that redefines a filter over a set W as a “propositional theory” (a set of W-propositions closed under pairwise conjunction and implication, not containing ∅), proves maximal ⇔ prime ⇔ ultra (Thm 2), the ultrafilter extension theorem as a propositional Lindenbaum’s lemma (Thm 3, assuming choice), Łoś’s theorem M_T,[A] ⊨ φ iff ⟦φ⟧_A ∈ T for ultraproducts of “intensional spaces” (Thm 9), and first-order compactness from it (Thm 12); “None of the results are novel” (p. 2). | Active — read in full 2026-09-26 (NOTE-177); worth reading as the most accessible philosopher’s route into ultrafilters and Łoś’s theorem, provided the reader supplies the fixes to a few slips in the proofs (listed under Limitations). |
| LIT-154 v3 | Criteria of Identity and Structuralist Ontology | Leitgeb & Ladyman (2007), DOI-10.1093/philmat/nkm039. Graph theory counts unlabelled graphs whose nodes, or whose isomorphic connected components, cannot be told apart even by an irreflexive relation. Examples are the edgeless two-node graph G′ and any graph with two or more isolated nodes (11 of the 156 unlabelled 6-node graphs). Mathematicians treat these as determinately distinct objects, so the authors conclude that the identity or difference of places in a structure is given by the structure itself, with identity counted among its relations, and not by any criterion of identity. The argument appeals to mathematical practice; it is not a proof. | Active — read in full 2026-09-26 (NOTE-133); worth reading as the short, standard counterexample to weak-discernibility versions of the PII for mathematical structuralism. It is widely cited, and its graph cases are simple enough to check by hand. Its physics coda is explicitly left open. |
| LIT-205 v2 | One equation to rule them all: a philosophical analysis of the Price equation | Luque (2016), DOI-10.1007/s10539-016-9538-y. Luque argues that the Price equation, Δz̄ = Cov(w, z) + E(wΔz) (eq. 1, p. 7), is a Kuhnian “generalization-sketch”, like f = ma. It is not a model, has no predictive power on its own, and becomes empirical only when a specific model interprets it. On that reading the critics’ charges (van Veelen 2005; van Veelen et al. 2012; Nowak & Highfield 2011) that it is “a tautology” and “lacks predictive power” are either false or ask too much, and he recommends that evolutionary biologists adopt it as their unifying framework. | Active — read in full 2026-09-26 (NOTE-101); worth reading as a clear statement of the generalization-sketch reading of the Price equation and as a map of its extensions up to 2015 (Tables 2–3). The philosophical argument is thin in places, notably the Gödel-based case that it is “not a tautology”, so it should be read as one side of the dispute and not as a settlement of it. |
| LIT-221 v2 | Yoneda lemma | nLab (2009), https://ncatlab.org/nlab/show/Yoneda+lemma. The page states that for a locally small category C, every presheaf X and every object c, there is a canonical bijection Hom_{[C^op,Set]}(y(c), X) ≅ X(c), given by η ↦ η_c(id_c). It proves the bijection by chasing id_c around a naturality square. From it the page derives that the Yoneda embedding y: c ↦ Hom_C(−,c) is fully faithful and that y(c) ≅ y(d) ⇔ c ≅ d. It also shows by counterexample that the hypothesis doing the work is naturality: objects whose hom-sets are pointwise in bijection need not be isomorphic. | Active — read in full 2026-09-26 (NOTE-195); Worth reading as a compact, correct statement and proof of the lemma with the embedding and the iso-determination corollary. It also has a “Necessity of naturality” section, which textbooks rarely include and which any structuralist reading of the lemma needs. Two caveats: its classical statement does not state naturality in c and X (the HoTT section does), and its finite counterexample is ill-formed as written. |
| LIT-229 v2 | Meaningful Information | Vitányi (2001; IEEE Trans. Inf. Theory 2006), arXiv:cs/0111053. With total recursive functions as models, the algorithmic minimal sufficient statistic, sophistication, splits a string’s information into structure and noise. The split collapses to zero for every string if partial recursive models are allowed. Random strings have sophistication about 0, and some strings of every length are almost all structure. The finite-set, probabilistic and total-function versions agree to within logarithmic terms. Sophistication is not computable, and a sufficient-statistic oracle would compute K. | Active — read 2026-10-14 (NOTE-790) in full; worth reading as the short proof that splitting an individual string’s information into structure and noise works only in a model class without a universal member: with partial recursive models every string’s minimal sufficient statistic is trivial (Lemma 6.2), while with total recursive functions sophistication runs from about 0 for random strings to above n − log n − 2 log log n for some string of every length (Theorem 6.5), and agrees with the finite-set and probabilistic versions to within O(log n) (§VII). Read it knowing that everything holds only to within additive constants or logarithmic terms, that sophistication is uncomputable and not usefully approximable, that two steps rest on cited work (Gács–Tromp–Vitányi; Vereshchagin–Vitányi), and that the structure-function characterisation of §IX is stated, not proved. |
| LIT-230 v2 | Riesz representation theorem (Wikipedia) | Wikipedia contributors (2026), https://en.wikipedia.org/wiki/Riesz_representation_theorem. A Hilbert space is isometrically (anti-)isomorphic to its continuous dual — every continuous linear functional φ is x ↦ ⟨x, f_φ⟩ for a unique f_φ with ‖f_φ‖ = ‖φ‖ — and this identification is what defines adjoints and the bra–ket correspondence. | Active — read 2026-10-14 (NOTE-861) in full; worth reading as a careful reference statement of the Riesz theorem in both the real and the complex case, with four constructions of the representing vector and the definition of the adjoint of a bounded operator through it. Read it knowing that it is Wikipedia, that it treats bounded operators only, and that it says nothing about kernels, RKHS or spectra. |
| LIT-232 v2 | Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding | Gebhart, Hansen and Schrater (2021; AISTATS 2023), ARXIV-2110.03789. A knowledge-graph embedding is a cellular sheaf of vector spaces on the schema graph, with restriction maps per relation type, plus a 0-cochain of its pullback to the knowledge graph. The summed Structured Embedding score is the sheaf Laplacian form, and TransE, TransR and RotatE are special cases; bilinear models are not. Since the zero cochain is always a section, training needs negatives and a margin. Composite multi-hop queries are scored without extra training by harmonic extension, a Schur complement, which beats naive TransE summation on ip and pi queries in untuned experiments. | Active — read 2026-10-14 (NOTE-843) in full; worth reading as the clean dictionary between coboundary-type knowledge-graph scores (Structured Embedding, TransE, TransR, RotatE) and cellular sheaves on the schema, with the summed score as a sheaf Laplacian form and multi-hop queries scored by harmonic extension, a Schur complement. Read it knowing that bilinear models fall outside it, that the zero cochain is always a section so all content is in the contrastive margin, that “approximate global section” means only small Laplacian energy, and that the experiments are preliminary, untuned and well below query-embedding baselines. It shares the word “sheaf” with the contextuality line and nothing else of substance. The account is THEORY-299. |
| LIT-237 v2 | Bra–ket notation (Wikipedia) | Wikipedia contributors (2026), https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation. Dirac’s bra–ket notation writes vectors as kets and continuous linear functionals as bras, and on a Hilbert space the Riesz representation theorem is what licenses treating every bra as the conjugate of a ket, so that ⟨φ|ψ⟩ is at once a functional applied to a vector and an inner product. | Active — read 2026-10-14 (NOTE-855) in full; worth reading for the conventions of Dirac notation and for its plain statement that the bracket is an inner product only where the Riesz theorem holds. Read it knowing that it is Wikipedia under a cleanup banner, that its lead was edited on 2026-09-30 to date the notation to a 1930 monograph while its own reference and later text say 1939, that its pointer to GNS for non-normalizable kets is weak, and that two of its sections contradict each other on |ψ⟩† = ⟨ψ|. |
| LIT-239 v2 | Kolmogorov’s Structure Functions and Model Selection | Vereshchagin & Vitányi (2002), arXiv:cs/0204037. For a string x of length n and every complexity bound α, Kolmogorov’s structure function h_x(α) + α, the minimal two-part code length λ_x(α) and the best achievable randomness deficiency β_x(α) + K(x) are one function to within O(log n). A set minimising code length at α is a best-fitting set there, though not conversely. Every non-increasing shape occurs for some x. h_x and λ_x can be approximated from above but not computed; β_x cannot be approximated at all. Finite-set models, extended to computable distributions at O(log n) cost. | Active — read 2026-10-14 (NOTE-787) in full; worth reading as the theorem that, for a single string and every bound on model complexity, the set with the shortest two-part code is within O(log n) of the best-fitting set in randomness deficiency (Theorem IV.8), with the complete catalogue of shapes the structure functions can take and the proof that fit cannot be approximated while code length can. Read it knowing that every result holds only to O(log n), that the models are complexity-bounded finite sets or computable distributions and not statistical families, which the authors say the results do not transfer to directly, and that the construction proving Theorem D.2 was not checked line by line here. |
| LIT-241 v2 | Gelfand–Naimark–Segal construction (Wikipedia) | Wikipedia contributors (2026), https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction. Any state (positive normalized linear functional) on a C*-algebra determines, essentially uniquely, a Hilbert space and a cyclic *-representation in which that state becomes a vector expectation ρ(a)=⟨π(a)ξ,ξ⟩, and the representation is irreducible exactly when the state is pure. | Active — read 2026-10-14 (NOTE-857) in full; worth reading as a short reference statement of the GNS construction: a state on a C*-algebra gives a cyclic representation, unique up to the unitary π(a)ξ ↦ π′(a)ξ′, irreducible exactly when the state is pure. Read it knowing that it is Wikipedia, that the uniqueness is stated with a reference to Kadison rather than proved, and that it is about states on complex C*-algebras, not kernels on sets: the kernel form the record uses comes from LIT-259. |
| LIT-243 v2 | Hilbert Spaces and the Riesz Representation Theorem | Adler (2021), https://math.uchicago.edu/~may/REU2021/REUPapers/Adler.pdf. Every continuous linear functional φ on a Hilbert space is v ↦ ⟨u, v⟩ for a unique u with ‖u‖ = ‖φ‖, and in any orthonormal basis u = Σ_k conj φ(e_k) e_k with ‖φ‖² = Σ_k |φ(e_k)|². Applied to f ↦ ∫ f dν on L²(ν + μ), it yields the Radon–Nikodym theorem for finite measures. | Active — read 2026-10-14 (NOTE-854) in full; worth reading as a gentle, complete proof of the Riesz theorem from first principles, done twice (by projection, then by an orthonormal basis), and of the Radon–Nikodym theorem from it. Read it knowing that it is student expository work with typographical slips, that its inner product is conjugate-linear in the first slot, that the representer is Σ conj φ(e_k) e_k, and that Radon–Nikodym is proved for finite measures only. |
| LIT-250 v3 | Duality of Bures and Shape Distances with Implications for Comparing Neural Representations | Harvey et al. (2023), ARXIV-2311.11436. The Procrustes shape distance between two representations (optimal rotational alignment of units) equals the Bures distance between their centred linear kernel matrices, so the rotation-invariant geometry of a representation is exactly the geometry of its kernel, and through Uhlmann’s theorem the fidelity of two kernels is the maximum overlap over all representations consistent with them. | Active — read 2026-10-14 (NOTE-753) in full; worth reading as the clean finite-dimensional statement that, for centred linear kernels, the Procrustes distance equals the Bures distance and normalized Bures similarity is the cosine of the Riemannian shape distance, so the kernel is a complete invariant of a representation up to translation and orthogonal maps, and as the proof that CKA and NBS are related only within a rank-dependent envelope. Read it knowing that it covers linear kernels only, assumes the orthogonal group rather than arguing for it, and treats the infinite limits by a law-of-large-numbers sketch and a citation; it does not reach an RKHS or GNS statement. |
| LIT-252 v2 | Spectral Inference Networks: Unifying Deep and Spectral Learning | Pfau et al. (2018), ARXIV-1806.02215. For a symmetric kernel k on a space with sampling density p, the span of the top-N eigenfunctions of Kf = E_x′[k(x,x′)f(x′)] on L²(p) maximizes the generalized Rayleigh quotient Tr(Σ⁻¹Π), with Σ = E[u uᵀ] and Π = E[k(x,x′)u(x)u(x′)ᵀ]; a Cholesky-masked gradient orders the eigenfunctions, argued informally, and moving averages of Σ and its Jacobian correct the minibatch bias. Slow Feature Analysis is the graph-Laplacian case on adjacent frames, where the slowest features are the bottom of the Laplacian’s spectrum. | Active — read 2026-10-14 (NOTE-824) in full; worth reading as the generic neural solver for eigenfunctions of a symmetric kernel operator on L²(p): the top-N span maximizes Tr(Σ⁻¹Π), a Cholesky-masked gradient orders the eigenfunctions, and moving averages of Σ and its Jacobian remove the minibatch bias. Slow Feature Analysis is the graph-Laplacian case. Read it knowing that symmetry of k is a precondition placed on the kernel, not derived from a pair distribution, that the ordering argument is informal by the authors’ own statement, that only the hydrogen-atom experiment has ground truth, and that it never mentions positive pairs, contrastive learning or the density ratio. |
| LIT-255 v2 | Information Theory with Kernel Methods | Bach (2022), ARXIV-2202.08545. With a kernel normalised to k(x,x) = 1, the uncentred covariance operator Σ_p = E_p[ϕ(x)ϕ(x)*] is a density operator, injective in p when k² is universal. Its relative entropy satisfies D(p̃‖q̃) ≤ D(Σ_p‖Σ_q) ≤ D(p‖q), with p̃ a kernel-smoothed p, and is within O(σ²) of KL on the torus. The empirical entropy is that of the normalised Gram matrix K/n. Kernel mutual information characterizes independence, and the dual gives upper bounds on log-partition functions. | Active — read 2026-10-14 (NOTE-837) in full; worth reading as the rigorous construction of the “density matrix” reading of kernel embeddings: with k(x,x) = 1 the uncentred covariance operator is a quantum state, its relative entropy lies between the KL of kernel-smoothed distributions and the KL itself, the empirical version is the matrix entropy of K/n, and kernel mutual information characterizes independence. Read it knowing that the kernel quantities only lower-bound Shannon ones and approach them at O(σ²) only on the torus, that conditional independence is characterized one way only, that Σ_p is uncentred, and that the paper never mentions fidelity, Bures distance, representation similarity or GNS. |
| LIT-259 v2 | Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations | Jorgensen & Tian (2014), ARXIV-1408.1164. The textbook calls building a Hilbert space from a positive definite function “the GNS construction” (Remark 1.34) and states (Cor. 1.35), leaving the proof as an exercise, that the feature map realising a kernel as inner products is unique up to a unitary among minimal realisations. It proves the parallel uniqueness for states and cyclic representations (Thm 4.35), says that in the abelian case states are probability measures, so that “the GNS construction is non-commutative measure theory” (Ex. 4.28), and writes the commutative GNS space out as L²(μ) (Ex. 4.86). | Active — read 2026-10-14 (NOTE-829) in part; worth reading for the passages that put the kernel construction and the GNS construction under one name: Remark 1.34 builds a Hilbert space from a positive definite function by the GNS recipe, Corollary 1.35 states that minimal realizations ϕ(x,y) = ⟨Φ(x), Φ(y)⟩ are unique up to a unitary, Theorem 4.35 proves the GNS theorem with its uniqueness, and Example 4.86 writes the commutative GNS space out as L²(μ). Read it knowing that Corollary 1.35 is not proved (the proof is “left as an exercise”; the argument of Theorem 4.35 transfers), that equation (1.24) carries a typo, that “GNS” for a kernel on a set is a naming and not a stated reduction, and that only the cited passages of this 442-page text were read. |
| LIT-260 v2 | Kernel Mean Embedding of Distributions: A Review and Beyond | Muandet et al. (2016), ARXIV-1605.09522. Point evaluation is bounded on an RKHS, so the Riesz theorem gives the reproducing kernel k_x. Likewise, when E√k(X,X) < ∞ the expectation functional f ↦ E_P f is bounded, so Riesz gives a unique mean embedding μ_P with E_P f = ⟨f, μ_P⟩. MMD is then the RKHS distance ‖μ_P − μ_Q‖, and it separates distributions exactly when the kernel is characteristic. | Active — read 2026-10-14 (NOTE-838) in full; worth reading as the one held text that runs the chain bounded evaluation → Riesz → k_x → Moore–Aronszajn → μ_P → MMD/HSIC in numbered statements, with Lemma 3.1 (μ_P exists and represents expectation when E_P√k(X,X) < ∞) proved in the text, and as a map of conditional mean embeddings and their limits. Read it knowing that it is a survey whose proofs are mostly sketched or cited, that every result assumes a fixed kernel, that it defines μ_P as a Bochner integral as well as by Riesz, so Riesz is one route to existence and not shown to be the only one, and that the unregularized conditional embedding C_YX C_XX⁻¹ often fails to exist. |
| LIT-261 v2 | A Generalized Representer Theorem | Schölkopf et al. (2001), DOI-10.1007/3-540-44581-1_27. For any nonempty set X, real positive-definite kernel k, strictly increasing g and any cost c on the training triples, even coupled, non-convex or infinite-valued, every minimizer of c + g(‖f‖) over the countable kernel expansions of finite RKHS norm is a finite expansion Σ α_i k(·,x_i) at the training points. The proof needs only the reproducing property and orthogonal decomposition onto span{k(·,x_i)}, not the Riesz theorem. Existence of a minimizer is assumed. | Active — read 2026-10-14 (NOTE-836) in full; worth reading as the short proof that any minimizer of an arbitrary cost on the training outputs plus a strictly increasing function of the RKHS norm is a kernel expansion on the training points, by the reproducing property and orthogonal decomposition, with no convexity and no Riesz theorem. Read it knowing that the theorem is stated over countable kernel expansions of finite norm rather than the completed RKHS, that it assumes a minimizer exists, and that its semiparametric Theorem 2 is stated without proof. |
| LIT-262 v2 | The Geometry of Information Retrieval | van Rijsbergen (2004), DOI-10.1017/CBO9780511543333. The book recasts IR’s vector-space, probabilistic and logical models in one Hilbert-space language. Documents are state vectors. Relevance and aboutness are Hermitian observables that need not commute. Subspaces carry a non-distributive logic whose conditional is shown to be a Stalnaker conditional equal to the Sasaki hook (ch. 5). Gleason’s theorem makes tr(ρP) the probability of a subspace, which ch. 6 uses to rewrite cosine matching, cluster representatives, relevance and pseudo-relevance feedback, dynamic clustering and ostensive retrieval. It is a proposed language, not a tested model: no experiment is reported, and its “content hypothesis” that the probability of a concept is cos²θ is argued, not tested. | Active — read in full 2026-09-27 (NOTE-239); worth reading as the founding statement of the Hilbert-space (quantum-probability) programme in IR. It is short, self-contained and mathematically careful at the level it works at, and its ch. 5 proof that the subspace conditional is a Stalnaker conditional and its ch. 6 trace-formula recastings of standard IR operations are the parts to take away. Read it as a language, not as evidence: it reports no experiment, and every IR payoff it names is a reformulation or a research direction. |
| LIT-265 v2 | The contextual fraction as a measure of contextuality | Abramsky, Barbosa & Mansfield (2017), Phys. Rev. Lett. 119, 050504. For any no-signalling empirical model in the sheaf-theoretic framework, the contextual fraction CF (one minus the largest weight of a noncontextual sub-model) is a linear programme. Its dual gives a generalised Bell inequality whose normalised violation equals CF, the largest attainable (Theorem 1). CF does not increase under relabelling, translation of measurements, coarse-graining or mixing, and is multiplicative in NCF under products (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC computing a non-linear function (Theorem 3). | Active — read 2026-10-09 (NOTE-236); worth reading as the reference definition of the contextual fraction CF. CF is the weight left over when the largest noncontextual sub-model is taken out of a no-signalling empirical model. It is a linear programme over the incidence matrix (Eq. 3). Its dual reads off a Bell inequality whose normalised violation is exactly CF, the maximum over all Bell inequalities (Theorem 1). CF is invariant under relabelling, cannot increase under translation of measurements, coarse-graining of outcomes or mixing, and composes as max under choice and as 1 − NCF·NCF′ under product (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC and of k-consistent games (Theorems 3–4). All proofs are complete in the supplement. It is defined only for models with compatible marginals. |
| LIT-267 v3 | The Lattice Representation Hypothesis of Large Language Models | Xiong (2026), ARXIV-2603.01227. Thresholding linear attribute directions (Fisher-LDA probes) gives a crisp object–attribute incidence, and the paper’s Theorem 1 is the standard fact that any such incidence has a complete concept lattice. The geometric meet is half-space intersection; the join is defined as the set union of two cones and “approximated by the conic hull” of their directions, which as written is not a lattice join. On five WordNet domains with GPT-4o-annotated attributes, the probes recover the incidence at F1 69.7–83.2 on three 7–8B models, and profile-based subsumption reaches F1 57.1–77.1. No lattice law is ever tested. | Proposed — read in full 2026-09-27 (NOTE-240); promising as a bridge between probe geometry and formal concept analysis, and unproven as a hypothesis about LLMs. It would be settled by three things. The attributes would have to come from something other than another LLM’s annotations. The region-level join would have to be defined correctly, as the Galois closure of the shared intent. And lattice identities (closure, absorption, meet/join agreement with the recovered context) would have to be tested, not pairwise subsumption and candidate ranking alone. |
| LIT-269 v2 | Attention Approximates Sparse Distributed Memory | Bricken & Pehlevan (2021), ARXIV-2111.05498. Kanerva’s SDM read weights each stored pointer by the number of neurons in the intersection of two Hamming balls of radius d. The paper shows that this weight is roughly log-linear in Hamming distance for close patterns. As a result, with L²-normalised vectors and a regression-fitted β, softmax attention reproduces SDM’s retrieval behaviour on random and learnt-projection data (App. B.7). Two things are weaker than they look. The analytic derivation (App. B.2, Eqs. 16–21) gets the exponent wrong: by my recomputation its β is ≈2–2.5× too small. The biological case is Kanerva’s 1988 cerebellar mapping, restated rather than re-argued. It also needs r = 2⁶⁴ ≈ 1.8×10¹⁹ neurons in the Attention setting, which the paper itself calls “biologically implausible” (App. B.5). | Active — read in full 2026-09-27 (NOTE-242); worth reading as the clearest statement of the SDM–attention correspondence. Its exact circle-intersection formula and its eight-algorithm convergence comparison are the parts to take away. Treat its analytic exponential derivation, its GPT-2 “confirmation” and its cerebellar implementation of attention as argument, not demonstration. |
| LIT-271 v2 | Neural Tangent Kernel of Matrix Product States: Convergence and Applications | Guo & Draper (2021), ARXIV-2111.14046. In the sequential infinite-bond-dimension limit, with tensor variances σ_i²/√(|α_i||α_{i+1}|) and per-tensor learning rates (|α_i||α_{i+1}|)^{-1/2}, the NTK of a periodic MPS tends to K(x,x′)=Σ_k φ(x_k)·φ(x′k) Π{l≠k} σ_l² φ(x_l)·φ(x′_l). By my reduction this is (Σ_k σ_k⁻²) times the MPS’s own GP covariance, so gradient flow is kernel regression with a fixed product kernel. The Born-machine solution P_x(t)=1/m−(1/m−P_x(0))e^{−4mKt/Z} is correct (I checked it; Z is in fact conserved). The lazy-training lemma is heuristic, the positive-definiteness “proof” shows at most semi-definiteness, several constants are off by factors of n or 2^{−n}, and there are no experiments. | Rejected — read in full 2026-09-27 (NOTE-244); not worth a reader’s time as a source. Its one sound calculation (the infinite-bond MPS is a linear model in a fixed product-kernel feature space, so its NTK is its GP kernel up to a constant) follows in a line from multilinearity. Tensor Programs II (ANTH-LIT-557) covers such architectures rigorously. The paper’s distinctive claims are the lazy-training proof, positive-definiteness “without any extra assumptions of the data set”, and “over-fitting is naturally avoided”. The first is circular, the second is false for the finite local dimensions it uses, and the third is contradicted by its own memorisation result. |
| LIT-272 v2 | Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras | Kartsaklis et al. (2014), ARXIV-1401.5980. The paper sets the sentence space equal to the noun space (F(s)=F(n)=W). It then uses the basis-copying map σ: v_i ↦ v_i⊗v_i of the Frobenius algebra fixed by W’s basis to lift corpus-built verb matrices Σ_i sbj_i⊗obj_i into rank-3 tensors. So a transitive sentence becomes sbj⊙(verb·obj) (“copy-subject”) or obj⊙(verbᵀ·sbj) (“copy-object”), a vector in W. On the Grefenstette–Sadrzadeh verb-disambiguation set, copy-object gets Spearman ρ=0.172, against 0.168 (Kron), 0.163 (Multp), 0.143 (copy-subject), 0.050 (additive) and an inter-annotator bound of 0.620. On a 112-term definition-matching task the best F1 is 0.21 (nouns) and 0.27 (verbs). No significance test is reported. The disambiguation numbers are identical to the authors’ COLING 2012 paper, and the definition numbers are the same data re-reported. | Active — read in full 2026-09-27 (NOTE-246); worth reading for the construction, not for its evidence. It is the citable categorical statement of the Frobenius lift of DisCoCat verbs: the functorial recasting, S=N=W, and the copy-subject and copy-object maps. Its experiments are the authors’ COLING 2012 numbers, re-reported without that paper’s caveat that copy-object’s lead is statistically insignificant. The best disambiguation ρ is 0.172, against a human bound of 0.620. The construction depends on a choice of basis, which the paper cites but never draws the consequence of. |
| LIT-273 v2 | Mathematical Foundations for a Compositional Distributional Model of Meaning | Coecke et al. (2010), ARXIV-1003.4394. The paper builds DisCoCat as a product category FVect × P of vector spaces and a free pregroup. A sentence’s meaning is f(w₁⊗…⊗wₙ), where f is the linear map got by putting vector spaces in place of the pregroup types in the reduction p₁…pₙ ≤ s. The ε maps become inner products and the η maps become Σᵢ eᵢ⊗eᵢ. So “John likes Mary” is Σ_ijk c_ijk⟨v|vᵢ⟩⟨w_k|w⟩ sⱼ ∈ S, and “not” is the 2×2 swap matrix placed on the sentence wire. There is no theorem and no experiment. The sentence space S is shown only as 1- or 2-dimensional truth values, over a toy model with one basis vector per individual, and the paper says how neither S nor the verb tensors are to be built from data. Its headline similarity numbers (3/4, 1/4, 3/8) are unnormalised inner products. Under its own Definition 5.1 they are 0.949, 0.316 and 0.6. | Active — read in full 2026-09-27 (NOTE-245); worth reading as the founding statement of DisCoCat, whose ε/η composition, S-for-every-sentence design and “not”-as-swap every later paper in the programme starts from. Read it as a specification, not a result. It proves nothing, runs no experiment, leaves S and the verb tensors unbuilt, and its categorical framing (a product with a posetal pregroup) cannot tell apart the parse ambiguities it says it can reason about. |
| LIT-274 v2 | A new description of orthogonal bases | Coecke et al. (2008), ARXIV-0810.0812. Theorem 5.1 proves that on a finite-dimensional complex Hilbert space, commutative †-Frobenius monoids in FdHilb and orthogonal bases are in bijection: the basis is the monoid’s copyable elements, and the monoid is the linear extension of copying and uniformly deleting that basis. §6 adds that the monoid is special (m∘δ = id) exactly when the basis is orthonormal. Corollaries 7.1 and 7.2 lift this to categories: with fully structure-preserving maps the category is equivalent to the groupoid of finite sets labelled by positive reals (the basis norms), and with comonoid homomorphisms it is equivalent to FinSet. | Active — read in full 2026-09-27 (NOTE-247); worth reading as the short, self-contained source of the fact that “a commutative †-Frobenius algebra” and “an orthogonal basis” are the same data in FdHilb, and “special” the same as “orthonormal”. The proof is complete in outline, and the heavy step is delegated to the spectral theorem for finite-dimensional commutative C*-algebras. Read §3’s displayed formulas (3) and (4) as the orthonormal case only, and the result as complex and finite-dimensional only. |
| LIT-275 v2 | H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics | Abramsky & Heunen (2010), ARXIV-1011.6123. In the category Hilb of complex Hilbert spaces of arbitrary dimension, a Frobenius algebra with a unit exists only in finite dimension (Lemma 3). So the paper drops the unit, and Theorem 22 proves that a nonunital commutative special †-Frobenius algebra corresponds to an orthonormal basis (the basis being the algebra’s copyable vectors) if and only if it is semisimple, if and only if it satisfies Ambrose’s H*-axiom (H), and if and only if it is a directed colimit of finite-dimensional unital ones. Whether (H) is automatic, i.e. whether a nonzero “radical” Frobenius algebra exists (Proposition 23), is left open. In categories of relations and of nonnegative ℓ²-matrices, Frobenius algebras decompose as disjoint unions of abelian groups and do satisfy (H) (Theorems 30–38). | Active — read in full 2026-09-27 (NOTE-248); worth reading as the precise statement of how far “a basis is a Frobenius algebra” survives in infinite dimensions. It gives the unit’s failure, the H*-algebra substitute and five equivalent conditions, all proved or reduced to Ambrose (1945), with one question left open. Read “arbitrary bases” in the abstract as “orthonormal bases of arbitrary dimension over ℂ”, and “observables” as discrete-spectrum observables only. |
| LIT-276 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | Ghose (2025), ARXIV-2512.12249. A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Rejected — read in full 2026-09-27 (NOTE-249); not worth a reader’s time for its formal claims. Every correct formal ingredient is Abramsky & Brandenburger’s (LIT-016) or Abramsky, Mansfield & Barbosa’s, and both are better read in the original. What the essay adds is either unconstructed (the sheafification, the site, the seven-element Heyting algebra, the link from λ to cohomology) or wrong as stated: v2’s obstruction class is zero by construction, a sheaf topos is not in general Boolean, and classical distributions do not form a sheaf. Its only value is as a record of a position, the Bohrian reading of contextuality plus the Jaina analogy. |
| LIT-277 v2 | The Cohomology of Non-Locality and Contextuality | Abramsky et al. (2011), ARXIV-1111.3620. For an empirical model e on a measurement cover 𝒰, a section s in the support of context C₁ gets a class γ(s) ∈ Ȟ¹(𝒰, F_{C̄₁}), relative Čech cohomology of the presheaf F = F_ℤ S_e of formal ℤ-linear combinations of support sections. γ(s) = 0 iff s belongs to a compatible family of such ℤ-combinations (Prop 4.2), so a non-vanishing γ(s) is a sufficient but not necessary witness of contextuality (Prop 4.3). γ is non-zero for every section of the PR box, GHZ, the triangle, the 18-vector Kochen–Specker set and the Peres–Mermin square, but it vanishes on the Hardy model and on one strongly contextual Kochen–Specker cover (§8). | Active — read in full 2026-09-27 (NOTE-250); worth reading as the short original source of the cohomological obstruction to contextuality. It gives the exact definition (a relative Čech class, with coefficients the free abelian group on support sections), the vanishing criterion and its two known failures. Its results are small propositions with complete proofs, plus machine-checked mod-2 computations that I reproduced. It calls itself “work in progress” (§1), and the conceptual development is in Abramsky et al.’s CSL 2015 “Contextuality, Cohomology and Paradox” (not read). A reading of that paper would show whether it supersedes this one. |
| LIT-278 v2 | Contextuality, Cohomology and Paradox | Abramsky et al. (2015), ARXIV-1502.03097. For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Active — read in full 2026-09-27 (NOTE-251); worth reading as the place where the cohomological witness is shown complete for a named, large class (every All-vs-Nothing model, over any ring). It is also where the obstruction is defined as a connecting homomorphism. It does not supersede LIT-277. It re-derives none of LIT-277’s computations, and it drops its negative results: the Hardy false positive and the strongly contextual cover with γ = 0. Its “paradox” title-word is supported by about one page of analogy and one worked identification. |
| LIT-279 v2 | On the Cohomology of Contextuality | Carù (2017), ARXIV-1701.00656. Carù gives a possibilistic model on the two-party, two-setting, four-outcome Bell cover that is strongly contextual, yet whose Čech obstruction vanishes on all 22 support sections (Fig 4, Table 1). That refutes Abramsky–Mansfield–Barbosa’s Conjecture 8.1 for any symmetry or connectedness condition the Bell cover meets. I confirmed the refutation over ℤ and ℤ/2. The rest is weaker. Odd-degree “higher obstructions” form a hierarchy (Prop 5.5), but every level above the first is empty for every no-signalling model (Prop 5.6). Prop 6.3 restates the known torsor description of Ȟ¹. Prop 6.1 is false in the “only if” direction; the PR box refutes it. | Active — read in full 2026-09-27 (NOTE-252); worth reading for one result, the first complete false negative of the cohomological witness (§4 and Appendix A, about two pages), which is correct as I verified it. §5 proves its own programme empty for no-signalling models, and §6 is a standard correspondence plus one false proposition. The follow-up, arXiv 1807.04203, may supersede it; that is unverified, since it was not read. |
| LIT-280 v2 | Towards a complete cohomology invariant for non-locality and contextuality | Carù (2018), ARXIV-1807.04203. For cyclic scenarios, where the contexts’ intersection graph is a single chordless N-cycle, Carù proves that ℤ/2 Čech cohomology of the (N−1)-th iterated “joint model” is a complete invariant for logical and strong contextuality (Thm 7.7). I confirmed this on the Hardy model, on his own 2017 counterexample (detected on all 22 sections at level 3) and on 659 random contextual cycle models. The extension beyond cycles does not hold as stated. Prop 8.2 is false on the paper’s own Table 5 model. Worse, the one non-cyclic case the paper claims to repair, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa, is never detected at any level under the paper’s definitions (my proof and computation). That refutes the paper’s closing Conjecture 9.1. | Active — read in full 2026-09-27 (NOTE-253); worth reading for §§4–7, a correct and genuinely new result: a complete cohomological test for contextuality on cyclic scenarios, which closes the Hardy and Carù-2017 gaps. Read it knowing that §8’s generalisation, its §8.1 Kochen–Specker example and Conjecture 9.1 are wrong as stated. |
| LIT-281 v2 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | Carù (2019), DOI-10.5287/ora-kznbp4k4y. The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Active — read in full 2026-09-27 (NOTE-254); worth reading as the consolidated source for Carù’s cohomology programme (Chs III–IV), for the valuation-algebra reformulation of contextuality (Ch V) and for the AvN triple theorem (Ch VI). Chs III–IV repeat, uncorrected, the errors recorded for LIT-279 and LIT-280, so read them with NOTE-252 and NOTE-253 at hand. |
| LIT-285 v2 | The Impossibility of a Paretian Liberal | Sen (1970), DOI-10.1086/259614. Sen proves that no social decision function can satisfy three conditions at once: unrestricted domain (U), the weak Pareto principle (P), and “minimal liberalism” (L*). A social decision function only has to yield a best element in every subset, and need not be transitive or satisfy independence. L* says at least two individuals are each decisive, both ways, over at least one pair of alternatives. I checked the two-case proof and it is correct: it builds a strict social cycle over 3 or 4 alternatives. The Lady Chatterley’s Lover example is a correct 3-alternative instance. Sen’s moral is that “in a very basic sense liberal values conflict with the Pareto principle” (p. 157). | Active — read in full 2026-09-28 (NOTE-255); worth reading as the original, six-page statement of the liberal paradox, with a complete and correct proof. Read it knowing two things. Its formal Condition L gives decisiveness over arbitrary pairs, not over a protected personal sphere, which the later rights-as-game-forms literature disputes. And Theorem I as stated tacitly needs at least two individuals. |
| LIT-299 v2 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Döring & Isham (2008), DOI-10.1063/1.2883777. Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Active — read in full 2026-09-29 (NOTE-290); worth reading as the source of the quantity-value object ℝ^⪰ and of the explicit statement that Gel’fand points of a context are ultrafilters of its projection lattice. Its results are small, mostly proved or cited from de Groote and Döring. The physically motivated parts (k(ℝ^⪰), “intrinsic dispersion”, unitaries) are definitions with no consequences drawn. |
| LIT-300 v2 | A closed set of normal orthogonal functions | Walsh (1923), DOI-10.2307/2387224. The paper defines a closed orthonormal set {φ} on (0, 1) of functions taking only the values ±1, recursively by (2) and explicitly by (3)–(4). In binary x = Σ a_i/2^i, the functions are signed products (−1)^{a_{i₁}+…+a_{i_r}}; the nth has exactly n sign changes. It proves Fourier-like properties: uniform convergence for continuous F with the terms grouped in dyadic blocks (Thm II); pointwise convergence to ½[F(a−0)+F(a+0)] for bounded variation (Thm IV); localisation (Thm V); uniform Cesàro summability for continuous F (Thm VII); a continuous function with a divergent φ-series at any chosen point (Thm VIII); and uniqueness of expansions (Thms IX–X). It never mentions groups, characters or (ℤ/2)ⁿ. | Active — read in full 2026-09-29 (NOTE-281); worth reading as the primary source of the Walsh system, and short. For Walsh functions as characters of the Cantor group / Boolean cube and the Walsh–Hadamard/Boolean-Fourier theory the map needs, it is superseded by later work (Paley 1932 for the Rademacher-product ordering; Fine 1949 for the dyadic-group view; O’Donnell, LIT-346, for Boolean Fourier analysis). |
| LIT-304 v2 | The Linear Representation Hypothesis and the Geometry of Large Language Models | Park et al. (2023), ARXIV-2311.03658. Three readings of “concepts are directions” are made precise with counterfactual pairs, and proved to line up. The shared direction of word-pair differences in the unembedding space is a logit-linear probe (Thm 2.2). The embedding-space direction is a steering vector that leaves causally separable concepts unchanged (Thm 2.5). Under any inner product making separable concepts orthogonal, the Riesz map sends the first to the second (Thm 3.2). Softmax training fixes the representation only up to γ ↦ Aγ + β, λ ↦ A⁻ᵀλ, so no inner product is identified, and the paper chooses M = Cov(γ)⁻¹ by setting a free diagonal D to the identity (Thm 3.4). Evidence: 27 concepts on LLaMA-2 7B (26 show a common direction; thing⇒part does not), read from histograms and heatmaps with no test statistic, plus a Gemma-2B heatmap comparison. | Active — read in full 2026-09-29 (NOTE-302); worth reading as the short, precise source of the counterfactual-pair formalisation of the linear representation hypothesis and of the observation that softmax training leaves the representation’s inner product unidentified. Read its theorems as statements about the final-layer context vector and the unembedding only. Read the choice M = Cov(γ)⁻¹ as a stipulation (D = I), not as a result, and the experiments as qualitative. |
| LIT-305 v2 | On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups | Kondor & Trivedi (2018), ARXIV-1802.03690. Theorem 1 says that a feed-forward network whose layers are linear maps followed by pointwise nonlinearities, with every layer indexed by a homogeneous space G/H_ℓ of a compact group G, is G-equivariant (every layer transforming by the induced action) if and only if each linear map is a generalised convolution f ∗ χ with a filter on the double coset space H_{ℓ−1}\G/H_ℓ. The “if” direction is a short coset computation. The “only if” direction goes through the group Fourier transform: equivariant maps preserve isotypic components (Lemma 6) and act on each Fourier matrix by right multiplication, M ↦ MB (Lemma 10), which the convolution theorem (Prop. 2) turns back into a convolution. | Active — read in full 2026-09-29 (NOTE-282); worth reading as the standard ML statement of “equivariance ⇔ convolution” for compact groups and as a compact introduction to isotypic decomposition and noncommutative Fourier analysis for neural networks. Read the “only if” proof as an outline. It is written out for finite or countable groups and scalar channels only, and one step (removing the pointwise nonlinearity) does not hold as stated for ReLU. |
| LIT-306 v2 | The logic of quantum mechanics | Birkhoff & Neumann (1936), DOI-10.2307/1968621. The paper argues heuristically that the experimental propositions of a quantum system correspond to closed linear subspaces of Hilbert space. Meet is intersection, join is closed linear span, and negation is orthogonal complement. The lattice satisfies L1–L4 and complementation L71–L73 but not the distributive law L6. The paper proposes the modular law L5 as its replacement. Under a finite-dimension assumption, it concludes that the propositional calculus of QM is an abstract projective geometry. The Appendix proves that orthocomplementations on P_{n−1}(F), n ≥ 4, are exactly the polarities of definite diagonal Hermitian forms with respect to an involutory anti-automorphism of F. | Active — read in full 2026-09-29 (NOTE-275); worth reading as the founding text of quantum logic, and for its proposed axiom, which is modularity, not orthomodularity. For the orthomodular Hilbert-lattice machinery that is used today (infinite dimensions, Piron, Solèr), later work supersedes it. The paper itself calls its main conclusions “admittedly heuristic”. |
| LIT-309 v2 | On hypercomplex numbers | Wedderburn (1908), DOI-10.1112/plms/s2-6.1.77. For associative algebras with a finite basis over an arbitrary field F, the paper proves the structure theory. A semi-simple algebra (no nilpotent invariant sub-algebra) is uniquely the direct sum of simple algebras (Thms 10, 17). Every simple algebra is the direct product of a primitive (division) algebra and a simple matric algebra, i.e. a full n×n matrix algebra (Thms 21–23). The maximal nilpotent invariant sub-algebra N contains all others (Thm 13), and A/N is semi-simple. When every non-invertible element is nilpotent, A = B + N with B primitive (Thm 28, with a proof correction added 1 Feb 1908). Section 8 lists the results as (i)–(vi). | Active — read in full 2026-09-29 (NOTE-280); worth reading as the primary source of the Wedderburn structure theorem over an arbitrary field, and of the matrix-unit argument (Thm 21) still used today. The notation of “complexes” and ”<” is dated. For the modern ring-theoretic form (Artinian semisimple rings ≅ Π M_{n_i}(D_i)), Artin 1927 (LIT-331, unread) or any algebra text supersedes it. |
| LIT-310 v2 | A topos foundation for theories of physics: IV. Categories of systems | Döring & Isham (2008), DOI-10.1063/1.2883826. Proposes axioms for a category Sys of physical systems, symmetric-monoidal under composite ⋄ and under disjoint sum ⊔, whose arrows are translations of local languages. Each arrow j : S₁ → S is realised by a left-exact functor τ(j) between the systems’ topoi, together with arrows φ(j) : Σ_{S₁} → τ(j)Σ_S and β(j) : τ(j)R_S → R_{S₁}. Classical physics satisfies the scheme exactly. In quantum theory the disjoint sum H₁⊕H₂ satisfies it (the pulled-back arrow of Â₁⊕Â₂ is δ̆(A₁)), but the composite H₁⊗H₂ does not: the natural pull-back of δ̆(A₁) differs from δ̆(A₁⊗1̂) off the image of V(H₁), which the authors attribute to “operator entanglement”. | Proposed — read in full 2026-09-29 (NOTE-285); promising and unproven. The rules are explicitly “experimental”, the quantum composite fails the commutativity condition the rules state, and the key negative claims ((4.46), and that no operator B̂ gives the pulled-back arrow) are asserted, not shown. What would settle it is a proof of (4.46) and of the non-existence of B̂, plus a variant of the rules (or of the context category) under which tensor-product composites satisfy (3.18). |
| LIT-311 v3 | Mathematical Methods of Statistics | Cramér (1946), DOI-10.1515/9781400883868. Per publisher description: a foundational treatise building mathematical statistics on measure-theoretic probability; it contains Cramér’s independent derivation of the variance lower bound for estimators (the Cramér half of Cramér–Rao). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-313 v2 | On the imbedding of normed rings into the ring of operators in Hilbert space | Gelfand & Neumark (1943), DOI-10.1090/conm/167/16. Theorem 1 is the first abstract characterisation of C*-algebras. Every normed -ring (a unital Banach -algebra with ‖xx‖ = ‖x‖·‖x‖, ‖x*‖ = ‖x‖, and e + x*x invertible) is isometrically *-isomorphic to a norm-closed -subalgebra of the bounded operators on some, not necessarily separable, Hilbert space. The proof builds, for each maximal left ideal M, a positive functional f vanishing on M. The inner product is (ξ, η) = f(yx) on R/M, and the construction takes the direct sum over all M. This is the GNS construction. Lemma 1 is the commutative case: such an algebra is isometrically *-isomorphic to C(𝔐) for its compact maximal-ideal space. §4 adds that the simple weakly closed rings on separable Hilbert space are exactly the factors of types I_n, II₁ and III (Theorem 3). | Active — read in full 2026-09-29 (NOTE-303); worth reading as the primary source of both Gelfand–Naimark theorems, commutative (Lemma 1) and non-commutative (Theorem 1), and of the GNS construction in embryo. It is compact and readable. Modern treatments state it with the single C*-identity. The authors themselves note that it suffices, and later work showed their extra axioms 5′ and 6′ redundant (not verified here). |
| LIT-314 v2 | Group Equivariant Convolutional Networks | Cohen & Welling (2016), ARXIV-1602.07576. Group convolution (eqs. 10–11) replaces translation by the elements of a discrete group G, so feature maps become functions on G and every layer is G-equivariant (eq. 12, with pooling and nonlinearities shown to commute with the action). With parameter counts held roughly fixed, p4-CNNs cut rotated-MNIST test error from 5.03% (planar baseline) to 2.28% (previous best 3.98%), and p4m convolutions cut plain-CIFAR10 error from 9.45% to 6.46% on a ResNet44 (Tables 1–2). | Active — read in full 2026-09-29 (NOTE-288); worth reading as the short, self-contained origin of the group-convolution layer and of the “equivariance as weight sharing over a symmetry group fixed in advance” design. Read the CIFAR “state of the art” claim as holding for un-augmented CIFAR10 only, and the paper as containing no representation theory (no irreps, characters or Fourier analysis on groups). |
| LIT-319 v2 | Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges | Bronstein et al. (2021), ARXIV-2104.13478. The authors cast CNNs, GNNs, Deep Sets, Transformers, spherical, mesh and gauge CNNs, and LSTMs as instances of one “blueprint” (p. 29). Each is a stack of linear G-equivariant layers, pointwise nonlinearities and local pooling, capped by a G-invariant global pooling, with the domain Ω and its symmetry group G given in advance (p. 27: “a symmetry group G”; p. 4: “exploiting the known symmetries”). The two priors are symmetry and scale separation. The Fourier transform is derived as the joint eigenbasis of shift-commuting (circulant) operators, which requires distinct eigenvalues (p. 37). Its group-theoretic generalisation via irreducible representations is announced and deferred to “future work” (p. 43). | Active — read in part 2026-09-29 (NOTE-274); worth reading as the standard systematisation of symmetry-as-architecture in deep learning and for its clean derivation of convolution and the Fourier basis from translation symmetry (§4.2). Read it as a proto-book that defers its representation theory (irreps, group Fourier transforms) and its learning theory to “future work”. It takes the symmetry group as known and never addresses discovering one. |
| LIT-321 v3 | Local Quantum Physics: Fields, Particles, Algebras (2nd ed.) | Haag (1996), DOI-10.1007/978-3-642-61458-3. Per publisher description: a monograph on algebraic (local) quantum field theory, treating physics through nets of local observable algebras, including the theory of superselection sectors and charges. | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-325 v3 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield (1998), DOI-10.1023/A:1026680806775. Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Active — read in full 2026-09-29 (NOTE-271); worth reading as the founding paper of the topos approach to quantum theory and the origin of “Kochen–Specker = no global section of a presheaf over contexts”, credited as such by Abramsky & Brandenburger (LIT-016). The reformulation itself is definitional and its proof is two paragraphs. The paper’s own contribution is the sieve-valued generalized valuation, whose later development (von Neumann algebras as base, daseinisation) lives in Parts III–IV and in Döring–Isham (LIT-343, LIT-335, LIT-299, LIT-310). |
| LIT-326 v2 | Sur une espèce de géométrie analytique des systèmes de fonctions sommables | Riesz (1907), https://archive.org/details/comptes-rendus-hebdomadaires-academie-des-sciences_1907-06-24_144_25. A priority note, not a paper of proofs. Riesz says his aim was an “analytic geometry” of square-summable functions, completed by his existence theorem (every square-summable coordinate sequence is the image of a square-integrable function). He states two consequences without proof. First, every linear operation U on square-summable functions that is continuous for convergence in mean is given by U(f) = ∫ f k for some square-summable k; uniqueness of k is not stated. Second, a completeness criterion for Schmidt’s problem. | Active — read in full 2026-09-29 (NOTE-289); worth reading only as the historical first statement of the Riesz representation theorem on L², and it takes three minutes. It states and does not prove. For the theorem itself, and for anything the record argues from it, a modern proof (LIT-243, or any functional-analysis text) supersedes it as a source. |
| LIT-329 v2 | Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe | Peter & Weyl (1927), DOI-10.1007/BF01447892. For a “closed continuous group”, a compact Lie group with invariant volume, the Fundamentalsatz (p. 752) proves the Parseval equality over all inequivalent irreducible representations E = (e_ik) of order n. It is n Σ_{i,k} |α_ik|² + … = V ∫ |x(s)|² ds for every continuous x, with α_ik = ∫ x ē_ik ds. For continuous class functions it reads |α|² + … = V ∫ |x|² ds with α = ∫ x χ̄ ds. The proof applies Hilbert–Schmidt theory to the convolution kernel z(st⁻¹), z = x x̃. The consequences are that finite combinations of matrix coefficients uniformly approximate every continuous function (Approximationssatz), that representations separate points (Satz I), and that characters separate conjugacy classes (Satz II). | Active — read in full 2026-09-29 (NOTE-305); worth reading as the primary source of the Peter–Weyl theorem, and it is short and constructive. Its setting is compact Lie groups and continuous functions. For arbitrary compact groups and the L²(G) = ⊕ (dim π)·π decomposition in modern form, a textbook statement (e.g. Weil 1940, or Folland’s A Course in Abstract Harmonic Analysis) supersedes it. |
| LIT-330 v2 | The approximation of one matrix by another of lower rank | Eckart & Young (1936), DOI-10.1007/BF02288367. For a real n×N matrix a, the rank-r matrix β nearest in the least-squares (Frobenius) distance is obtained by writing a = u′λU (u, U orthogonal, λ diagonal ≥ 0, “Theorem I”) and zeroing all but the r largest diagonal elements. The solution is unique unless λ_r = λ_{r+1}. The squared error is Σ_{i>r} λ_i², so the error is at most l(1−r/R)^{1/2}, with l the length of a and R its rank. The paper also argues that the resulting canonic components cannot straightforwardly be read as psychological factors. | Active — read in full 2026-09-29 (NOTE-301); worth reading as the short, readable origin of the Frobenius-norm low-rank approximation theorem in the form still used. The proof is a first-order stationarity argument plus an unproved simultaneous-diagonalisation theorem, and it covers only the least-squares norm. For the general unitarily-invariant-norm statement, cite Mirsky (1960, not in the record). |
| LIT-331 v3 | Zur Theorie der hyperkomplexen Zahlen | Artin (1927), DOI-10.1007/BF02952526. Per standard description: extends Wedderburn’s structure theorem from finite-dimensional algebras to rings with chain conditions (Artinian semisimple rings are products of matrix rings over division rings). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-332 v3 | Facing up to arrangements: face-count formulas for partitions of space by hyperplanes | Zaslavsky (1975), DOI-10.1090/memo/0154. Per standard description: counts the regions (and faces) into which a hyperplane arrangement divides space via the Möbius function / characteristic polynomial of its intersection lattice (Zaslavsky’s theorem). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-333 v2 | The theory of topological commutative groups | Pontryagin (1934), DOI-10.2307/1968438. For countable discrete abelian 𝔊, the character group X = Hom(𝔊, K), K = ℝ/ℤ with pointwise convergence, is compact and second countable (Thm 1). Characters separate points and subgroups, with annihilator correspondence and quotient/subgroup duality (Thms 2–4). An orthogonal pair (X, 𝔊) with trivial annihilators has each the character group of the other (Thm 5). By the First Fundamental Theorem, every compact second-countable abelian group is the character group of its discrete character group; the proof uses Peter–Weyl via Haar. Hence duality between countable discrete and compact second-countable abelian groups. Chapters III–IV give structure theorems for locally compact connected abelian groups (compact ⊕ ℝⁿ). Duality for general locally compact abelian groups is not proved here. | Active — read in full 2026-09-29 (NOTE-284); worth reading as the primary source of Pontryagin duality in its compact/discrete form, and it is careful and self-contained given Peter–Weyl and Haar. For the full locally compact abelian duality (each LCA group is the dual of its dual), van Kampen (1935) and later textbook treatments supersede it. |
| LIT-335 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | Döring & Isham (2008), DOI-10.1063/1.2883742. A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Active — read in full 2026-09-29 (NOTE-292); worth reading as the core reference for “quantum propositions as clopen sub-objects of the spectral presheaf” (daseinisation, Sub_cl(Σ), truth objects). It is a cleaned-up restatement of Isham–Butterfield (LIT-325) plus de Groote’s operation, not a new no-go result. Several proofs are sketched or omitted, and the “no global elements” statement it relies on needs dim H > 2, which it never says. |
| LIT-337 v3 | Boolean Concept Logic | Wille (2000), DOI-10.1007/10722280_22. Extends Formal Concept Analysis with negation by introducing semiconcepts and protoconcepts, which form double Boolean algebras, as a Boolean concept logic (per the standard description of the paper; abstract not retrieved). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-343 v2 | A topos foundation for theories of physics: I. Formal languages for physics | Döring & Isham (2008), DOI-10.1063/1.2883740. The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Active — read in full 2026-09-29 (NOTE-277); worth reading as the statement and vocabulary of the Döring–Isham programme (neo-realism, PL(S) and L(S), state object and quantity-value object, representation in a topos). It is discursive and contains no results, and every piece of quantum mathematics is deferred to papers II–III (LIT-335, LIT-299). |
| LIT-344 v5 | Formal Concept Analysis: Mathematical Foundations | Ganter & Wille (1999), DOI-10.1007/978-3-642-59830-2. The standard mathematical monograph on Formal Concept Analysis: formal contexts, concept lattices and their structure theory (per publisher record). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read; a second attempt on 2026-10-09 again reached no lawful full text. Springer sells the 1999 book and the 2024 second edition (OpenAlex: both closed access, no repository copy); HathiTrust holds the 1999 edition search-only; the Internet Archive has no copy. The front matter Springer serves freely (preface and contents) was read; the main text was not. No NOTE is filed. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-346 v2 | Analysis of Boolean Functions | O’Donnell (2014), DOI-10.1017/CBO9781139814782. Every f: {−1,1}ⁿ → ℝ has a unique multilinear (Fourier–Walsh) expansion f = Σ_S f̂(S)χ_S over the parities χ_S(x) = ∏{i∈S} x_i, which are an orthonormal basis (Thm 1.1, Thm 1.5) and are the characters of the group F₂ⁿ, χ_S(x+y) = χ_S(x)χ_S(y) (eq. 1.5; §3.2; §8.5). The central operators are all diagonal in this basis with eigenvalues depending only on |S|: the noise operator T_ρχ_S = ρ^{|S|}χ_S (Prop. 2.47) and the Laplacian Lχ_S = |S|χ_S (Prop. 2.37). Influence, noise stability and Arrow’s Theorem (via Stab{−1/3}) become statements about the Fourier weight at each level (Thms 2.20, 2.38, 2.49, 2.56). | Active — read in part 2026-09-29 (NOTE-291); worth reading as the standard, freely available text on Fourier analysis of functions on the Boolean cube. Chapters 1–2 alone give the character basis, Plancherel and Parseval, convolution, and the level-by-level diagonalisation of the noise operator and Laplacian. Read it as abelian harmonic analysis only: it has no nonabelian representation theory, no irreps of dimension > 1 and no isotypic decomposition. |
| LIT-349 v3 | Information and the Accuracy Attainable in the Estimation of Statistical Parameters | Rao (1945), DOI-10.1007/978-1-4612-0919-5_16. Proves the lower bound on the variance of unbiased estimators in terms of Fisher information (the Cramér–Rao bound), and in the same paper introduces the Fisher-information Riemannian metric and the Rao–Blackwell improvement idea. | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-352 v2 | A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming | Murota et al. (2010), DOI-10.1007/s13160-010-0006-9. Given real symmetric A₁…A_m, the method finds an orthogonal P that block-diagonalises them all as simultaneously and finely as possible. That is the decomposition of the matrix *-algebra 𝒯 they generate into simple components (Algorithm 4.1) and then irreducible components (§4.2), with no group or algebra supplied in advance. The simple components are the connected components of the graph “Q_iᵀA_pQ_{i′} ≠ O” between the eigenspaces of one random combination A(r) = Σ r_pA_p. This is correct with probability 1 when span{I, A₁…A_m} = 𝒯 ∩ Sⁿ (Props. 3.2–3.4, 4.2). The structure theorem it relies on, Theorem 3.1(C) “if 𝒯 is irreducible then 𝒯 = Mₙ”, is false over ℝ as stated. The companion paper (Maehara & Murota 2010) handles the complex and quaternion types. | Active — read in full 2026-09-29 (NOTE-276); worth reading as the clearest short account of numerical block-diagonalisation of a *-algebra from generators alone, via eigenspaces of a random element. For the general real case, i.e. irreducible components of complex or quaternionic type, the companion Maehara & Murota (JJIAM 27:263–293, 2010) is needed, and is not in the record. For the map’s use, both should be cited. |
| LIT-353 v3 | The Theory of Representations for Boolean Algebras | Stone (1936), DOI-10.2307/1989664. The paper identifies Boolean algebras with Boolean rings with unit (xx = x), and generalized Boolean algebras with Boolean rings (Thms 1–4). Theorem 67 proves that every Boolean ring A is isomorphic to the algebra B(A) of classes 𝔈(a) = {prime ideals p : a ∉ p}, under symmetric difference and intersection. Its ideal lattice is isomorphic to the system of classes 𝔈(𝔞) under arbitrary unions and finite intersections. The existence of enough prime ideals (Thm 63) is proved by transfinite induction, and Theorem 70 shows that “every Boolean ring has an isomorphic algebra of classes” and “every ideal is the product of its prime divisors” are equivalent without choice. Theorems 68–69 classify all set-representations as restrictions of B(A) to subclasses of prime ideals. The topology, i.e. Stone spaces and duality, is explicitly deferred to “another paper”. | Active — read in full 2026-09-29 (NOTE-272); worth reading as the primary source of the Stone representation theorem, via prime ideals of Boolean rings, and of the Boolean-ring viewpoint. It is long and technical, and for the representation alone Chapter IV plus Theorems 1–2, 33–40 and 49 suffice. For the topological duality (Boolean algebras ↔ totally disconnected compact Hausdorff spaces), the map’s actual need, Stone 1937 (Trans. AMS 41, LIT-358) is the source. |
| LIT-354 v4 | Algebraic Geometry and Statistical Learning Theory | Watanabe (2009), DOI-10.1017/CBO9780511800474. Per publisher description: develops statistical learning theory for singular models (neural networks, mixtures, hidden Markov models) in which the Fisher information is degenerate, using resolution of singularities to show that the real log canonical threshold governs Bayesian free energy and generalization error. | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read; no lawful full text could be reached, and it stays Unavailable until a copy is supplied. Its core results are now read in a lawful substitute by the same author, Watanabe’s A Widely Applicable Bayesian Information Criterion (LIT-616; arXiv 1208.6338, JMLR 2013). That paper restates the book’s standard representation theorem, the real log canonical threshold and the free-energy asymptotics F = nL_n(w0) + λ log n − (m − 1) log log n, citing the book for each. Read it for the results; the book holds their proofs, the generalization-error theory and WAIC. |
| LIT-355 v2 | An Introduction to Hyperplane Arrangements | Stanley (2006), DOI-10.1090/pcms/013/08. The notes define the regions of a real arrangement (connected components of the complement) and the intersection poset L(A) with its Möbius function and characteristic polynomial χ_A(t) = Σ μ(x) t^{dim x}. They prove Zaslavsky’s theorem (Thm 2.5): r(A) = (−1)ⁿ χ_A(−1) and b(A) = (−1)^{rank A} χ_A(1), so the numbers of regions and bounded regions depend only on L(A), not on the face structure (Cor. 2.1, Fig. 2). The same machinery counts faces of every dimension (Thm 2.6: f_k = Σ_{x≤y, dim x=k} |μ(x,y)|). Later lectures add Whitney’s theorem, deletion–restriction, matroids and geometric lattices, broken circuits, supersolvability, the finite-field method, and the metric d(R, R′) = number of separating hyperplanes, whose zero set shows that a region is determined by its separating set (p. 89). | Active — read in full 2026-09-29 (NOTE-313); worth reading as the standard open introduction to real hyperplane arrangements. It is the best reachable source for “regions”, the intersection poset and Zaslavsky’s region-count theorem, with complete proofs of the region formula. Read it knowing that it never uses the oriented-matroid vocabulary of “topes” or “covectors” (it cites Björner et al., Oriented Matroids, only in passing) and that it does not cite Zaslavsky’s own texts. |
| LIT-357 v2 | Studies on the Foundation of Quantum Mechanics. I | Husimi (1937), DOI-10.11429/ppmsj1919.19.0_766. Husimi derives Birkhoff and von Neumann’s lattice axioms from statistical premises. Implication is P̄ ≤ Q̄ in every state (and requires compatibility), negation is Q̄ = 1 − P̄, and compatible propositions obey classical logic. Meet is obtained from “Ehrenfest’s principle” (additivity of mean values) via the top eigenvalue of aP + bQ (pp. 778–779), and the modular law, equivalently the parallelogram law |P| + |Q| − |P∩Q| = |P∪Q|, is proved under a finite-chain assumption. The orthomodular identity is used but never named or stated as an axiom. It appears as the step “since we admit traditional logic to any classical part … S + P∩Q = R” for R ⊃ P∩Q (p. 784), and as the case “(Q − P) + P = Q provided Q ⊃ P” of a classical identity (p. 780). | Active — read in full 2026-09-29 (NOTE-312); worth reading as the first derivation of the Birkhoff–von Neumann propositional calculus from operational premises (states, mean values, compatibility), and for its physical account of meet and join. It is not, however, a clean citation for “the orthomodular law”. The law is an unnamed lemma resting on “classical logic within compatible sets”, and the paper’s axiomatic target is modularity under a finite-chain assumption. A text that states orthomodularity as an axiom (e.g. Kalmbach, Orthomodular Lattices, 1983; unverified here) would settle row 1’s citation. |
| LIT-358 v2 | Applications of the theory of Boolean rings to general topology | Stone (1937), DOI-10.1090/S0002-9947-1937-1501905-7. Stone topologises the set 𝔈 of prime ideals of a Boolean ring by taking the sets 𝔈(a) as a basis. 𝔈 is then a totally disconnected, locally bicompact Hausdorff space, bicompact exactly when the ring has a unit (Thm 1), and every such space arises this way (Thm 2). Theorem 4 states that “the algebraic theory of Boolean rings is mathematically equivalent to the topological theory of Boolean spaces”: isomorphism ↔ homeomorphism, ideals ↔ open sets, quotients ↔ closed subsets, unit ↔ bicompactness, and subrings ↔ continuous images (Thm 7). The same paper also contains what are now called the Stone–Čech compactification (Thms 78–79, 88), the Stone–Weierstrass theorem (Thm 82), the Banach–Stone theorem (Thm 83) and closed ideals of C(X) ↔ closed sets (Thm 85). | Active — read in full 2026-09-29 (NOTE-311); worth reading as the primary source of Stone duality and the correct citation for map row 8’s “Stone duality”, replacing LIT-353. For that purpose Chapter I (pp. 378–394) is enough. The rest is a long general-topology memoir whose best-known results (Stone–Čech, Stone–Weierstrass, Banach–Stone) are stated in 1937 vocabulary (bicompact, H-space, analytical isomorphism) and are now better learned from textbooks. |
| LIT-359 v2 | Counting the faces of cut-up spaces | Zaslavsky (1975), DOI-10.1090/S0002-9904-1975-13885-7. A supplemental reading for the owner’s prior-art novelty map, where an earlier reading found the map’s citation wrong or unreachable. | Unavailable — seeded 2026-09-29 as a supplemental reading for the owner’s prior-art novelty map; a close reading was attempted and no legitimate full text could be reached |
| LIT-361 v2 | Superselection Rules for Philosophers | Earman (2008), DOI-10.1007/s10670-008-9124-z. A supplemental reading for the owner’s prior-art novelty map, where an earlier reading found the map’s citation wrong or unreachable. | Unavailable — seeded 2026-09-29 as a supplemental reading for the owner’s prior-art novelty map; a close reading was attempted and no legitimate full text could be reached |
| LIT-362 v2 | Learning the Irreducible Representations of Commutative Lie Groups | Cohen & Welling (2014), arXiv:1402.4437. Proposes “Weyl’s principle” (via Kanatani 1990) as a definition of disentangling: the elementary components of data are the irreducible subspaces of the representation of a symmetry group acting on it. It learns such a decomposition for one class of group fixed in advance, compact commutative subgroups of SO(D) (“toroidal”), from pairs (x, y) of raw data vectors related by an unobserved group element, y = W R(φ) Wᵀx + ε. The orthogonal basis W (2-D invariant subspaces) is fitted by SGD on a closed-form marginal likelihood, and the integer weights ω_j are then read off a batch rotated by a known 0.1°. The only experiment trains 100 filters on 250,000 white-noise 16×16 patches paired with random rotations of themselves (learned ω from −11 to 12), and uses the invariant √κ̂ for 1-NN on rotated MNIST, reported in a figure with no numbers. Spectral degeneracy plays no role. | Active — read in full 2026-09-29 (NOTE-307); worth reading as the earliest ML paper in this batch to state that the “elementary components” of a representation are the irreps of a symmetry group and to learn them from data; read it as learning the irreducible decomposition of an assumed abelian group class from transformation pairs on raw pixels, not as detecting an unknown symmetry in a trained network, and with an experiment that is a one-group proof of concept on noise patches. |
| LIT-364 v2 | The Theory of Signal Detectability. Part I: The General Theory; Part II: Applications with Gaussian Noise | Peterson & Birdsall (1953), https://deepblue.lib.umich.edu/handle/2027.42/7068. Peterson and Birdsall show that the criterion approach (maximise P_SN(A) − β·P_N(A), or maximise detection at fixed false-alarm k) and the Woodward–Davies a-posteriori approach are all solved by one receiver that computes the likelihood ratio ℓ(x) = f_SN(x)/f_N(x) and thresholds it (Part I, Theorems 1–7). They introduce the receiver operating characteristic, whose slope at each point is the threshold β (Theorem 8, Eq. 2.51). For a signal known exactly in white Gaussian noise, ln ℓ is normal with equal variance 2E/N₀ under both hypotheses and means 2E/N₀ apart, so the ROC is a one-parameter family indexed by the “detection index” d = (M_SN − M_N)²/σ² = 2E/N₀ (Part II, Eqs. 4.1–4.8, Fig. 4.1). The receiver is a correlator or matched filter (Eq. 5.3, 4.10). | Active — read in part 2026-09-30 (NOTE-315); worth reading as the founding engineering statement of signal detection theory: the likelihood-ratio receiver, the ROC and its slope property, and the equal-variance Gaussian case with an explicit d = 2E/N₀. Read the 1953 report knowing that the scan is poor and that the 1954 IRE paper (with W. C. Fox) is the citable journal form, which was not read here. |
| LIT-365 v2 | On the Problem of the Most Efficient Tests of Statistical Hypotheses | Neyman & Pearson (1933), DOI-10.1098/rsta.1933.0009. Registered to cover signal detection theory, which the record lacked. | Unavailable — seeded 2026-09-30 at the owner’s request to cover signal detection theory; a close reading was attempted and no legitimate full text could be reached |
| LIT-373 v4 | The two dragons of cognition: recursive condensation for predictive processing | Li (2026), DOI-10.3389/fncom.2026.1778902. A “Hypothesis and Theory” essay proposing that the neocortex is a “biological Savitch machine”. Alternating expansion (“odd parity”) and contraction (“even parity”) phases manufacture the disjoint closed sets of Urysohn’s Lemma, and storing (“memoizing”) the condensed tokens converts exponential search into polynomial “navigation”. Its one non-standard formal result, Theorem 2, R(N) ∝ b^{αN} (§5.2.2, p. 11), is stated without proof. There is no data, simulation or model. The two mathematical pillars are misdescribed: Savitch’s theorem saves space at the cost of time and does not memoize, and Urysohn’s Lemma gives a continuous separator, not a linear one, and is not an “if and only if”. | Rejected — re-read jointly with LIT-374 and arXiv 2602.07974 on 2026-10-01 (NOTE-321); Read jointly with LIT-374 and the companion StrLT preprint (arXiv 2602.07974), the essay is the cognitive motivation of a coherent programme: structure is discovered by costly search, condensed into frozen tokens, and reused cheaply. It is not just a misuse of two theorems. Read charitably, its mathematical slips are repairable and are not the reason to set it aside: the Urysohn “iff” and linear readout, Savitch as memoisation, the Definition 2 clash. The reason is what is left after repair: an amortised-inference and complementary-learning-systems picture already in the literature it cites; a neural mapping whose predictions do not discriminate it from standard consolidation accounts; one original quantitative claim, Theorem 2, that is not argued. A reader who wants the programme should read LIT-374 and the companion, where it is made precise. |
| LIT-375 v2 | Structural Learning Theory: A Metric-Topology Factorization Approach | Li (2026), arXiv:2602.07974. >- The formal core of Xin Li’s structural-learning programme. Width w(P; γ, δ) is the least number of open cells on each of which some predictor is γ-contractive with conditional risk ≤ δ. The paper proves that width is incomparable with the per-cell VC dimension (Thm 2.2). It shows that with K < w cells there is an n-independent error floor η(w, K), under a non-degeneracy assumption stated in the body (Thm 2.4, Assumption 2.3). Width is at least c·β₁·L/D₀ under a bounded-overlap assumption (Thm 2.5), and discovering all basins costs Ω(w log w) samples (Thm 2.6). The CS-Laplacian eigenvalue count converges uniformly to the predictor-relative count w_G(P) when n ≳ (H_G + d_X log(1/r_x) + log(1/δ′))/(r_x^{d_X} g_eff²) (Thm 3.5); no result shows that w_G equals width. There are no experiments. | Proposed — read in full 2026-10-01 (NOTE-324); >- Promising and unproven. It is the programme’s fullest and most careful formal statement, and it carries the learning theory of LIT-374 §3 with the assumptions in the body. Its routine results hold, and its bouquet slip is repairable. What would settle it: - a theorem that the CS count for a learned predictor equals w(P; γ, δ), not w_G(P); - a K-sweep on benchmarks of known width, showing the floor at K < w for routed learners, and testing whether a capacity-matched monolithic model shows it; - the decoupled-versus-joint training comparison it motivates but does not run. |
| LIT-376 v2 | Local Urysohn Width: A Topological Complexity Measure for Classification | Li (2026), arXiv:2603.15412. >- It defines local Urysohn width uw_{D₀}(P, γ): the least number of connected sets of diameter ≤ D₀, each carrying a continuous classifier correct on the margin-γ safe region it meets, needed to cover that safe region. It proves: - uw = w on a bouquet of w circles with safe balls at the antipodes, when 3γ/2 ≤ D₀ < L/2 − 3γ/4 (Thm 4.1); - uw ≥ w·m, with m = Θ(L/D₀) safe balls per loop, so uw = Ω(w·L/D₀) (Cor 5.1); - two-way non-determination with VC dimension (Thm 6.1); - an Ω(w log w) sample lower bound via label-permuted coupon collection (Thm 7.1); - uw ≥ 2β₁/Δ₀ for good, convex, bounded-adjacency covers (Thm 8.5). The proofs are correct. The obstruction is metric, as the paper concedes in Remark 8.11: patches cannot reach across the wedge point. Because Urysohn’s lemma makes local correctness automatic on any patch, the measure is a connected covering number of the safe region that does not depend on the labels. | Proposed — read in full 2026-10-01 (NOTE-323); >- Promising and unproven. It is the cleanest and most honest paper in Li’s programme: definitions are precise, the main theorems are correctly proved, a dependency table separates proved from conjectural results, and earlier errors are corrected openly. It is also the consistent construction that LIT-374’s and arXiv 2602.07974’s bouquet lacked. What would settle it: a result showing that this width governs error or sample cost for learners not built from local experts (for example local-learning or nearest-neighbour rates), or a proof of Conj 8.9 under conditions its motivating examples actually satisfy. Without one of these, it is a covering number with a topological headline. |
| LIT-378 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | Döring & Isham (2008), arXiv:0803.0417. A 212-page restatement of the topos programme of papers I–IV. A theory of a system S is a representation of a typed language L(S) in a topos. In quantum theory the topos is presheaves on the poset V(H) of abelian von Neumann subalgebras, propositions are clopen sub-objects of the spectral presheaf Σ via daseinisation, a pure state gives sieve-valued truth values ν(A ε Δ; ψ)V = {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}, and each bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose values are monotone intervals over subcontexts. Classical physics fits its axioms for composite systems exactly; quantum theory fits the disjoint sum but not the tensor-product composite. | Active — read in full 2026-10-01 (NOTE-326); worth reading as the one-volume statement of the Döring–Isham topos programme. It is, by the authors’ own account (p. 14), “partly an amalgam” of papers I–IV (LIT-343, LIT-335, LIT-299, LIT-310), with corrections and some new material. The new material is the value of a quantity in a pseudo-state and the inverse-image link between L(S) and PL(S) (§§8.5–8.8, Thm 8.6), the algebra of the candidate quantity-value objects (§9.3), the Boolean-subalgebra base category (§5.5.3), Kochen–Specker as a lifting obstruction (§6.6), and a long speculative §14–15. It adds no new no-go result. Its proofs are as sketchy as the series’ at the same places: Sub_cl(Σ) is Heyting (Thm 16.1) with ⇒ left unwritten, and ι : O → P_clΣ natural and monic is left to the reader. A reader who has read I–IV gains only §§8.5–8.8, 9.3 and 14; a reader new to the programme should start here. |
| LIT-380 v2 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton et al. (1999), arXiv:quant-ph/9911020. Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Superseded — by LIT-382; read in full 2026-10-01 (NOTE-327); covered better by part IV (Butterfield & Isham 2002, read alongside), which restates this paper’s V-framework definitions nearly verbatim, proves the G ≅ CloΣ isomorphism that this paper only asserts and develops its interval valuations into theorems, and by Döring–Isham II (LIT-335), whose spectral presheaf over V(H) and clopen propositions are the mature form; it remains the citation for introducing V as the base category (as LIT-335 credits it). |
| LIT-381 v2 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham (1998), arXiv:quant-ph/9808067. Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Active — read in full 2026-10-01 (NOTE-329); worth reading as the series’ conceptual companion and the only place it states the classical analogue and the partial-truth motivation; the mathematics is light (one short theorem), and the sieve semantics it motivates is carried on in part III, part IV and Döring–Isham (LIT-343, LIT-335). |
| LIT-382 v2 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham (2001), arXiv:quant-ph/0107123. Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Active — read in full 2026-10-01 (NOTE-328); worth reading as the finished statement of the series’ von Neumann-algebra framework. It proves the G ≅ CloΣ isomorphism (coarse-graining = direct image of clopen regions) that Döring–Isham’s daseinisation relies on, and it shows that a state’s sieve-valued truth values are fixed by its support, a global element of G. Its two theorems are correct but close to definitional, so its value lies in the framework rather than in depth. |
| LIT-389 v2 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield (1999), arXiv:gr-qc/9910005. Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Active — read in full 2026-10-01 (NOTE-334); worth reading only as the programme’s short, informal manifesto as of 1999, written before parts III–IV appeared. It joins the Kochen–Specker valuations to the continuum scepticism (real-valued quantities, space and probabilities) that the Döring–Isham programme later develops, and the Literalism/Extra-Values framing appears nowhere else in the record. Its one technical section is a recap of KS I–II, and a reader of LIT-325, LIT-381 and LIT-378 loses little by skipping it. |
| LIT-390 v2 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham (1996), arXiv:gr-qc/9607069. Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Active — read in full 2026-10-01 (NOTE-333); worth reading as the origin of the Isham school’s sieve-valued, coarse-graining truth values, which the Kochen–Specker series names as its motivation (KS I, ref. [13]); it is the only place the construction is applied to consistent histories, it is short and elementary, and its constructions are correct as stated, but read it knowing that the Heyting values measure only whether α can be embedded in a consistent set (the probability d(α, α) is the same in every window), so it says nothing about the conflicts between consistent sets that motivate it. |
| LIT-497 v2 | The Genetical Evolution of Social Behaviour. I | Hamilton (1964), J. Theor. Biol. 7:1–16. A one-locus model in which an individual’s genotype affects relatives’ fitness. Weighting each effect by Wright’s coefficient of relationship r gives “inclusive fitness”, which tends to be maximized as classical fitness is. An action that costs the actor is favoured only if the benefit to a relative exceeds 1/r times the cost: more than two brothers, four half-brothers or eight first cousins (§5). The increase of mean inclusive fitness is proved only for giving traits; the model is not about morality. | Active — read in full 2026-10-02 (NOTE-401); worth reading as the source of inclusive fitness and of the rule the moral-psychology literature calls Hamilton’s rule. It is a population-genetic model with non-overlapping generations, random mating and weak selection, and its rule appears as −k > 1/r (§5), not in the later rb > c form. It proves that mean inclusive fitness increases only for genes whose total effect on others is non-negative (giving traits); for harming traits it shows only that selection points uphill. It says nothing about morality: it is the kin domain of morality-as-cooperation in its biological form, read as biology. |
| LIT-498 | The Evolution of Cooperation | Axelrod & Hamilton (1981), DOI-10.1126/science.7466396. In an iterated Prisoner’s Dilemma with probability w of meeting again, TIT FOR TAT won two computer tournaments (14 and 62 entries) and an ecological simulation, and no strategy can do better against it when w ≥ max((T−R)/(T−P), (T−R)/(R−S)). ALL D is always stable; reciprocity can still enter through kinship or a cluster, after which a nice stable strategy resists clusters. Applications range from cleaner fish to disease and nondisjunction. | Active — read in full 2026-10-02 (NOTE-411); worth reading as the formal statement of how reciprocity can start, spread and hold: tit-for-tat is robust in Axelrod’s tournaments, cannot be bettered by any mutant when the continuation probability w is high enough, and can enter an all-defect population through kinship or clustering. “Evolutionarily stable” is stronger than what is proved: the proof shows no strategy does better than TIT FOR TAT, and always-cooperate does as well, which Boyd & Lorberbaum (1987) later turned into a proof that no pure strategy is evolutionarily stable in the repeated game. It grounds the exchange domain of morality-as-cooperation; it is not about morality. |
| LIT-501 | Five Rules for the Evolution of Cooperation | Nowak (2006), DOI-10.1126/science.1133755. Five mechanisms let natural selection favour cooperators who pay c to give b: kin selection (r > c/b), direct reciprocity (w > c/b), indirect reciprocity (q > c/b), network reciprocity (b/c > k) and group selection (b/c > 1 + n/m). Each is cast as a 2×2 game, from which conditions for cooperators to be evolutionarily stable, risk-dominant or advantageous follow. Punishment is said not to be a mechanism. Cooperation is proposed as a third principle of evolution. | Active — read in full 2026-10-02 (NOTE-409), main text only; the online supplement with the derivations was not read. Worth reading as the standard short map of mechanisms for the evolution of cooperation (kin selection, direct, indirect and network reciprocity, group selection), each reduced to a benefit-to-cost threshold from a 2×2 game. Every mechanism is modelled with the donation game, so it covers altruism only: mutualism, coordination and conflict resolution, four of morality-as-cooperation’s seven domains, are outside it. Its one sentence on morality credits indirect reciprocity with “the evolution of morality”, citing Alexander (1987). |
| LIT-505 | The Logic of Animal Conflict | Maynard Smith & Price (1973), DOI-10.1038/246015a0. Why do animals with dangerous weapons fight conventionally? A simulation of five strategies (Mouse, Hawk, Bully, Retaliator, Prober-Retaliator) shows Hawk is not an evolutionarily stable strategy while Retaliator nearly is, so “limited war” can be favoured by individual selection. In a pure war of attrition no fixed persistence is stable; the ESS is an exponential mixture. There is no Hawk–Dove game and no ownership asymmetry here. | Active — read in full 2026-10-02 (NOTE-404); worth reading as the paper that introduced the evolutionarily stable strategy and showed that “limited war” in animal contests can be favoured by individual selection. It is not the source of the Hawk–Dove game or of owner–intruder conventions, though morality-as-cooperation cites it for both: its strategies are Mouse, Hawk, Bully, Retaliator and Prober-Retaliator, its contestants are symmetric, and its “Retaliator is an ESS” ties with Mouse, so it fails the paper’s own second ESS condition. What it does ground for MAC’s contest domains is restraint backed by retaliation, and, informally, retreat before a visibly superior opponent and honest signals of formidability. |
| LIT-527 | Pattern formation outside of equilibrium | Cross & Hohenberg (1993), DOI-10.1103/RevModPhys.65.851. A 262-page review classifying non-equilibrium patterns by the linear instability of a uniform state: type I_s (finite wave-number, stationary: convection rolls, Turing patterns), III_o (uniform oscillation: the stirred BZ reaction) and I_o (travelling waves). Near threshold, universal amplitude equations describe them, and phase equations describe them further above it. It finds no general variational principle for non-equilibrium steady states, only perturbative ones near threshold. | Active — skimmed 2026-10-02 (NOTE-418), about 30 of 262 pages read closely; worth reading as the physicists’ standard account of pattern formation away from equilibrium, and as the sober counterweight to the dissipative-structure programme. It treats Rayleigh–Bénard convection, Turing patterns and the BZ reaction in one framework, classifying patterns by the linear instability of a uniform state and describing them near onset by universal amplitude equations. Its conclusion finds “no evidence for the existence of any global minimization principles controlling the structure, except as a perturbative statement near threshold”, and quotes Glansdorff and Prigogine conceding as much. |
| LIT-535 | The Chemical Basis of Morphogenesis | Turing (1952), DOI-10.1098/rstb.1952.0012. Chemicals (“morphogens”) that react and diffuse through tissue can turn a stable homogeneous state unstable, with random disturbances setting which pattern grows. On a ring of cells the linearized onset takes six forms. The most important is stationary waves of finite “chemical wave-length”, which (9.4a, b) allow for two morphogens only if they diffuse at different rates. A 20-cell example computed on the Manchester computer gives a three-lobed pattern. Hydra tentacles, whorled leaves, dappling and gastrulation are suggested applications, not tested ones. | Active — read in full 2026-10-02 (NOTE-427); worth reading as the founding paper of reaction–diffusion pattern formation and the origin of what the Brussels school later called the “Turing bifurcation”. A homogeneous, stable steady state of reacting and diffusing chemicals can be destabilized by diffusion itself and settle into a stationary pattern with an intrinsic “chemical wave-length”. The analysis is linear and near onset by design, every biological application is offered as suggestion, and the paper says outright that its patterns are “dynamic equilibria” fed by “a continual supply of free energy”, which places them among dissipative structures before the term existed. |
| LIT-608 | Deep Equals Shallow for ReLU Networks in Kernel Regimes | Bietti & Bach (2021), ICLR 2021. Dot-product kernels on the sphere are diagonal in spherical harmonics, with one eigenvalue µ_k per degree k shared by N(d, k) ~ k^(d−2) harmonics. Theorem 1 reads the asymptotic decay of µ_k off the kernel function’s expansion at ±1: a non-integer exponent ν there gives µ_k ~ k^(−d−2ν+1). The arc-cosine kernels have ν = 1/2 and 3/2, and composing them preserves ν, so the NTK of a deep ReLU network decays as k^(−d) and its random-feature kernel as k^(−d−2), exactly as for two layers. Deep and shallow ReLU kernels therefore have the same RKHS, the same as the Laplace kernel’s for the NTK. | Active — read 2026-10-03 (NOTE-475): main text and Appendix A in full, the proofs in Appendices B and C followed at the level of their lemmas. Worth reading as the general rule behind the NTK eigenvalue decays the record holds: for a dot-product kernel on the sphere, the decay of its spherical-harmonic eigenvalues is fixed by the kernel function’s non-integer exponent at the endpoints ±1 (Theorem 1). Deep ReLU kernels keep the shallow exponent, so depth leaves the spectrum’s shape unchanged: µ_k ~ C(d, L) k^(−d) for the NTK and k^(−d−2) for the random-feature kernel, with only the constant growing in L (Corollaries 2–3). |
| LIT-616 | A Widely Applicable Bayesian Information Criterion | Watanabe (2012), arXiv:1208.6338; JMLR 14:867–897 (2013). A model is singular when the parameter-to-distribution map is not one-to-one or the Fisher information is not positive definite: neural networks, mixtures, reduced-rank regression, HMMs. Its Bayes free energy is nL_n(w0) + λ log n − (m − 1) log log n + O_p(1), with λ the real log canonical threshold (RLCT), not d/2. WBIC, the posterior average of nL_n(w) at inverse temperature 1/log n, has the same leading terms even for singular, unrealizable models, equals BIC for regular ones, and needs no knowledge of the true distribution. Two temperatures give an estimate of λ. In reduced-rank regression WBIC picked the true rank in 100 of 100 trials and recovered theoretical RLCTs closely. | Active — read in full 2026-10-03 (NOTE-473); worth reading as the record’s lawful, short statement of the core of singular learning theory, standing in for Watanabe’s unread book (LIT-354). It states the Bayes free energy asymptotics F = nL_n(w0) + λ log n − (m − 1) log log n + O_p(1), with λ the real log canonical threshold and m its multiplicity (Theorem 2), and defines λ through resolution of singularities (Theorem 1, Eq. 18). Its own result is WBIC, the posterior mean of nL_n(w) at inverse temperature 1/log n, which matches F to the λ log n term even for singular and unrealizable models (Theorem 4), equals BIC for regular ones (Theorem 5), and estimates λ from two temperatures (Corollary 3). The prior results are stated with citations to the book, not re-proved. |
| LIT-655 | Understanding Mode Connectivity via Parameter Space Symmetry | Zhao, Dehmamy, Walters & Yu (2025), ICML 2025. Relates the topology of a loss’s symmetry group to that of its minimum. The minimum of a full-rank l-layer linear regression is homeomorphic to GL_h(ℝ)^{l−1}, so it has 2^{l−1} components; a skip connection reduces 4 to 3 in a 1-D case, and permutations connect all components when h ≥ 2. Rescaling symmetry yields minima in one component with arbitrarily large barriers on the straight line. Group actions give explicit constant-loss curves, and a curvature bound κ on such a curve bounds the chord’s distance from the minimum by about κ‖w₂ − w₁‖²/8. | Active — read 2026-10-03 (NOTE-528); worth reading as the account of mode connectivity from continuous parameter symmetries. The minimum of a full-rank l-layer linear network is homeomorphic to GL_h(ℝ)^{l−1} and so has 2^{l−1} connected components, a skip connection merges some of them, and a permutation with negative determinant joins the rest. Rescaling symmetry gives minima in the same component with an arbitrarily large loss barrier on the straight line between them, even after a restricted permutation. A group element g gives an explicit constant-loss curve exp(t log g)·w, and bounded curvature of such curves bounds the barrier on the chord. The theory is for linear networks and homogeneous last layers; the experiments are small synthetic regressions. |
| LIT-679 | A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent | Liao, Couillet & Mahoney (2020), NeurIPS 2020 and J. Stat. Mech. (2021). For random Fourier features ΣX = [cos(WX); sin(WX)] with n, p, N large and comparable, the expected resolvent of ΣXᵀΣX/n is asymptotically Q̄ = ((N/n)(Kcos/(1+δcos) + Ksin/(1+δsin)) + λI)⁻¹, with δσ = (1/n)tr KσQ̄, not the Gaussian-kernel prediction. Closed forms for training and test MSE follow and match MNIST, Fashion-MNIST and Kannada-MNIST. In the ridgeless limit the δ jump at 2N = n, separating an under- and an over-parameterized phase; the test MSE diverges there unless test data sit within about λ of the training data, which is the double-descent peak. A tuned λ removes it. | Active — read 2026-10-03 (NOTE-519); worth reading as an exact random-matrix account of double descent in random Fourier feature ridge regression when n, p and N are large together: the RFF Gram matrix does not converge to the Gaussian kernel in spectral norm, but its resolvent has a deterministic equivalent built from the cos and sin kernels with corrections δcos, δsin solving a fixed point (Theorem 1). Training and test MSE follow in closed form (Theorems 2–3). As λ → 0 the δ jump from O(1) to order 1/λ across 2N = n, so the model has two phases, and the test error diverges at the boundary when test data differ from training data; that singularity is the double-descent peak. Data need only be concentrated random vectors, not Gaussian. |
| LIT-680 | Proving Linear Mode Connectivity of Neural Networks via Optimal Transport | Ferbach, Goujaud, Gidel & Dieuleveut (2024), AISTATS 2024. Finding the permutation that aligns two networks’ neurons layer by layer is an optimal-transport problem between empirical weight distributions, so Wasserstein convergence rates bound when the permuted straight line stays low-loss. Wide two-layer nets trained by SGD in the mean-field regime are linearly connected modulo permutation with high probability. Deep nets with i.i.d. neuron weights (as at initialization) are too, once each layer is wider than about (1/ε) to the power of the previous one, a rate shown tight. Low-dimensional weight distributions need less width, and a covariance-weighted weight matching beats plain weight matching. | Active — read in full 2026-10-03 (NOTE-543); the first proof of linear mode connectivity modulo permutation beyond a single hidden layer. Aligning neurons is an optimal-transport problem between the empirical distributions of their weights, so convergence rates of empirical measures in Wasserstein distance bound how wide a network must be for the permuted straight line to stay low-loss. It proves the result for wide two-layer nets trained by SGD in the mean-field regime, and for deep nets whenever neuron weights within a layer are i.i.d., which holds at initialization. The width needed grows like (1/ε) to the power of the previous layer’s width, a rate shown to be tight, and falls if weights concentrate near a low-dimensional subspace. The proofs rest on independence of neurons, which the authors call the central and least realistic assumption. |
| LIT-683 | Exact expressions for double descent and implicit regularization via surrogate random design | Dereziński, Liang & Mahoney (2019), NeurIPS 2020. For the Moore–Penrose estimator X⁺y in linear regression, a determinantal surrogate design (density ∝ pdet(XXᵀ), Poisson sample size with mean n) gives exact MSE: σ² tr((Σ + λnI)⁻¹)(1 − αn)/(d − n) + wᵀ(Σ + λnI)⁻¹w·(d − n)/tr((Σ + λnI)⁻¹) for n < d, σ² tr(Σ⁻¹) at n = d, σ² tr(Σ⁻¹)(1 − βn)/(n − d) for n > d, where n = tr(Σ(Σ + λnI)⁻¹). For n < d, E[X⁺y] equals the population ridge solution with that λn: over-parameterization is implicit ridge regularization whose strength makes n the effective dimension. The surrogate agrees with the i.i.d. design as d, n → ∞ for sub-Gaussian data. | Active — read 2026-10-03 (NOTE-518); worth reading as the exact, non-asymptotic account of double descent and implicit regularization for minimum-norm least squares: replacing the i.i.d. design by a determinantal “surrogate” design gives closed-form MSE in every regime (Theorem 1) and shows that, below the interpolation threshold n < d, the expected minimum-norm solution equals the population ridge solution with the λn for which the ridge effective dimension tr(Σ(Σ + λnI)⁻¹) equals n (Theorem 2). The surrogate matches the i.i.d. design asymptotically for sub-Gaussian data (Theorem 3), at an empirically O(1/d) rate. The tool is a class of random matrices for which determinant and expectation commute. |
| LIT-714 v2 | The Method of Characteristics Revisited: A Viability Approach | Aubin (1999), lecture notes on arXiv. A set K is viable under a system if from each state in K at least one evolution stays in K, and invariant if every evolution does. By Nagumo’s theorem K is viable exactly when the dynamics point into K’s contingent cone at each state. The viability kernel of K is the largest viable subset of K; its complement in K is a repeller. The capture basin of a target is the set of states from which it can be reached in finite time. | Active — read 2026-10-05 (NOTE-576): the introduction and chapters 1–2 of the author’s arXiv lecture notes, the parts that define viability. Worth reading as the shortest first-hand statement the record holds of Aubin’s viability concepts: a set is viable if from every state at least one evolution stays in it, invariant if all do, and the viability kernel is the largest viable subset of a constraint set. Chapters 3 onward apply these to Hamilton–Jacobi equations and were not read. Stands in for the unread monograph LIT-738. |
| LIT-738 v3 | Viability Theory | Aubin (1991; reprinted 2009), Birkhäuser: the monograph that founds viability theory, the study of which evolutions of a dynamical or controlled system can remain in a given constraint set, and of the set of initial states from which one can. Unread: registered from metadata; the definitions are read in Aubin’s 1999 mini-course notes LIT-714. | Unavailable — registered 2026-10-05 from Crossref and the author’s own list of his books, not read: no lawful full text could be reached. Springer serves the 2009 reprint behind a bot challenge, which was not passed. Aubin’s 1999 mini-course notes (LIT-714, read) state the definitions this book is cited for: a viable evolution, viability and invariance, the viability kernel and the capture basin. Filed because an essay the owner is writing cites “Aubin’s mathematical theory of viability” for the difference between stability and viability. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-772 | A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics | Fritz (2020), Adv. Math. 370:107239. Markov categories as a framework for synthetic probability and statistics: conditioning, conditional independence, almost-sure equality and sufficient statistics stated in categorical terms, with the Fisher–Neyman, Basu and Bahadur theorems proved there. Basu needs no extra axiom; Fisher–Neyman and Bahadur need strict positivity, not conditionals. Fisher–Neyman is matched to the classical theorem only in FinStoch. | Active — read 2026-10-09 (NOTE-588); worth reading as the reference account of Markov categories and of how much probability and statistics follows from copying, discarding and composing kernels: determinism, four notions of conditional independence with the semigraphoid rules (weak union only with conditionals), almost-sure equality, and sufficiency, completeness and ancillarity. Basu’s theorem holds in every Markov category, and Fisher–Neyman and Bahadur under strict positivity, which Stoch has without conditionals. The abstract Fisher–Neyman is matched to the classical theorem only for finite sets. |
| LIT-773 | A Noether theorem for Markov processes | Baez & Fong (2013), J. Math. Phys. 54:013301. For a Markov process, an observable commutes with the Hamiltonian if and only if both its expected value and its standard deviation are constant in time for every state; in quantum mechanics the mean alone suffices. On a finite state space such observables are exactly those constant on the connected components of the transition graph. | Active — read 2026-10-09 (NOTE-598); worth reading as the stochastic counterpart of the commutator form of Noether’s theorem: for a Markov process, an observable commutes with the generator iff its mean and second moment are conserved in every state. On a finite state space that holds iff it is constant on each connected component of the transition graph. A conserved mean alone is not enough, unlike quantum mechanics, as an explicit 3-state example shows. Only diagonal observables are treated, not symmetries that permute states. |
| LIT-776 | Conditional Simulation Using Diffusion Schrödinger Bridges | Shi, De Bortoli, Deligiannidis & Doucet (2022), UAI, PMLR 180. Extends the Schrödinger-bridge formulation of diffusion generative modelling from unconditional to conditional simulation, applied to super- resolution, filtering in state-space models and refining pre-trained networks. Read: the extended-space bridge is proved; the empirical gains are at small step counts and mixed across metrics. | Active — read 2026-10-09 (NOTE-589); worth reading for the result that posterior sampling from simulation alone, when (X, Y) can be drawn but the likelihood cannot be evaluated, is a Schrödinger bridge between the joint p(x, y) and a reference p_ref(x|y)p(y) with y held fixed along the path (Proposition 1), solvable by iterative proportional fitting with learned drifts (CDSB). Its first iteration is the ordinary conditional score-based model, so later ones refine it, and the reference end can be a cheap posterior approximation rather than noise. Gains over the conditional score model are at equal small step counts and are not uniform across metrics; it filters the Lorenz-63 model with 20 steps where the score model diverges. |
| LIT-777 | Context–content systems of random variables: The Contextuality-by-Default theory | Dzhafarov & Kujala (2016), J. Math. Psychol. 74:11–33. Contextuality-by-Default for finite systems of categorical random variables, indexed by content and context. A system is noncontextual when some coupling of its context bunches makes every connection a maximal coupling; the test is a linear program, the binary cyclic criterion is stated (proved elsewhere), and the minimal total variation of a signed maximally connected quasi-coupling, minus 1, measures the degree of contextuality. | Active — read 2026-10-09 (NOTE-600); worth reading as the reference statement of Contextuality-by-Default: each measurement gets a separate random variable in each context, and a system is noncontextual when some coupling of its contexts makes every content’s copies coincide as often as their own distributions allow (maximal couplings). This separates context-dependent marginals (“direct influences”) from contextuality, turns the test into linear-programming feasibility (Theorem 4.1), and, since a signed such coupling always exists (Theorem 6.1), measures contextuality as the least total variation above 1. The definition was already being replaced by “multimaximal” couplings when it appeared (footnote 10). |
| LIT-778 v3 | Comparison of Statistical Experiments | Torgersen (1991), Cambridge University Press. The comprehensive treatment of comparing statistical experiments, from Wald, Blackwell and Le Cam: when one experiment is more informative than another, Le Cam’s deficiency as the measure of the difference, and its relations to sufficiency and randomisation. Not read: no lawful full text. | Unavailable — filed 2026-10-09 from Crossref and the publisher’s pages, not read: no lawful full text was reachable. Cambridge Core sells the book and its chapters (OpenAlex: closed access, no repository copy); its public pages carry only the summary and each chapter’s opening paragraph. The Internet Archive copy is lending-only (access-restricted, printdisabled), and HathiTrust has no record for the ISBN. No NOTE is filed. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-781 | Equivalent Comparisons of Experiments | Blackwell (1953), Ann. Math. Statist. 24(2):265–272. Completes the Blackwell order: for finitely many states, one experiment is at least as informative as another for every decision problem exactly when the other is a garbling of it. The easy direction is Blackwell (1951) and the finite converse Sherman and Stein’s; this paper re-proves the converse by minimax, extends it to arbitrary outcome spaces by a martingale argument, introduces comparison by k-action problems, and reduces two-state comparison to the type I/type II error trade-off. | Active — read 2026-10-09 (NOTE-595); worth reading as the paper in which the comparison of experiments is settled in general: for experiments with finitely many states, being at least as informative in every decision problem (Bohnenblust, Shapley and Sherman’s order) and being able to reproduce the other by a stochastic transformation (Blackwell’s 1951 sufficiency, later called garbling) are the same order. The finite-outcome converse is Sherman and Stein’s, re-proved here by a minimax argument; the extension to arbitrary outcome spaces, by martingale convergence, is new. It adds comparison by k-action problems, and shows that for two states testing problems, and so the type I/type II error curves, decide the order. |
| LIT-788 | The Geometry of Causality | Gogioso & Pinzani (2023), arXiv 2303.09017, Part 3 of the trilogy. Empirical models on any cover of a space of input histories are the points of a causaltope, a product of simplices cut by linear causality equations, so causal fractions are linear programs. Causal separability relative to a given ambient space lets entangled or contextually controlled quantum switches witness indefinite order once known no-signalling constraints are imposed, with the causally separable fraction bounded below by the separable local fraction. | Active — skimmed 2026-10-09 (NOTE-624); worth reading as the geometric half of the trilogy and the successor of the withdrawn LIT-813’s polytope and causal-fraction results: the empirical models on any cover of a space of input histories are exactly the points of a “causaltope”, the product of simplices sliced by linear causality equations derived from the space’s topology, so supported fractions and causally separable fractions are linear programs. Causal separability is defined relative to an ambient space rather than to the indiscrete one, which lets quantum switches with entangled or contextual control witness indefinite causal order once known no-signalling constraints are imposed (and not otherwise); the causally separable fraction is bounded below by the separable local fraction, and in the switch examples it tracks, and sometimes equals, the local fraction (“contextual causality”). The BFW model is 0% supported by every causally complete 3-event space and 100% by each of three indefinite orders. Read in part: introduction, causaltopes, standard causal separability and the examples. |
| LIT-790 | Contextuality in Canonical Systems of Random Variables | Dzhafarov, Cervantes & Kujala (2017), Phil. Trans. R. Soc. A 375:20160389. Proposes that contextuality be judged only on a system’s canonical representation: every variable, and every joining or coarsening of interest, replaced by its binary splits, with connections coupled multimaximally (pairwise maximally). For two content-sharing k-valued variables with all splits kept, the canonical system is noncontextual if and only if one variable nominally dominates the other (Pr below the other’s for at most one value; Theorem 4.6). For continuous densities with all splits, any difference in distribution is contextual. | Active — read 2026-10-09 (NOTE-640); worth reading as the step that makes Contextuality-by-Default representation-dependent. Every variable is replaced by binary splits (dichotomizations) and connections are coupled multimaximally, so a verdict belongs to a chosen set of splits, not to the measurements alone. Its one worked result is striking: if every split of two content-sharing k-valued variables is kept, that pair alone is noncontextual exactly when one variable “nominally dominates” the other, i.e. its probabilities fall below the other’s for at most one value (Theorem 4.6). For continuous variables the analogue makes any difference in distribution count as contextuality. The authors present this as opening behavioural contextuality, not as a reductio. Proofs are in the supplement and contain small slips that do not affect the results. |
| LIT-800 | The Combinatorics of Causality | Gogioso & Pinzani (2022–23), arXiv 2206.08911 (v1 “The Topology and Geometry of Causality”). Generalises causal orders to spaces of input histories, sets of partial input assignments on which outputs may depend, so that causal constraints can depend on inputs. Defines causal completeness, tightness and sequential, parallel and conditional composition; proves the maximal complete spaces are the causal switch spaces; and counts 2644 causally complete spaces on 3 binary-input events (19 from definite orders), with about a billion estimated on 4. | Active — read 2026-10-09 (NOTE-635); worth reading as the paper that replaces causal orders by “spaces of input histories”: finite, join-prime sets of partial functions from events to inputs, each history being data on which the output at its tip event may depend. Order-induced spaces are a special case, and input-dependent (dynamical) causal constraints, such as a switch where one party’s input fixes the order of the other two, are first-class. Spaces form a lattice under inclusion of their extended histories; causal completeness (one tip event per history) recovers definite order; causal switch spaces are exactly the maximal complete spaces. On 3 events with binary inputs there are 2644 causally complete spaces in 102 symmetry classes, of which order-induced spaces account for 19; most are non-tight. It is the first of the three papers that supersede LIT-813, and its spaces replace that paper’s defective “locale of inputs”. |
| LIT-802 v2 | Le structuralisme | Piaget (1968), Presses Universitaires de France, Que sais-je? no. 1311; English as Structuralism (1970), Basic Books. By its catalogue description, an attempt at a synthesis of the senses structuralism has taken in the contemporary sciences, finding a common ideal of intelligibility beneath variable critical intentions. Not read: no lawful full text was reachable. | Unavailable — filed 2026-10-09 from catalogue records, not read: no lawful full text was reachable. The Internet Archive holds the 1968 PUF edition and the 1970–1973 English editions only as access-restricted lending items, and HathiTrust holds the 1970 Basic Books translation as Limited (search-only); the book is in copyright and the publishers offer no open access. No NOTE is filed. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-803 | Seven Sketches in Compositionality: An Invitation to Applied Category Theory | Fong & Spivak (2018; Cambridge University Press 2019), ARXIV-1803.05316. A textbook invitation to applied category theory in seven chapters, each pairing an application with a structure: generative effects with orders and Galois connections, resources with monoidal preorders and enrichment, databases with functors and adjunctions, co-design with profunctors, signal-flow graphs with props, circuits with hypergraph categories and operads, behaviour with sheaves and toposes. Its first chapter defines a generative effect as an observation that preserves order but not joins, and shows that such maps are exactly the non-left-adjoints. | Active — skimmed 2026-10-09 (NOTE-601): the preface and Chapter 1 read closely, Chapter 3’s opening and its section on adjunctions and data migration read closely, the openings and summaries of Chapters 2 and 4–7 read; Chapters 2 and 4–7 otherwise not read. Worth reading as an introduction to applied category theory organised around compositionality: an observation of systems is a monotone map, a generative effect is its failure to preserve joins, and a monotone map out of a preorder with all joins has no generative effect exactly when it is a left adjoint. Later chapters treat resource theories, databases as functors, co-design, signal-flow graphs, circuits and temporal logic, each through a categorical structure. |
| LIT-808 | The Topology of Causality | Gogioso & Pinzani (2023), arXiv 2303.07148, Part 2 of the trilogy. Extends Abramsky–Brandenburger from no-signalling to arbitrary, possibly dynamical or indefinite, causal structure: causality is continuity in the lowerset topology, causal functions form a presheaf that fails to be a sheaf exactly on spaces with “solipsistic contextuality”, empirical models live on any open cover, and standard empirical models on causal switch spaces (total orders included) are always local, so the Leggett–Garg model is local for its own total order. | Active — read 2026-10-09 (NOTE-612); worth reading as the direct successor of the withdrawn LIT-813’s sheaf construction, now over spaces of input histories (LIT-800) rather than a “locale of inputs”: a joint input–output function is causal when each event’s output depends only on the histories with that event as a tip, causal functions are exactly free assignments of outputs to tips (up to tip-equivalence in non-tight spaces), and an extended function is causal iff it is continuous for the lowerset topology. Causal functions form a separated presheaf that is a sheaf exactly when the space admits no “solipsistic contextuality”, so tight spaces give a sheaf and some dynamical causal structures do not, letting deterministic local data fail to glue. Empirical models are compatible families of distributions over causal functions on any open cover, from the fully solipsistic to the classical; standard empirical models on causal switch spaces, total orders included, are always local. Source of THEORY-171. |
| LIT-813 v2 | The Sheaf-Theoretic Structure of Definite Causality | Gogioso & Pinzani (2021), QPL 2021, EPTCS 343:301–324; withdrawn by the authors in 2024. Extends Abramsky–Brandenburger to a finite causal order. Inputs live on lower sets, sections are causal functions (an output depends only on inputs in its past), and an empirical model is a causal conditional distribution. Locality is a global section, equivalently a mixture of deterministic causal functions (Proposition 21). The models form a polytope cut out by linear causality equations, and pre-orders sketch indefinite causality. The construction’s “locale of inputs” is not a locale, as the authors’ withdrawal says. | Superseded — by LIT-800, LIT-808, LIT-788; read 2026-10-09 (NOTE-629), and superseded by its authors’ trilogy: LIT-800 (The Combinatorics of Causality, arXiv 2206.08911), LIT-808 (The Topology of Causality, 2303.07148) and LIT-788 (The Geometry of Causality, 2303.09017). The authors withdrew it: the arXiv record’s v3 (3 April 2024) states that the paper “has been superseded by arXiv:2206.08911v4, arXiv:2303.07148 and arXiv:2303.09017”, that its Definition 3 (the locale of inputs) “is not fit for purpose”, and that it is “unlikely to be the right reference”. The reading confirms the defect: the meet given in Proposition 5 can leave the poset, and the poset is not distributive, so it is not a locale (NOTE-629 gives a two-event counterexample). The successors replace the locale of inputs with spaces of input histories under the lowerset topology, a genuine topology; the sheaf of causal functions and locality as a global section are redone in LIT-808, and the polytope and the BFW causal fractions are proved in LIT-788. |
| LIT-819 | Getting to the Bottom of Noether’s Theorem | Baez (2020; chapter 2022), ARXIV-2006.14741. In Poisson algebras, complex *-algebras and Banach–Lie algebras, “a generates symmetries of b iff b generates symmetries of a” follows from the antisymmetry of the bracket and uniqueness of solutions; for a bilinear bracket, antisymmetry is the self-conservation principle {a, a} = 0. The nontrivial content is a map from observables to generators: in complex quantum mechanics multiplication by i, and in a unital JB-algebra Alfsen and Shultz’s dynamical correspondence, whose second condition Baez links to “inverse temperature is imaginary time”. | Active — read 2026-10-09 (NOTE-630); worth reading as an account of what Noether’s theorem, in its Hamiltonian form, actually assumes: given unique solutions of the flow equations, “a generates symmetries of b iff b generates symmetries of a” reduces to the antisymmetry of a bilinear bracket, which is equivalent to each observable conserving itself. The content lies in identifying observables with generators; in quantum theory that identification is multiplication by i, and Alfsen and Shultz’s dynamical correspondence recovers it from a JB-algebra under the self-conservation principle and a second condition that Baez reads as “inverse temperature is imaginary time”. The paper proves nothing new, by its own statement. |
| LIT-827 | Backprop as Functor: A compositional perspective on supervised learning | Fong, Spivak & Tuyéras (2017; LICS 2019), ARXIV-1711.10455. Supervised learners compose only when each carries a request function that passes a corrected input upstream; learners (P, I, U, r) then form a symmetric monoidal category Learn, and gradient descent with backpropagation, for a fixed step size ε and an error function e with ∂e/∂x(x₀, −) invertible, is a faithful, injective-on-objects strong symmetric monoidal functor Para → Learn. Functoriality is the chain rule. | Active — read 2026-10-09 (NOTE-625); worth reading as the paper that makes supervised learners composable: a learner A → B is a parameter set with implement, update and request functions, learners form a symmetric monoidal category Learn, and for a fixed step size and an error function whose derivative in its first argument is invertible, gradient descent with backpropagation is a faithful strong symmetric monoidal functor from differentiable parametrised functions to Learn. The backward message is a requested input, not a gradient step; cross-entropy falls outside the theorem. |
| LIT-831 | Contextuality-by-Default 2.0: Systems with Binary Random Variables | Dzhafarov & Kujala (2017), Quantum Interaction 2016, LNCS 10106:16–32. CbD 2.0 for binary random variables: a system is noncontextual when its bunches are compatible with the multimaximal couplings of its connections, couplings in which every subset of a connection is maximally coupled. For binary variables that coupling exists, is unique and has an explicit staircase form (Theorem 1), so subsystems of a noncontextual system are noncontextual (Theorem 3). Cyclic and consistently connected systems are judged as in CbD 1.0, and the total-variation measure carries over. For non-binary variables the construction fails, and dichotomization is proposed. | Active — read 2026-10-09 (NOTE-643); worth reading as the paper that defines “CbD 2.0”: for systems of binary random variables, the copies of a content across contexts are coupled multimaximally (every subset, equivalently every adjacent pair in the order of their means, maximally coupled) rather than maximally as a whole, and such a coupling exists and is unique (Theorem 1). The change buys heredity (a noncontextual system’s subsystems are noncontextual, Theorem 3) and removes the partial/complete distinction, and it leaves cyclic systems, consistently connected systems and the negative-probability measure as they were. It shows by example that for non-binary variables multimaximal couplings may fail to exist, may not be unique, and are not stable under coarse-graining, and ends by proposing to dichotomize every variable, the step the canonical-systems paper (LIT-790) takes. |
| LIT-836 | Probabilistic Foundations of Contextuality | Dzhafarov & Kujala (2017), Fortschr. Phys. 65:1600040. Being jointly distributed is transitive in Kolmogorovian probability. So if a measurement were the same random variable in every context, overlapping contexts would force a joint distribution, and Kochen–Specker and Bell contradictions would show that such variables do not exist. The assumption to drop is “Noncontextual Identification”. Contextuality becomes the impossibility of a coupling of context-indexed variables in which same-content copies satisfy a property C. For inconsistently connected binary systems, C is multimaximality (Definition II.1). | Active — read 2026-10-09 (NOTE-623); worth reading as the short foundational argument for Contextuality-by-Default. The paper argues that the traditional reading of a Kochen–Specker or Bell-type contradiction (no joint distribution exists) cannot be right within Kolmogorovian probability, because being jointly distributed is transitive, so overlapping contexts already force one. The assumption a reductio refutes is “Noncontextual Identification”, that a content is the same random variable in every context. The paper then restates contextuality as the impossibility of a coupling with a specified property C. It gives multimaximality as C for binary measurements, with the pairwise characterization (Theorem II.3) that CbD 2.0 (LIT-831) cites. It ends with Specker’s three boxes treated without assuming consistent connectedness. |
| LIT-837 v2 | Structuralism | Piaget (1970), Structuralism, trans. and ed. Chaninah Maschler, Basic Books (London: Routledge and Kegan Paul, 1971; reissued 2015). The English edition of Le structuralisme (1968), held as LIT-802. Skimmed thinly from a publisher’s preview (17 pages of chs. I–IV): a structure is a whole under laws of composition, a system of transformations and self-regulating. Logico-mathematical structures are fully reversible (groups); linguistic and social ones are regulated in time but not fully reversible; and a structure is real only if governed from within. | Active — skimmed 2026-10-09 (NOTE-618), thinly: the publisher’s Google Books preview of the 2015 reissue served only 17 pages of text, pp. 3–63 at one page in four and p. 69, all in chs. I–IV, and none of the chapters on linguistics, the social sciences or philosophy, nor the conclusion. Worth reading as Piaget’s definition of structure by wholeness, transformation and self-regulation, and for his distinction between logico-mathematical structures, whose operations all have inverses (groups), and linguistic, social and psychological structures, whose transformations unfold in time under regulations that are “not entirely reversible”. He argues that a structure must be “governed from within” to be real, and defines the subject as “the center of functional activity”. Read it with the caveat that the skim saw the argument in fragments. |
| LIT-840 v2 | Asymptotic Methods in Statistical Decision Theory | Le Cam (1986), Springer Series in Statistics, xxvi + 742 pp. Asymptotic statistics organised around Blackwell’s experiments, Wald’s decision theory and Le Cam’s deficiency distance between experiments, with approximation by Gaussian shift and exponential families as the central theme. Not read: no lawful full text; known here from its front matter. | Unavailable — filed 2026-10-09 from catalogue records and the publisher’s free front matter, not read: no lawful full text was reachable. Springer sells the book and its chapters (OpenAlex: closed access, no repository copy); the Internet Archive copy is lending-only (access-restricted, printdisabled); HathiTrust holds it search-only. The front matter Springer serves freely (26 pages: contents, acknowledgments and the author’s “To the Reader” with a chapter-by-chapter summary) was read; the main text was not. No NOTE is filed. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-841 | Semantic Unification: A Sheaf Theoretic Approach to Natural Language | Abramsky & Sadrzadeh (2014), ARXIV-1403.3351, in the Lambek Festschrift (LNCS 8222, pp. 1–13). Models basic Discourse Representation Structures as sections of a presheaf of consistent literals over (vocabulary, variables), an anaphoric resolution as a cover by variable maps, and the meaning of a discourse as the gluing of its sentences’ sections, unique when it exists. Failure to glue is inconsistency, not contextuality in the Abramsky–Brandenburger sense; competing resolutions are competing covers, ranked in the probabilistic section by corpus counts. | Active — read in full 2026-10-09 (NOTE-644); worth reading as the first paper to model discourse meaning as sheaf-theoretic gluing, and as the record’s earliest bridge between the Abramsky–Brandenburger framework and natural-language semantics. A basic Discourse Representation Structure is a section of a presheaf of consistent literals; a choice of anaphoric resolution is a cover, a jointly surjective family of variable maps; and the meaning of the discourse is the gluing, which is unique when it exists (Proposition 1, its only result). Its “contextuality” is context dependence, not the failure of a compatible family to glue: the only obstruction it exhibits is inconsistency of literals. Its probabilistic section ranks candidate covers by summed corpus counts rather than gluing distributions. Short and preliminary by its own account. |
| LIT-843 | Combining contextuality and causality: a game semantics approach | Abramsky, Barbosa & Searle (2024), ARXIV-2307.04786, Phil. Trans. R. Soc. A 382:20230002. Adds an enabling relation to a measurement scenario and reads contextuality as a game: the Experimenter chooses measurements, Nature chooses outcomes, and Nature’s deterministic strategies replace the event sheaf. Recovers flat contextuality and Gogioso–Pinzani causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic strategies can fail to glue or glue non-uniquely. | Active — read in full 2026-10-09 (NOTE-646); worth reading as the version of sheaf-theoretic contextuality in which measurements are performed in a causal order and the choice of the next one may depend on earlier outcomes. A causal measurement scenario adds an enabling relation (Kahn and Plotkin’s information matrices renamed); Nature’s deterministic strategies over it replace the event sheaf, and the Abramsky–Brandenburger definitions of empirical model and contextuality then go through unchanged. Flat scenarios and Gogioso and Pinzani’s Bell scenarios with a causal order come out as special cases, and Experimenter strategies express adaptive measurement, as in the Anders–Browne GHZ OR-gate. Its sharpest point is that the strategy presheaf is not a sheaf: deterministic, compatible local strategies can fail to glue (Example 7.2) or glue non-uniquely (Example 7.3). A short paper; several results are deferred to an extended version. |
| LIT-845 | Closing Bell: Boxing black box simulations in the resource theory of contextuality | Barbosa, Karvonen & Mansfield (2021), ARXIV-2104.11241, in Samson Abramsky on Logic and Structure in Computer Science and Beyond (Springer, 2023). Characterises which convex maps between the empirical models of two scenarios are induced by classical (non-adaptive) procedures: exactly those given by a non-contextual model of a “hom” scenario [S, T] whose outcomes are procedures. Non-local games are procedures into a one-bit scenario, and the hom construction makes scenarios with predicates a closed category. | Active — read in full 2026-10-09 (NOTE-650); worth reading as the result that says which transformations between the empirical models of two different scenarios are classical. A probabilistic procedure S → T (a mixture of deterministic procedures, each answering every measurement of T by a context of S and a post-processing of outcomes) induces a convex map EMP(S) → EMP(T). A convex map is so induced exactly when it comes from a non-contextual model, satisfying a simpliciality predicate, on a new “hom” scenario [S, T] whose outcomes are procedures. Compatible local procedures that do not glue into one global procedure are contextual models of [S, T]: “contextual simulations”. The construction makes scenarios with predicates a closed category. Non-adaptive only; the arXiv v2 corrects Theorem 44 of the published chapter. |
| LIT-846 | Categories of Empirical Models | Karvonen (2018), ARXIV-1804.01514, QPL 2018, EPTCS 287:239–252 (2019). Defines simulations between empirical models on different scenarios: a simplicial relation from target measurements to sets of source measurements, and a stochastic outcome map natural in the context. Non-contextual models are those simulable from the terminal model, the non-contextual fraction is monotone along simulations, strong contextuality is reflected, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-651); worth reading as the paper that first gives empirical models on different measurement scenarios a notion of morphism: a simulation d → e answers each measurement of e by a set of jointly measurable measurements of d (a simplicial relation, so the cover may change) and a stochastic map of outcomes natural in the context. With it, non-contextual models are exactly those simulable from the terminal (empty) model, the non-contextual fraction can only grow along a simulation, strong contextuality is reflected backwards, Graham reductions are simulations (Vorob’ev’s theorem), and no contextual model can be cloned. It is restricted to no-signalling models and has no preprocessing. |
| LIT-847 | A comonadic view of simulation and quantum resources | Abramsky, Barbosa, Karvonen & Mansfield (2019), ARXIV-1904.10035, LICS 2019. Measurement protocols form a comonad on empirical models; a simulation of e by d is a co-Kleisli map from d ⊗ c (c non-contextual) to e. Simulability coincides with convertibility by a term in the free operations of the resource theory of contextuality, including a new conditional measurement; the non-contextual fraction is monotone, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-647); worth reading as the paper that makes simulation between empirical models adaptive and shows it is the same thing as convertibility by free operations. Measurement protocols (decision trees of compatible measurements) form a comonad MP on scenarios and on empirical models; a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Such a simulation exists exactly when e is obtained from d by a term in the free operations (relabelling and translation of measurements, coarse-graining, mixing, choice, tensor, and the new conditional measurement), with an equational theory and normal forms. Along any simulation the non-contextual fraction cannot fall, and no contextual model can be cloned. |
| LIT-856 | Small Singular Values Matter: A Random Matrix Analysis of Transformer Models | Staats, Thamm & Rosenow (2024; NeurIPS 2025). Taking the Marchenko–Pastur law as the zero-information null for a weight matrix, trained transformer matrices (BERT, Pythia-410M, Llama-3.1-8B) show outliers above the bulk and, in rectangular matrices only, below it, since only there does the null spectrum have a positive lower edge. In the MLP projections the singular vectors of both kinds of outlier align with activation-covariance eigenvectors, and zeroing the smallest decile of a rectangular matrix can hurt more than zeroing most larger ones. A linear teacher–student ensemble yields a below-bulk outlier when the noise is suppressed along a learned direction. | Active — read 2026-10-09 (NOTE-659); worth reading here as the paper that uses the Marchenko–Pastur law as a null model for where a trained weight matrix holds information, and so notices what a magnitude-ordered reading misses: a rectangular matrix’s null spectrum has a lower edge above zero, so learning can push singular values below the bulk as well as above it. In three transformers the departures line up with activation-covariance directions, at the lower end in the MLP up and gate projections, and zeroing the smallest decile of a rectangular matrix can cost more than zeroing most larger deciles; in square matrices it never does. A two-layer linear teacher–student ensemble produces such a lower outlier when the loss suppresses the noise along a learned direction. The evidence is measured, not proved, and it is uneven: overlap and ablation importance disagree for several matrix types, and the decile ablation does not isolate the outliers it is meant to test. |
| LIT-857 | Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws | Boncoraglio, Erba, Troiani, Xu, Krzakala and Zdeborová (2025), arXiv. For a tied single-head softmax attention layer trained by ridge-penalized ERM on Gaussian token sequences labelled by an attention-indexed target, in the limit n/d², p/d fixed, a six-parameter variational problem gives the exact train and test error, interpolation and perfect-recovery thresholds, and the singular-value law of the learned query–key map: η·ReLU(S0 + δZ − ε), Z a GOE matrix. Weight decay is a nuclear-norm penalty on that map, so the factorized model beats direct training of the product. Power-law targets give sequential mode recovery and LASSO- rate scaling laws. Non-rigorous (replica/AMP), checked against Adam. | Active — read 2026-10-09 (NOTE-661); worth reading as a solvable model in which the spectrum of trained attention weights is derived, not just measured: for empirical risk minimization of one tied softmax attention layer on Gaussian sequences, with n of order d², the learned query–key map is, in law, a rescaled ReLU of the target plus a semicircle noise bulk shifted by a threshold, so its outliers are recovered target directions and its bulk is finite-sample noise. Weight decay on the factor is exactly a nuclear-norm penalty on the product. With a power-law target, modes are recovered one by one as the threshold moves, which gives heavy-tailed learned spectra and power-law learning curves whose exponents match the LASSO rates. The main results are replica/AMP “Claims”, not theorems, checked against Adam runs at d = 100–400. |
| LIT-860 | Symmetry in language statistics shapes the geometry of model representations | Karkada, Korchinski, Nava, Wyart and Bahri (2026), ICML 2026 (PMLR 306:56137–56172). If co-occurrence among words on a latent continuum depends only on their separation, a full-rank spectral word embedding places them on Fourier modes of the latent coordinate: circles for cyclic concepts, open curves with harmonic ripples for intervals, curved sheets for maps, all proved for that model with the exponential kernel solved exactly. Wikipedia’s month and year statistics fit the symmetry; Gemma 2 and EmbeddingGemma show matching PCA pictures; the geometry survives removing the words’ mutual co-occurrences, which a seasonal latent variable shared by many words explains. | Active — read 2026-10-09 (NOTE-664); worth reading as a derivation of the circles, rippled curves and curved sheets found in language-model representations of months, years and places from one property of the data: when the co-occurrence of two words on a latent continuum depends only on their separation, the matrix the embedding factorises is circulant or Toeplitz, so its eigenvectors are sinusoids of the latent coordinate. For a full-rank spectral word embedding the paper proves the principal components are Fourier modes with amplitudes set by the kernel’s Fourier transform, solves the open-boundary exponential kernel exactly (a Sturm–Liouville problem), and bounds linear decoding of the coordinate by the missing Fourier tail (ε² ≲ r^(−1/D)). It measures the symmetry in Wikipedia statistics, shows the month circle survives zeroing every month–month co-occurrence, and explains that by a seasonal latent variable that couples many words. The “word embeddings” are explicit factorisations of the co-occurrence matrix, not trained networks; the language-model evidence is PCA pictures of Gemma 2 2B and EmbeddingGemma; the robustness argument is a perturbation sketch. |
| LIT-861 | Topological Invariance and Breakdown in Learning | Yang, Poggio, Chuang and Ziyin (2025), ARXIV-2510.02670. For any permutation-equivariant update rule whose map is K-Lipschitz, one step of size η changes the distance between any two neurons by a factor in [1 − ηK, 1 + ηK], and coincident neurons stay coincident. Below η* = 1/K the induced map on the set of neuron vectors is therefore a homeomorphism, measure-preserving for the pushed-forward neuron distribution and a C¹ diffeomorphism under a smoothness condition; above it only a continuous surjection is guaranteed. The paper reads this, with the edge of stability, as two phases of training, the second a topological simplification; the proofs hold, but that reading and the Betti-number experiments go beyond them. | Active — read 2026-10-09 (NOTE-663); worth reading as a short, correct proof that permutation symmetry constrains how neurons can move under training: if the update rule is permutation-equivariant (gradient descent, SGD or Adam on a permutation-symmetric loss), neurons that coincide stay coincident at every step size, and if the update map is K-Lipschitz each step moves every pair of neurons apart or together by at most a factor 1 ± ηK. So below η = 1/K the induced map on the set of neuron vectors is a bi-Lipschitz homeomorphism (a C¹ diffeomorphism under a smoothness condition) and distinct neurons cannot merge in finitely many steps; above it only a continuous surjection is guaranteed. The proofs are a few lines and hold. What they do not carry is the paper’s larger picture: for finitely many neurons a homeomorphism of the neuron set is only a bijection, “topological simplification” above 1/K is permitted, not shown, and the experiments measure Betti numbers of point clouds at a fixed scale, which the theorem does not protect. The large-step runs show the clouds fragmenting (more components), not merging, and in the Adam runs the step size times the measured top Hessian eigenvalue stays below 0.05 where the topology changes. |
| LIT-866 | Hyperbolic Entailment Cones for Learning Hierarchical Embeddings | Ganea, Bécigneul and Hofmann (ICML 2018), PMLR 80:1646–1655. Represents a partial order in the Poincaré ball by nested cones: v is below u when v lies in u’s cone. Nesting forces apertures of at most π/2 that narrow toward the boundary (r sin ψ/(1 − r²) non-increasing), so cones are undefined near the origin; the widest family is sin ψ = K(1 − r²)/r, with a closed-form membership test. Overlapping cones admit several parents. On WordNet noun closure link prediction in 5–10 dimensions it beats Poincaré and order embeddings once some closure edges are in training; with none, all methods fail. | Active — read 2026-10-09 (NOTE-671); worth reading as the paper that represents a partial order, not only a tree, by containment of regions in hyperbolic space. Each concept is a point x of the Poincaré ball and owns a cone S_x, the image under the exponential map of the tangent vectors within an angle ψ(x) of the outward ray through x; v counts as entailed by u when v lies in S_u. Requiring the cones to be axially symmetric, rotation invariant, continuous in aperture and nested (v in S_u implies S_v inside S_u) forces ψ ≤ π/2 (Lemma 2) and makes r sin ψ(r)/(1 − r²) non-increasing (Theorem 3), so no cone can be defined near the origin; the widest family is sin ψ(x) = K(1 − ‖x‖²)/‖x‖ outside a ball of radius ε, and membership is a closed-form angle test (Theorem 5). That family is transitive (Theorem 4), but the proof is only said to be “similar” and is not given. Cones of incomparable points can overlap, so a node may sit below several parents. The evidence that learned cones fit a DAG is one WordNet-noun experiment: held-out edges of the transitive closure, test F1 at 5 and 10 dimensions, single runs, where hyperbolic cones lead once 10% or more of the closure is in training but every method fails (F1 27–43%) when only the transitive reduction is, and Euclidean cones initialised from Poincaré embeddings match the hyperbolic ones. |
| LIT-867 | Emergent Hyperbolic Network Geometry | Bianconi and Rahmede (2017), Scientific Reports 7, 41974. Growing simplicial complexes, made by gluing d-simplices to (d − 1)-faces with probability ∝ 1 + s n_α (flavor s = −1, 0, 1), have a tree as their dual and, drawn with equal links as ideal simplices in the Poincaré ball, fill hyperbolic d-space: the hyperbolic geometry is an outcome of a purely combinatorial rule rather than its cause. Degrees are exponential for d + s = 1 and power-law with γ = 2 + 1/(d + s − 1) ≤ 3 for d + s > 1; clustering and modularity are high for d ≥ 2. Hyperbolicity is shown by construction, conditional on equal link lengths, not proved. | Active — read 2026-10-09 (NOTE-678); worth reading as the model that turns the hidden-metric account of complex networks around: a simplicial complex grown by gluing d-simplices to (d − 1)-faces, with no geometry in the rule, comes out small-world, clustered, modular and (for d + s > 1) scale-free, with γ = 2 + 1/(d + s − 1) from a master equation. Its dual is a tree, so with every link given equal length each simplex is drawn as an ideal simplex of the Poincaré ball and each new node at the normalised sum of its parent face’s positions. That embedding, not a proof, is the paper’s case that the geometry is hyperbolic, and the authors concede it rests on the equal-length assumption (unequal lengths fit a sphere) and that the ideal-simplex distances are infinite. A fitness on faces gives a numerically observed transition in which the complex grows in one direction. |
| LIT-870 | Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning | Lin, Coifman, Mishne and Talmon (2023), ICML 2023 (PMLR 202), ARXIV-2305.18962. Recovers a latent tree from point-cloud or graph data without learning: diffusion densities at dyadic time scales are embedded in a product of Poincaré half-spaces, and the ℓ1 product distance, a scale-weighted Hellinger sum, is proved equivalent up to constants to a thresholded snowflake min{1, d_T^(2α)} of the tree metric when the operator behaves like the heat kernel on a tree. It keeps local tree neighbourhoods (best MAP on the benchmarks) but not global distances (higher distortion than hyperbolic MDS). | Active — read 2026-10-09 (NOTE-675); worth reading as a construction that recovers a latent tree from unlabelled data by distances alone, with a proof of what is recovered. Each point is represented by its diffusion densities P^(2^−k) e_i at dyadic times, each density is placed in a Poincaré half-space at a height set by its scale, and the hyperbolic diffusion distance is the ℓ1 sum over scales of the hyperbolic distances, 2 sinh⁻¹(2^(1−kα)‖φ_i^k − φ_j^k‖) with φ the square roots of the densities, a scale-weighted Hellinger distance. Theorem 1 (Appendix C, after Leeb and Coifman) says that for the heat kernel on a tree this is equivalent up to constants to min{1, d_T^(2α)}, 0 < α < 1/2: a thresholded snowflake of the tree metric, not the tree metric, and saturated beyond a fixed radius; the main text states it without the threshold. The tree heat-kernel bounds are stated, not proved, and one step of the Hölder bound (Proposition C.5) holds only for tree distances of at least 1, so the admissible range of α is not secured. Empirically it gives the best mean average precision on five graph benchmarks and two single-cell datasets and worse average distortion than hyperbolic MDS or Sarkar’s construction: local neighbourhoods are kept, global distances are not. Nothing in it concerns attributes or linear directions. |
| LIT-872 | Hyperbolic geometry of the olfactory space | Zhou, Smith & Sharpee (2018), Science Advances 4(8):eaaq1458. In four natural odour sources the rank order of correlations between compound concentrations has the clique-topology signature of points near the surface of a 3D hyperbolic ball and not of a uniform Euclidean cube, and human perceptual descriptions of 127 odorants have that of a full 3D hyperbolic ball. Pleasantness, boiling point and acidity read out as linear axes of the fitted fruit-odour space. The hierarchy that motivates the hyperbolic model is argued from biochemistry, not recovered from the data. | Active — read 2026-10-09 (NOTE-672); worth reading as the first application of clique topology to stimulus statistics rather than neural data, and as the Sharpee lab’s first claim of a hyperbolic geometry in biology. For four natural odour sources (strawberry, tomato, blueberry, mouse urine; 45–78 compounds over 50–101 samples), the rank order of absolute concentration correlations gives Betti curves that match points sampled near the surface of a 3D hyperbolic ball (R_max = 7, R_min = 0.9 R_max, the same for all four) and reject uniform Euclidean cubes of any dimension tried. Human descriptor profiles for 127 odorants (Dravnieks) match a full 3D hyperbolic ball and no Euclidean cube. Linear axes for pleasantness, boiling point and acidity are found in the fitted fruit-odour space. The tree that motivates the hyperbolic model is argued, not recovered: compounds do not cluster by chemical class, and the only null geometry is a uniform Euclidean cube. |
| LIT-874 | Representation Tradeoffs for Hyperbolic Embeddings | Sala, De Sa, Gu and Ré (ICML 2018), PMLR 80:4460–4469. Trees embed in the hyperbolic plane with distortion 1 + ε (Sarkar’s construction), but a hyperbolic distance d costs about d bits, so the embedding needs Θ((ℓ/ε) log deg_max) bits per coordinate in the plane: logarithmic in branching, linear in depth (lower bound proved for H², with an approximation). Higher dimension, via coding-theoretic child placement, divides the upper bound by r. Also an exact hyperbolic MDS by PCA after centring at a new mean, with a perturbation bound growing as sinh²(diameter), and a local-convexity condition for hyperbolic PGA. | Active — read 2026-10-09 (NOTE-679); worth reading as a quantitative account of why trees fit hyperbolic space and what the fit costs. Following Sarkar, any finite tree embeds in the hyperbolic plane with worst-case distortion at most 1 + ε by placing each node’s children evenly on a hyperbolic circle of radius τ around it; but a hyperbolic distance d needs about d bits to write down, so the coordinates need O((ℓ/ε) log deg_max) bits, ℓ the longest path, and an explicit tree of deg_max chains shows Ω((ℓ/ε) log deg_max) is needed in the plane. Precision grows with the logarithm of the branching and linearly with depth: hyperbolic space is cheap for short bushy hierarchies and dear for long chains. In r dimensions, children placed on hypercube vertices chosen by a Hadamard code divide the cost by r up to r ≈ log deg_max (upper bound only). For general metrics it gives an exact hyperbolic MDS: cosh of the distances is uuᵀ − XXᵀ once the points are centred at a new pseudo-Euclidean mean, so PCA recovers them. The lower bound’s proof uses a small-angle approximation and covers only the plane; the PGA guarantee is local convexity, not global recovery; the experiments, including the WordNet MAP of 0.989 in two dimensions, are reported at ε = 0.1, where WordNet takes 2,877 bits per coordinate, while the introduction quotes the ε = 1 figure of about 500. |
| LIT-875 | An argument for hyperbolic geometry in neural circuits | Sharpee (2019 online), Curr. Opin. Neurobiol. 58:101–104. A four-page review arguing that neural and other biological circuits should have hyperbolic geometry, because hyperbolic space approximates trees, hyperbolic networks are navigable, and Zipf’s law, read as states growing exponentially with log-improbability and taken as the mark of maximally informative, latent-driven systems, shares that exponential growth. Its evidence is the olfactory embedding of Zhou, Smith & Sharpee (2018) and earlier perception work; the Zipf-to-hyperbolic link is an analogy, not a derivation. | Active — read 2026-10-09 (NOTE-670); worth reading as the short programmatic statement, by the lab behind the olfactory and hippocampal hyperbolic findings, of why hyperbolic geometry should be expected in neural circuits, and as a map of the literatures it joins: hyperbolic random graphs and greedy routing, Zipf’s law and statistical criticality, and maximally informative representations. Its own argument is a chain of analogies, not a derivation. Zipf’s law is rewritten as a count of states growing exponentially with energy (−log probability), and exponential growth of states is a hallmark of trees and of hyperbolic space, so the two are said to be connected; but exponential growth of states holds for any system with extensive entropy, and what singles out Zipf (unit slope) plays no part in the step to geometry. A rigidity argument for three dimensions cites Mostow’s theorem, which concerns finite-volume hyperbolic manifolds of every dimension ≥ 3 and says nothing about point clouds. The one piece of data, a 3D Poincaré-ball embedding of fruit odorants with a pleasantness gradient, is reproduced from Zhou, Smith & Sharpee (2018). |
| LIT-876 | Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience | Zhang, Rich, Lee & Sharpee (2022 online, Nat. Neurosci. 26, 2023). The pairwise correlations of rat dorsal CA1 place cells have the topological signature of points sampled uniformly from a three-dimensional hyperbolic ball, not of a low-dimensional Euclidean cube, and the ball’s radius grows with the logarithm of exploration time. Place-field sizes are near exponential, and in a Poisson model exponential sizes carry more positional Fisher information than uniform or log-normal ones, with an optimal radius that matches the measured one for the number of CA1 neurons. | Active — read 2026-10-09 (NOTE-668); worth reading as, on its own account, the first evidence that neural responses, not stimuli, carry a hyperbolic geometry. Pairwise spike-train correlations of a few dozen active dorsal CA1 cells per session (34 to 56 in the sessions shown), in rats on a novel 48-m track, a 2.5-m track and a 1.8-m box, give Betti curves (clique topology, invariant to any monotone transform of the correlations) that match points sampled uniformly from a 3D hyperbolic ball and fail to match uniform samples from Euclidean cubes of low dimension or shuffled spikes. The fitted radius (10.5–15.5 in units of inverse curvature) grows with the logarithm of exploration time, on scales from seconds to days, as the maximal information acquirable does. Place-field sizes are close to exponential, as the model requires, and in simulation exponentially distributed field sizes give more Fisher information than uniform or log-normal ones, with an optimal radius growing as log N that extrapolates to the radius measured. The “hierarchy” is one of nested place-field sizes; no tree is recovered from the data, and the only null geometry tested is Euclidean with uniform sampling. |
| LIT-879 v2 | Hyperbolic Neural Networks | Ganea, Bécigneul and Hofmann (NeurIPS 2018), ARXIV-1805.09112. Builds neural-network layers in the Poincaré ball from Möbius gyrovector operations: closed-form exponential and logarithmic maps, parallel transport from the origin as a scaling, geodesic hyperplanes with a closed-form point-to-hyperplane distance, and from them hyperbolic logistic regression, feed-forward layers and GRUs, all reducing to the Euclidean ones as the curvature goes to zero. Linear maps and nonlinearities are lifted through the origin’s tangent space, so the curvature acts only through biases and the output distances. Gains over Euclidean models are clear only on a synthetic tree-like task and on separating WordNet subtrees in low dimensions. | Active — read 2026-10-09 (NOTE-686); worth reading as the paper that gives the Poincaré ball the operations a neural network is built from, and so shows exactly where hyperbolic geometry enters such a network and where it does not. From Ungar’s gyrovector formalism (Möbius addition and scalar multiplication) it derives the exponential and logarithmic maps at any point, parallel transport from the origin as the scaling λ₀/λ_x (Theorem 4), geodesic hyperplanes as exp_p of a tangent orthogonal complement, and the closed-form distance to one (Theorem 5); all of these check out numerically. A linear map or pointwise nonlinearity is lifted as exp₀ ∘ f ∘ log₀, so a stack of such layers is one Euclidean network in log₀ coordinates; the curvature acts only through the bias translations between layers and the distance-based softmax. The evidence is small: 5-dimensional sentence encoders on SNLI, where hyperbolic models are on par or worse, and on a synthetic noisy-prefix task, where they win most when the noise is lowest, each cell the best of three runs; and hyperbolic logistic regression separating WordNet subtrees in pre-trained Poincaré embeddings better than Euclidean hyperplanes do, mostly in 2 to 5 dimensions. |
| LIT-890 | The power of deeper networks for expressing natural functions | Rolnick and Tegmark (2017; ICLR 2018). Extends LIT-887’s 2^n result: one hidden layer of smooth, bias-free units needs exactly ∏(rᵢ+1) neurons to Taylor-approximate x₁^r₁⋯xₙ^rₙ, a deep network O(Σ log rᵢ); sparse polynomials inherit the gap up to a factor 1/c; at depth k a product of n inputs costs at most about 2^(n^(1/k)), conjectured tight. The uniform-approximation lower bounds rest on an unproved step, and the gap grows with degree, not with the number of inputs. | Active — read 2026-10-09 (NOTE-693); worth reading as the sequel that turns LIT-887’s one depth-separation theorem into a family, and for where the family stops. Proved, for networks without biases and smooth activations with nonzero Taylor coefficients: one hidden layer needs exactly ∏(rᵢ+1) neurons to Taylor-approximate the monomial x₁^r₁⋯xₙ^rₙ, against Σ(7⌈log₂ rᵢ⌉+4) for a deep network; a sparse polynomial with c monomials costs at least 1/c of its dearest monomial; xᵈ costs d+1 shallow against 7⌈log₂ d⌉ deep; and a product of n inputs costs O(n^((k−1)/k)·2^(n^(1/k))) at depth k. The move to uniform approximation, its headline over LIT-887, rests for the lower bounds on a step the proof asserts and that is false in general (a network can be uniformly within ε of a polynomial while its Taylor coefficients diverge), so only the Taylor-sense lower bounds are established. That depth k needs 2^Θ(n^(1/k)) neurons is a conjecture, backed by one trained-network figure at n = 20 with tanh, outside the theorems’ hypotheses. The gap is exponential in a monomial’s degree, not in the number of inputs: polynomials of bounded degree, such as the low-order Hamiltonians LIT-887 calls natural, show none. |
| LIT-893 v3 | Linear Representations of Finite Groups | Serre (1977), DOI-10.1007/978-1-4684-9458-7, Graduate Texts in Mathematics. The standard textbook statement that, over a field of characteristic zero, characters classify finite-dimensional representations of a finite group up to isomorphism, with complete reducibility and the orthogonality of irreducible characters. Registered unread, as the source a new claim names for results it checks by reasoning. | Unavailable — registered 2026-10-10 from Crossref’s record, not read. Filed because a new claim grounds on it as the standard source for the classification of finite-group representations by characters; the results that claim uses were checked by reasoning, not against this book. It stays Unavailable until it is read, not on merit. Looked for on 2026-10-10: Springer sells it, and the only full text found online is an uploaded scan on a chat server with no sign of the publisher’s permission, so it was not used. |
| LIT-929 v2 | Generalized Stability Theory. Part II: Nonautonomous Operators | Farrell & Ioannou (1996), J. Atmos. Sci. 53, 2041. For a time-dependent operator A(t) there are no modes; the propagator’s singular vectors, the Lyapunov exponents and the Lyapunov vectors take their place. Its main result: an operator stable or neutral at every instant can be asymptotically unstable when it varies in time, only if it is non-normal, and random variation destabilizes generically, needing less variation the more non-normal and less damped the mean operator. Examples: a switched oscillator, random neutral matrices, the Lorenz system and a three-layer baroclinic model. | Active — read 2026-10-14 (NOTE-842) in full; worth reading for its main result: a time-dependent linear operator that is stable, or neutral, at every instant can be asymptotically unstable, only if it is non-normal, and random variation of the operator does this generically, with less variation needed the more non-normal and less damped the mean operator. Read it knowing that genericity is cited to stochastic stability theory rather than proved, that its examples are low-dimensional, and that in its deterministic examples variation much faster than the transient growth destabilizes nothing. Filed from it: THEORY-325. |
| LIT-930 v3 | Will a Large Complex System be Stable? | May (1972), Nature 238, 413. A large system of n components with randomly signed interactions of strength s and connectance C is almost surely stable when s√(nC) < 1 and unstable when it exceeds 1, with a sharp transition: past a point, more components or more connections destabilize. It challenged the view that diversity brings stability. Not read. | Unavailable — looked for a lawful full text on 2026-10-14 and found none, so it is not read and no NOTE is filed. Unpaywall: closed, no open location. OpenAlex: two locations, doi.org and PubMed 4559589, neither open. Europe PMC: PMID 4559589, not open access, no PDF. Semantic Scholar: closed. nature.com: the article page is paywalled (“Access through your institution”), and its .pdf URL returned that HTML page. A web search on the title in quotes found no author-posted or institutional copy; May died in 2020 and has no personal page with papers. The Internet Archive could not be reached from this client (connection reset), and a 1972 Nature paper would not be public domain there. It stays Unavailable until a copy is supplied. |
| LIT-931 v2 | Generalized Stability Theory. Part I: Autonomous Operators | Farrell & Ioannou (1996), J. Atmos. Sci. 53, 2025. For du/dt = Au with A non-normal, the eigenvalues fix only the t → ∞ behaviour. Growth over time t is the top singular value of e^{At}, set at t → 0 by the numerical abscissa, the top eigenvalue of (A + A†)/2. Systems with every mode damped amplify perturbations by large factors before decaying, and their forced and white-noise responses exceed the equivalent normal response. Worked in a 2 × 2 example and damped Eady models. | Active — read 2026-10-14 (NOTE-831) in full; worth reading as the compact statement of non-modal stability for a fixed linear operator: growth over a finite time is the top singular value of the propagator, the eigenvalues govern only the long-run limit, and forced and stochastic responses exceed those of the normal system with the same spectrum. Read it knowing that it is a synthesis of earlier results with worked atmospheric models, that non-normality is relative to the chosen norm, and that it is linear and autonomous. Filed from it, with LIT-932: THEORY-323. |
| LIT-932 v2 | Hydrodynamic Stability Without Eigenvalues | Trefethen et al. (1993), Science 261, 578. Plane Couette and Poiseuille flow become turbulent at Reynolds numbers where eigenvalue analysis says they are stable. Their linearized operators are far from normal: with every mode decaying, perturbations grow transiently by O(R) and resonate by O(R²), and an operator perturbation of O(R⁻²) makes them eigenvalue-unstable. Pseudospectra, not eigenvalues, show it. That nonlinear mixing then carries the amplified disturbance into turbulence is conjectured, from a 2 × 2 model with threshold O(R⁻³). | Active — read 2026-10-14 (NOTE-839) in full; worth reading as the short statement of why eigenvalue analysis fails for shear flows: the linearized Couette and Poiseuille operators are far from normal, so perturbations grow transiently by O(R), resonate by O(R²), and an operator perturbation of O(R⁻²) destabilizes, all read off the pseudospectra. Read it knowing that the transient growth results are credited to earlier work, and that its account of transition, linear growth plus energy-neutral nonlinear mixing with threshold O(R^γ), γ < −1, is a conjecture from a 2 × 2 model. Filed from it: THEORY-304, and with LIT-931, THEORY-323. |
| LIT-948 v2 | Distances for Markov chains from sample streams | Calo et al. (2025). The bisimulation metric between two finite Markov chains, in its bicausal optimal-transport form (LIT-962), is the value of a new linear program whose constraints use each chain’s discounted occupancy measure over consecutive state pairs in place of its transition kernel. A stochastic primal-dual method on it, SOMCOT, estimates the metric to accuracy ε with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) independent transition samples per chain, with no model of the dynamics. The samples are independent draws from the occupancy measures, which needs restarts from the initial state. The states must be observed and enumerable, and the ground cost known. Experiments on block-structured random walks and a discretised pendulum. | Active — read 2026-10-14 (NOTE-763) in full; worth reading as the first estimator of the bisimulation metric between finite Markov chains with a guarantee and no transition model. A reformulated LP needs only each chain’s sampled occupancy of consecutive state pairs, and a stochastic primal-dual method reaches accuracy ε with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) samples per chain. Read it knowing that the samples are independent draws from the discounted occupancy measure, which needs restarts, not one running stream. The states must be observed and enumerable, and the cost on state pairs given. The chains have no actions. Filed as THEORY-280. |
| LIT-955 v2 | Tangled webs: A practical investigation of graph tangles. | Salter-Duke (2024), PhD thesis, Victoria University of Wellington. Tangles (Robertson and Seymour 1991) define a cohesive region as one every low-order cut points towards. The thesis gives an exact but exponential algorithm for edge-connectivity tangles and tests them on eleven small networks and small images. Tangle communities match standard ones only where community structure is strongly delineated; three networks have no distinguishable tangles at the orders reached; tangles of one to three vertices recur; one-pixel lines cannot be distinguished. The author finds the notion “not as compelling as I had hoped”. | Active — read 2026-10-14 (NOTE-814) in full; worth reading as an early empirical test of tangles, the Robertson–Seymour notion of a region that cannot be pulled apart, as a criterion of cohesion: an exact low-order algorithm, eleven small real networks and some small images. Read it knowing that the result is mostly negative (tangle communities match standard ones only where structure is strongly delineated, tangles of one to three vertices recur, thin shapes cannot be seen), that turning a tangle into a vertex set needs the author’s assignment rule, and that only small graphs and low orders were computable. Registered for QUESTION-035; THEORY-254 is filed from it. |
| LIT-956 v2 | Back and forth between algebraic geometry, algebraic logic, sheaves and forcing | Sayed Ahmed (2018). Distributive lattices with join-preserving idempotent operators are represented, after Comer, as continuous sections of a sheaf of quotient algebras over the Priestley space of their fixed-point part; a small site is said to give a notion of forcing, with a Łukasiewicz conjunction forced through a tensor product, and many-valued theories are interpreted in a quasi-topos. Most theorems are stated with sketched proofs or none, and negation is never written out. | Active — read 2026-10-14 (NOTE-817) in full; worth reading only as a pointer to Comer’s sheaf duality for algebras of formulas, extended here to distributive lattices with operators, and to presheaf forcing with an added Łukasiewicz clause. Read it knowing that its main theorems are stated with sketched proofs or none, that it never writes negation or a pseudo-complement, and that it concerns distributive lattices and forcing posets, not contexts. It does not supply the Heyting-over-contexts machinery QUESTION-055 was filed for; no THEORY is filed from it. |
| LIT-962 v2 | Bisimulation Metrics are Optimal Transport Distances, and Can be Computed Efficiently | Calo et al. (2024). For two finite Markov chains without actions, with real-valued state labels r and discount γ, the bisimulation metric of Desharnais et al., van Breugel and Worrell, and Ferns et al. equals the optimal-transport distance between the chains’ path laws over bicausal couplings, with ground cost (1−γ)|r(x)−r(y)|. That distance is 1/(1−γ) times the value of a finite linear program over discounted “occupancy couplings” of state-pair transitions. Entropy regularisation of the program gives Sinkhorn Value Iteration, with an O((1−γ)⁻⁵ε⁻²) iteration bound under exact inner projections, and in experiments it is faster than earlier dynamic-programming solvers. | Active — read 2026-10-14 (NOTE-757) in full; worth reading as the paper that makes approximate bisimulation a transport cost. For finite action-free chains with real-valued labels, the discounted bisimulation metric solves the same fixed-point equation as optimal transport over bicausal couplings with ground cost (1−γ)|r(x)−r(y)|, so the two are one object, and that cost is the value of a finite linear program over occupancy couplings. Read it knowing three things. Its chains have no actions, so it does not reach systems driven by utterances, which is what QUESTION-008 asks about. Its only statement about trace equivalence is second-hand and one-way. Its Sinkhorn Value Iteration guarantee needs exact inner fixed points. Filed as THEORY-247. |
| LIT-981 v3 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps | Coifman et al. (2005). A symmetric, positivity-preserving, positive semi-definite affinity kernel is normalized into a Markov operator that is conjugate to a symmetric one, ã. Its eigenfunctions, scaled by powers of their eigenvalues, embed the data so that Euclidean distance equals the diffusion distance D_m induced by ã^(m), exactly; truncation is claimed to relative accuracy. A one-parameter normalization yields the graph Laplacian, the backward Fokker–Planck operator, or the Laplace–Beltrami operator, the last independent of sampling density. The convergence theorem and the α-family are stated without proof. Non-symmetric kernels are not treated. | Active — read 2026-10-30 (NOTE-1441) in full, from the published PDF on co-author Zucker’s Yale page (www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_1.pdf), with every equation seen; it replaces the 2026-10-14 skim of the PubMed Central text (NOTE-784), whose equations were not visible. Worth reading for QUESTION-052 as the classical symmetric case: from a symmetric, positivity-preserving, positive semi-definite affinity, the normalised walk is conjugate to a symmetric operator, and its eigenfunctions scaled by λ^m embed the data so that Euclidean distance equals a diffusion distance; a density-corrected normalisation (α = 1) has the Laplace–Beltrami operator as its small-ε limit, independent of the sampling density. Read it knowing that its diffusion distance is the one induced by the symmetric kernel ã^(m), not the inverse-stationary- weighted distance of Coifman and Lafon’s companion (LIT-1246); that its truncation is claimed to relative accuracy, the defect THEORY-745 records for the companion; that Theorem 2.1 and the α-family are stated without proof, with the proofs in LIT-1246; that Eq. 5 and Eq. 14 give the α = 0 potential opposite signs; and that its “directed diffusion” is anisotropic, not a non-reversible chain. |
| LIT-993 v2 | Functional maps | Ovsjanikov et al. (2012). A map between two shapes is represented as the linear operator it induces between their function spaces, a matrix in Laplace–Beltrami eigenbases. Volume-preserving bijections give orthogonal matrices, and isometries map each eigenspace to the matching one, so near-isometries are compact and nearly block-diagonal in about 30–100 eigenfunctions. Descriptor, landmark, segment and commutativity constraints become linear, and matching is one least-squares solve plus spectral ICP, which beat the 2011 state of the art on SCAPE and TOSCA. | Active — read 2026-10-14 (NOTE-822) in full; worth reading for QUESTION-052 as a worked construction of a correspondence as a linear operator between two function spaces, truncated to low Laplace–Beltrami eigenspaces: a volume-preserving bijection gives an orthogonal matrix, and an isometry maps each eigenspace to the matching one. Read it knowing that every structural result assumes a bijection, so it offers no construction for one-way or non-invertible maps, and that the stability of truncated eigenspaces is cited, not shown. |
| LIT-1051 | A Pseudorandom Generator from any One-way Function | Håstad, Impagliazzo, Levin and Luby (1999), SIAM Journal on Computing 28(4). There is a pseudorandom generator if and only if there is a one-way function (Theorem 6.2.2), in the uniform setting: adversaries are algorithms, and every reduction is an oracle Turing machine that turns an adversary for the constructed generator into an inverter for the given function. The route runs one-way function → false entropy generator → pseudoentropy generator → pseudorandom generator, where “computational entropy” is the Shannon entropy of some P-samplable distribution that no adversary of the given strength tells apart from the output. The general reduction is only weak-preserving: composed literally it needs seeds of length n^34, and the paper’s direct construction n^10. The paper attributes to Goldreich [G : 89] the equivalence of pseudorandom generators with P-samplable pairs that are computationally indistinguishable but statistically very different; with Theorem 6.2.2 that gives the one-way function form, but the paper never writes that combined sentence. | Active — read in full 2026-10-15 from the authors’ copy on Johan Håstad’s KTH page (www.csc.kth.se/~johanh/prgfromowf.pdf, a 46-page typescript, linked from ~johanh/papers.html; not the typeset SIAM pages 1364–1396, which were not checked). Unpaywall and OpenAlex report the DOI closed; OpenAlex lists two CiteSeerX records (10.1.1.35.3930, 10.1.1.36.6283), not needed once the KTH copy was found. Worth reading as the source of the uniform theorem “There are one-way functions iff there are pseudorandom generators” (Theorem 6.2.2) and of computational entropy: the Shannon entropy of a P-samplable distribution that no adversary of a given time-success ratio tells apart from the output, with pseudoentropy (more than the input’s entropy) and false entropy (more than the output’s own) as the intermediate primitives. All adversaries are uniform and every reduction is an oracle Turing machine; there are no circuits and no min-entropy in the paper, and the general reduction is only weak-preserving (seed length n^10 even in the direct construction). For CLAIM-302 and CLAIM-304 it shows that “if pseudorandom generators exist” may be read “if one-way functions exist”, and that the claims’ rendering G(U_n) is a false entropy generator with false entropy at least n, which names the gap they construct. For LIT-1053 it states, in Section 8, that Goldreich’s note [cited as ICSI TR-89-051, July 1989] shows pseudorandom generators equivalent to “a pair of P-samplable distributions which are computationally indistinguishable but statistically very different”; the one-way-function form follows only by combining that with Theorem 6.2.2, which the paper does not write out. For QUESTION-110 its time-success ratio is a worked cost-of-recovery measure that folds the success criterion into the cost, and for QUESTION-103 its distinguishing advantage is hit rate minus false-alarm rate, described by the paper itself as “the computationally restricted adversary definition of statistical distance”; it says nothing about real observers. |
| LIT-1053 v2 | A note on computational indistinguishability | Goldreich (1990), Information Processing Letters 34(6), 277–281. As cited in Goldreich’s Foundations of Cryptography (§3.8.2 of the 1995 draft): a pair of polynomial-time constructible ensembles that are computationally indistinguishable yet statistically far apart exists if and only if one-way functions exist. Not yet read; the summary is the book’s description. | Unfetched — filed 2026-10-15, not read. A lawful full text exists but could not be fetched from this environment on 2026-10-15: found but could not be fetched. Goldreich posts the paper on his Weizmann pages as www.wisdom.weizmann.ac.il/~oded/PSX/iplnote.pdf, linked from his list of papers (~oded/papers.html, where it is dated 1989) and from his annotated bibliography (~oded/annot/node47.html). Web search indexes that PDF; the search tool’s summary of it, which was not checked against the text, says it began as a Technion technical report of January 1990 and that it states the existence of pseudorandom generators to be equivalent to the existence of efficiently constructible distributions that are computationally indistinguishable yet statistically far apart. That differs from the book’s paraphrase, which says one-way functions. It cannot be settled without the text. Places looked, and what each gave: Unpaywall reports the DOI closed, with no OA location and no repository copy; OpenAlex reports it closed, with no repository full text, listing only the DOI and two CiteSeerX records (10.1.1.145.7487, 10.1.1.39.458); Semantic Scholar gives openAccessPdf status CLOSED; the Elsevier article API reports openaccess 0 and openArchiveArticle false, and ScienceDirect returned 403, so IPL 34(6) is not in Elsevier’s open archive; Crossref lists only text-mining links under Elsevier’s TDM licence; DBLP served a bot challenge. Goldreich’s PDF itself was requested 22 times over about 20 minutes: https://www.wisdom.weizmann.ac.il gave an empty reply every time, http://www.wisdom.weizmann.ac.il gave 503 every time (“upstream connect error … connection termination”), and https://wisdom.weizmann.ac.il was refused by the proxy (502). WebFetch also got 503 for the PDF. The Internet Archive’s availability API reports a capture of the PDF dated 2026-01-23 (web.archive.org/web/20260123154034/https://www.wisdom.weizmann.ac.il/~oded/PSX/iplnote.pdf). web.archive.org and wayback.archive.org reset every connection from here, about 25 attempts in all; plain http to web.archive.org gave 403; WebFetch refuses web.archive.org; archive.ph also reset. The availability API has no capture of papers.html or of a .ps version. CiteSeerX’s download links now redirect to the Internet Archive, so they failed in the same way. The ECCC hosts (eccc.weizmann.ac.il, eccc.uni-trier.de) do not mirror Goldreich’s ~oded pages; they return their generic HTML page. The paper can probably be read once the Weizmann server or web.archive.org can be reached from this environment, or from any ordinary browser. What it says is known here only through Goldreich’s Foundations of Cryptography (LIT-1061, §3.8.2), which cites it for the equivalence with one-way functions without proving it. |
| LIT-1061 | Foundations of Cryptography | Goldreich (2001), Cambridge University Press. Volume 1 of the standard treatment of the complexity-theoretic foundations of cryptography: one-way functions, computational indistinguishability, pseudorandom generators and zero-knowledge proofs. | Active — read in part 2026-10-15 (Skimmed), and not in its printed form. The text read is the author’s 1995 draft, “Foundations of Cryptography (Fragments of a Book)” (23 February 1995, with the “Additional Fragment (Nr. 1)” of 9 February 1996), which Goldreich posted through ECCC (eccc.weizmann.ac.il/static/books/Foundations_of_Cryptography_Fragments_of_a_Book/; PDF at eccc.weizmann.ac.il/resources/pdf/foc.pdf, 305 pp.). The draft’s Chapters 1–3 were read in full; its zero-knowledge chapter, the 1989 annotated references and the 1996 addendum were not. The 2001 Cambridge text (DOI 10.1017/CBO9780511546891) is closed: Unpaywall and OpenAlex list no open location. The author’s own pages (wisdom.weizmann.ac.il/~oded/foc-book.html and related) and the Internet Archive’s capture of foc-book.html could not be reached from this environment. Worth reading as the standard source of the definitions of computational indistinguishability (Def. 3.2) and of a pseudorandom generator (Def. 3.9), and for the result that the first is strictly coarser than statistical closeness. For CASE-083 it confirms each ingredient: the generator definition, the 2⁻ⁿ image count (§3.3.1, stated there only as Δ ≥ ½ with Δ the unhalved sum; the count reappears in the proof of Prop. 3.13), and closure under efficient post-processing, which is set as Exercise 1 of §3.8.4 rather than proved. For CLAIM-302 it shows that the separation itself needs no assumption: Prop. 3.3 gives, by counting, an ensemble on at most 2^(n/2) strings that no polynomial-time test tells from uniform. Generators, or equivalently one-way functions, are needed only for an efficiently samplable pair (Goldreich 1990, cited in §3.8.2). Indistinguishability is defined only for ensembles indexed by n, never for a single pair of distributions. For CLAIM-304 it supplies the definitions; the encoding is the record’s own. Section numbers are the draft’s; the printed volume (Ch. 3 at pp. 101–183) was not checked. |
| LIT-1086 v2 | The difficulty of optimum index selection | Comer (1978), ACM TODS 3(4). Registered for QUESTION-114 as the result usually cited as “optimal index selection is NP-complete”. Unfetched, so that description is unverified: a secondary account describes the problem as choosing at most p of a file’s attributes that together tell every pair of records apart, which has no workload or cost model. | Unfetched — filed 2026-10-15, not read. No NOTE is filed, and it stays Unfetched until it is read, not on merit. Lawful full texts exist but could not be fetched from this environment on 2026-10-15 (“found but could not be fetched”). Unpaywall, OpenAlex (W2058765797) and Semantic Scholar (CorpusId 1606952) all report it bronze open access, publishedVersion, at the ACM Digital Library (dl.acm.org/doi/pdf/10.1145/320289.320296); OpenAlex reports no repository full text. ACM returned HTTP 403 to every request from this environment (the /doi/pdf/, ?download=true, /doi/epdf/ and landing-page forms, by curl and by WebFetch). The Wayback Machine has no capture of the ACM PDF URL. A precursor exists as a Purdue technical report: Douglas E. Comer, “The Difficulty of Optimum Index Solution” [sic], Purdue Department of Computer Science Technical Report 77-248, 11 pp., in Purdue’s institutional repository (docs.lib.purdue.edu/cstech/182; the record’s date field, 1975, conflicts with the 1977 report number). The record page loaded, but its PDF (docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1181&context=cstech, and the /context/cstech/article/1181/viewcontent forms) returned a Cloudflare challenge (403) to curl, with and without browser headers and the landing page’s cookie, and to WebFetch. The Wayback Machine holds a capture of that Purdue PDF (snapshot 20240708123022), but web.archive.org is blocked by this environment’s egress policy (“Blocked by egress policy”; connection resets), and WebFetch refuses that host. Whether the technical report is the same text as the TODS article was not checked. Also tried: scholar.archive.org (no matching result), CORE (403 and 429), CiteSeerX (redirect, no record found), Google Scholar (no version other than ACM’s surfaced), pdfs.semanticscholar.org (no file), cs.purdue.edu technical-report paths (404), and web searches for a course-hosted copy (none found). No shadow library was consulted. Finkelstein, Schkolnick and Tiberio 1988 (LIT-1084, read) cite it only as “Even a restricted version of the index-selection problem is in the class of NP-hard problems”, so the record’s description of it as proving “optimal index selection NP-complete” rests on secondary accounts, and its model, reduction and conditions are unconfirmed. |
| LIT-1089 | The optimal selection of secondary indices is NP-complete | Piatetsky-Shapiro (1983), ACM SIGMOD Record 13(2). For one relation, a known mix of equality queries and updates, single-attribute secondary indexes, and a fixed upkeep charge per index on each update, choosing the index set that minimizes average transaction cost is NP-hard: minimum set cover reduces to it, each subset becoming an attribute and each element a query that any one of its attributes’ indexes can answer. The paper calls the problem NP-complete but states only the optimization version, and the proof is a sketch of about half a page. Storage is not in the model; the only price of an index is upkeep on updates. A variant with off-line updates under an update-cost budget is asserted NP-complete without proof. The paper does not cite or discuss Comer 1978. | Active — read in full 2026-10-15 from the scan SIGMOD serves openly on sigmodrecord.org (sigmodrecord.org/publications/sigmodRecord/8301/pdfs/984523.984530.pdf, linked from the January 1983 issue page), 4 pp., printed pages 72–75; the ACM Digital Library refused requests from this environment, and Unpaywall and OpenAlex list no open copy. Worth reading as the first proof that choosing secondary indexes for a known mix of queries and updates, to minimize average transaction cost, is NP-hard, by a half-page reduction from minimum set cover: attributes are the sets, queries the elements, and each index pays a fixed upkeep on updates. For QUESTION-114 it shows that the trade of upkeep against per-query savings under a known workload can be stated exactly and cannot be solved exactly in general, but only as the number of attributes and query types grows; storage, build cost and the file’s own order are outside its model, and it does not mention Comer 1978 (LIT-1086), so it is the citation for workload-based hardness and says nothing about what Comer proved. |
| LIT-1093 | Amortized efficiency of list update and paging rules | Sleator and Tarjan (1985), CACM 28(2). For a list searched sequentially from the front, where reaching the i-th item costs i, moving each accessed item to the front costs at most twice what any other rule costs on the same sequence, including rules that know the whole sequence in advance (four times, counting move-to-front’s own moves), with no probabilistic assumption. A fixed list sorted by access frequency, and the transpose and frequency-count rules, are not within any constant factor of that. Rearranging ahead of need never pays in this model (Theorem 3). For paging, where only a small fast memory is cheap, no rule that cannot see the future can do better in the worst case than n/(n − n_MIN + 1) times the offline optimum MIN, and LRU and FIFO attain that bound up to an additive n_MIN; with equal memories the ratio is the memory size. The model is a single linear list (or a two-level memory) with uniform items and access cost depending only on position; it says nothing about partitions, bins or search that is not a scan. | Active — read in full 2026-10-15 from the scan of the published CACM article (pp. 202–208, with ACM’s running heads and copyright line) on Sleator’s CMU page (www.cs.cmu.edu/~sleator/papers/amortized-efficiency.pdf); the ACM Digital Library’s open-access PDF refused direct requests from this environment. Worth reading as the founding worst-case comparison of an online rule with the best offline one: on a sequentially scanned list where reaching position i costs i, move-to-front costs at most twice what any rule in the class costs on the same sequence, even one that knows the future (four times counting its own moves), while a fixed list sorted by frequency, transpose and frequency count are not within any constant; and for paging no online rule beats n/(n − n_MIN + 1) times Belady’s MIN, a bound LRU and FIFO meet up to an additive n_MIN. For QUESTION-114 it shows that in a single pile searched from the top, putting the found item on top is within 2 of the best dynamic policy, not of the best arrangement, and that sorting ahead of need never pays (Theorem 3); bins are outside its model. For QUESTION-095 it prices one term, retrieval misses under a fixed cache size and the cost of not knowing future requests, and nothing on precomputation, upkeep or staleness. |
| LIT-1096 | Competitive snoopy caching | Karlin, Manasse, Rudolph and Sleator (1988), Algorithmica 3. For a shared-bus multiprocessor whose caches can either keep a replicated block (and pay one bus cycle for every write to it) or drop it (and pay p to fetch it again if it is read), the paper gives counter-based on-line algorithms whose total bus cost on every request sequence is at most twice that of the best off-line algorithm, and proves that no deterministic on-line algorithm can guarantee better than 2. The rule is: drop a copy once the writes it has caused since it was last read have cost as much as refetching it. With a third option (reading or writing through without fetching) the lower bound rises to 3, and the algorithm given is 3-competitive ignoring writebacks, 4 counting them; with associative caches the bound becomes the paging ratio. The paper never names rent-or-buy or ski rental; its rent is upkeep paid on an idle copy, and its purchase is the refetch after dropping, the reverse orientation of building a bin. Costs are fixed constants, the guarantees are worst case only, and the authors call the practical significance “unclear”. | Active — read in full 2026-10-15 (NOTE for this LIT) from a scan of the published Algorithmica article (pp. 79–119, Springer-Verlag copyright line) posted among the reading handouts of MIT course 6.896, Spring 2004 (courses.csail.mit.edu/6.896/spring04/class/www/handouts/papers/snoopy-cache.pdf); no author or repository copy was found (Unpaywall and OpenAlex: closed; not on Sleator’s CMU or Karlin’s UW pages). Worth reading as the first competitive analysis of a rent-or-buy decision: a cache keeping a shared block pays 1 per write to keep its copy current, or drops it and pays p to fetch it again if it reads; dropping once the writes charged since the last read have cost p is within 2 of the off-line optimum on every sequence (Theorem 4.1), and no deterministic on-line rule does better (Theorem 3.1); with a third option the bound rises to 3 (Theorem 3.3). The paper states the principle generally (change state when the extra cost of staying equals the cost of changing) but never names rent-or-buy or ski rental, and its instance is the decision to stop maintaining a copy, the mirror of building a bin. For QUESTION-114 and CASE-094 it gives a worst-case rule for when a bin pays without knowing future use: make it when the extra search already spent equals its cost; for CLAIM-303 it shows that a recurrence count weighted by reconstruction cost is, in the worst case, the right criterion and within 2, and that the second-occurrence rule is that rule only when filing costs about one reconstruction; for QUESTION-095 it prices upkeep against re-retrieval and nothing on precomputation or staleness, which it forbids rather than prices. |
| LIT-1097 | Should Tables Be Sorted? | Yao (1981), JACM 28(3). For storing a set of n keys from a universe {1,…,m} in a table so that the membership query (is x in S?) is answered with the fewest worst-case adaptive probes, Yao proves that when the table cells may hold only keys of S or a bounded number of pointer symbols, sorted order with binary search (⌈lg(n+1)⌉ probes) is optimal once m exceeds a Ramsey-number threshold N(n, p, q), whatever the arrangement rule, table size or hashing scheme. The threshold is astronomically large, and for small universes sorted order is beaten (one probe suffices for m = 3 with n = 2, and for m ≤ 2n − 2 with n > 2). The optimality also fails as soon as one cell may hold an arbitrary value from the universe: with a single extra “directory” cell, two probes suffice for m ≥ 2^{16n²}. Only query cost is counted; building the table is free and unconstrained, so the paper says nothing about the cost of arranging or about trading arrangement cost against lookups. | Active — read in full 2026-10-15 from the published Journal of the ACM version (28(3), pp. 615–628), which ACM serves free (OpenAlex/Unpaywall: bronze) but refused to this environment with 403, fetched instead as the copy on a University of Maryland course page (www.cs.umd.edu/users/gasarch/COURSES/858/S13/tables.pdf) and compared with the author’s Stanford report STAN-CS-79-753 (i.stanford.edu/pub/cstr/reports/cs/tr/79/753/CS-TR-79-753.pdf), which has the same results. Worth reading as the paper that set up what is now called the cell-probe model, with lookup cost counted as the worst-case number of cells read and arrangement and search strategy left arbitrary: for static membership, sorted order with binary search (⌈lg(n+1)⌉ probes) is optimal once the key space exceeds a Ramsey-number threshold, provided cells hold only stored keys or pointer symbols; it is beaten for small key spaces (one probe for m ≤ 2n − 2), and for large ones as soon as one cell may hold an encoded value (two probes for m ≥ 2^{16n²}). For QUESTION-114 it shows that “founding cell-probe paper” holds but “when storage in sorted order beats other arrangements” does not as worded: the theorem says nothing beats sorting under a no-encoding restriction for huge universes, and the rest of the paper shows other arrangements beating it. It counts lookup probes only; building the arrangement is free, so it bears nothing on whether sorting is amortized search or on the cost of organising set against lookups later. |
| LIT-1099 | When Is “Nearest Neighbor” Meaningful? | Beyer et al. (1999), ICDT ‘99. For a fixed number n of data points and a query drawn independently of them, if the variance of the p-th power of the query-to-point distance, divided by its mean, goes to zero as dimension m grows, then the farthest point is, with probability tending to one, within any factor 1 + ε of the nearest (Theorem 1). The condition holds for IID dimensions and for several correlated or non-identical workloads, and fails when data and queries lie on a low-dimensional subspace or when queries fall inside well-separated clusters. Simulations put the collapse of contrast within 10 to 20 dimensions for IID data, and two real image-feature sets showed weak contrast. The condition is sufficient, not shown necessary; the theorem is about the ratio of distances with n held fixed, and says nothing about real data of low intrinsic dimension except that such data can escape it. | Active — read in full 2026-10-15 from the authors’ University of Wisconsin–Madison technical report of the same title (CS TR #1377, June 1998, 19 pp.), served by the department’s archive (research.cs.wisc.edu/techreports/1998/TR1377.pdf) and, byte-identical, by the MINDS@UW repository (minds.wisconsin.edu/handle/1793/60174), which OpenAlex lists as the submitted version. The ICDT ‘99 chapter itself is closed (Unpaywall, OpenAlex, Semantic Scholar) and was not seen; the CiteSeerX copy redirects to the Internet Archive, which was unreachable. The report is a scan, read by OCR and page images. Worth reading as the source of the sufficient condition for nearest-neighbour contrast to vanish: if the variance of the p-th power of the query-to-point distance, divided by its mean, goes to zero, then with a fixed number of points the farthest is within any factor 1 + ε of the nearest with probability tending to one. IID coordinates satisfy it, and so do several correlated or non-identical workloads. Data on a low-dimensional subspace, and queries that land in well-separated clusters, do not. Simulations put the loss of contrast within 10 to 20 dimensions for IID data. The condition is sufficient and not shown necessary, n is held fixed, the result concerns the ratio of distances, and the paper measures no index. For CLAIM-309 it has no bearing, and the filing’s “no arrangement by similarity discriminates” overstates it. For CLAIM-310 it corroborates that what serves retrieval depends on the query distribution: the same clustered data is meaningful for queries inside clusters and not for uniform ones, and the meaningful answer there is the whole cluster. For QUESTION-114 it sets a precondition: under its condition there is no nearest neighbour worth organizing to find. |
| LIT-1102 | Lower Bounds on Near Neighbor Search via Metric Expansion | Panigrahy et al. (2010), FOCS 2010. Cell-probe lower bounds for near-neighbour search follow from the expansion of the graph joining points within distance r: a deterministic t-probe structure with m cells of w bits must have (mwt/n)^t at least the vertex expansion, and a randomized one must have m^t w/n at least the robust expansion at measure 1/m^t, on random data sets with weakly overlapping neighbourhoods. For the hypercube this recovers known bounds and gives, for example, space n^{1+Ω(1/(ct))} for approximation c and n^{Ω(1/(tε²))} for 1 + ε; for symmetric (Cayley) graphs a one-probe structure matches the bound, and for dynamic low-contention structures the update time must be at least the robust expansion over poly(t). The bounds are average case, for approximate decision search, and weak in t: they fall off exponentially in the number of probes, so they are tight only for a constant number of probes and say little about structures that probe n^ρ cells. | Active — read in full 2026-10-15 from the authors’ arXiv preprint (arXiv:1005.0418v1, 3 May 2010, 29 pp.); the FOCS 2010 text was not seen and has no open copy. Worth reading as the framework that turns the expansion of a metric’s neighbourhood graph into cell-probe lower bounds for approximate near-neighbour search: (mwt/n)^t ≥ vertex expansion for deterministic t-probe structures, m^t w/n ≥ robust expansion at measure 1/m^t for randomized ones, on random data with weakly overlapping neighbourhoods, giving for the hypercube space n^{1+Ω(1/(ct))} at approximation c and n^{Ω(1/(tε²))} at 1 + ε, a matching one-probe upper bound on Cayley graphs, and an update-time lower bound for dynamic low-contention structures. The bounds are average case, approximate, and fall off exponentially in the number of probes, so they are tight only for constant t. For THEORY-340 it is the similarity analogue of Yao’s lookup floor, holding with arbitrary cell encoding where Yao’s sorted-order optimality did not; for QUESTION-114 it puts a floor under fast per-query search priced in space, and the only lower bound on upkeep among the readings, in a restricted class, with no workload, amortization or build time; for CLAIM-310 its dependence on expansion is consistent with attribute bins working for discrete lookup (the record’s reading); no bearing on THEORY-345 or CLAIM-309. |
| LIT-1103 | The Small-World Phenomenon: An Algorithmic Perspective | Kleinberg (2000), STOC 2000. On an n × n lattice where each node also gets q long-range links drawn with probability proportional to d^(−r), short paths exist for r = 0, but no algorithm that sees only the links of nodes the message has visited finds them faster than about n^(2/3) steps in expectation. At r = 2 the greedy rule (pass to the contact nearest the target in lattice distance) delivers in O((log n)²) expected steps (proved for p = q = 1); for 0 ≤ r < 2 every such algorithm needs Ω(n^((2−r)/3)) and for r > 2 Ω(n^((r−2)/(r−1))). So efficient decentralized search depends on how the long-range links are distributed relative to the lattice, not on the links’ existence or on the graph’s diameter. The k-dimensional version (r = k) is stated without proof, and the model is a lattice with a known distance that every node can compute to the target. | Active — read in full 2026-10-15 (NOTE for LIT-1103) from the author’s own copy on Kleinberg’s Cornell page (www.cs.cornell.edu/home/kleinber/swn.pdf), a 14-page manuscript linked there under the STOC 2000 citation and footnoted as the version that is also Cornell CS Technical Report 99-1776; the ACM typeset proceedings (pp. 163–170) returned 403, and the Nature 2000 letter is a different work, not read. Worth reading as the source of the result that a network can have short paths that no searcher using local information can find: on an n × n lattice with long-range links drawn with probability proportional to d^(−r), greedy routing delivers in O((log n)²) expected steps at r = 2 (proved for one local and one long-range contact per node), every decentralized algorithm needs Ω(n^((2−r)/3)) steps for 0 ≤ r < 2 and Ω(n^((r−2)/(r−1))) for r > 2, and r = k in k dimensions is stated without proof. For CLAIM-310 it shows that findability, with nodes, lattice and link count fixed, turns on how the links are arranged relative to a distance every node can compute, which supports the claim’s “as arrangements” half; it narrows “only one distribution” to this family of inverse power laws and to polylogarithmic delivery, and it prices no cost of building the arrangement, so it adds nothing to QUESTION-114’s organizing cost. |
| LIT-1106 | Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality | Indyk and Motwani (1998), STOC 1998. Approximate nearest-neighbour search in d dimensions reduces, with small overhead, to a decision problem (is there a point within r, or none within (1+ε)r), and that decision problem is solved by locality-sensitive hashing: if hash functions make near pairs collide with probability at least p1 and far pairs with at most p2, then n^ρ random hash tables with ρ = ln(1/p1)/ln(1/p2) answer queries with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, with constant success probability. For Hamming space by bit sampling, and for ℓ_p with p in [1, 2] by embedding, ρ ≤ 1/(1+ε) in the 1999 text (1/ε, for ε > 1, in STOC). The paper also gives the other end of the trade: Õ(d) query time for n·O(1/ε)^d or (nd)^{O(1)} preprocessing. It is an upper bound only: it proves no lower bound, does not claim its exponents are best, solves only the approximate problem with constant probability, and counts preprocessing as space. | Active — read in full 2026-10-15 from the authors’ expanded “preliminary version” of December 30, 1999, served from Indyk’s MIT directory (people.csail.mit.edu/indyk/nndraft.ps), with its statements compared against the STOC 1998 printed text (pp. 604–613) as posted on a Princeton course page (cs.princeton.edu/courses/archive/spr04/cos598B/bib/IndykM-curse.pdf); the ACM copy refused requests and Unpaywall and OpenAlex list no open copy. The 2012 Theory of Computing journal version (with Har-Peled) was not read. Worth reading as the origin of locality-sensitive hashing and of the reduction from approximate nearest neighbour to its decision version: hash functions that make near pairs collide more often than far ones give n^ρ tables, O(dn + n^{1+ρ}) space and O(n^ρ) hash evaluations per query, ρ = ln(1/p1)/ln(1/p2), which for Hamming bit sampling is at most 1/(1+ε) in the 1999 text (1/ε, for ε > 1, in STOC), with constant success probability; bucketing gives Õ(d) queries for n·O(1/ε)^d preprocessing. Upper bounds only, approximate only, and preprocessing is counted as space. For QUESTION-114 it is the first price list for organizing a collection for similarity queries, space paid once against hash evaluations per query, with no amortization over queries and no workload; for CLAIM-310 it shows the single-arrangement constraint escaped by keeping many random partitions and storing every point in each, paid for in space; it neither extends nor narrows THEORY-340 and has no bearing on THEORY-345 or CLAIM-309. |
| LIT-1110 | Provenance Semirings | Green et al. (2007), PODS 2007. Annotating each input tuple with an element of a commutative semiring, and computing union and projection with + and join with ·, gives one positive relational algebra (RA+) that is set semantics over the Booleans, bag semantics over the naturals, c-tables over positive Boolean expressions, event tables over sets of outcomes, and lineage over sets of tuple ids. The polynomials N[X] in the tuple ids are the most general such annotation: for any RA+ query and any commutative semiring K, the K-annotated answer is the provenance polynomial evaluated at the input’s K-annotations (Theorem 4.3). For datalog the paper needs ω-continuous semirings and formal power series N∞⟦X⟧, with the analogous factorization (Theorem 6.4) and a criterion for infinite coefficients (Theorem 6.5). Everything is positive: no negation, difference or aggregation, and revaluing inputs can only remove output tuples, never add them; trust and deletion propagation are not treated. | Active — read in full 2026-10-15 (NOTE-913) from the authors’ postprint in the University of Pennsylvania’s ScholarlyCommons repository (repository.upenn.edu/handle/20.500.14332/8764, file pods07.pdf, 10 pp., posted by permission of ACM), found through Unpaywall and OpenAlex (green, submittedVersion) and fetched through the repository’s API because the landing page is script-only. Worth reading as the source of semiring provenance: for positive relational algebra over any commutative semiring the annotated answer is the provenance polynomial in N[X] evaluated at the input annotations, and for datalog over ω-continuous semirings the same holds with formal power series in N∞⟦X⟧ (THEORY-373, THEORY-380). For QUESTION-105 it shows that retaining one polynomial per output suffices for every revision that re-annotates input facts in a commutative semiring, deletion included, without rerunning the query; it proves no minimality, and it excludes added facts, negation and contraction. Its “why-provenance” is one set of contributing tuples per output, coarser than the witness basis of LIT-1104 that it cites as the same notion, and its structure for it fails its own semiring axioms. Deletion propagation and trust are not in the paper; deletion follows from Theorem 4.3 as the record’s application. For CLAIM-293 the polynomial separates joint from alternative support but, like topology, leaves the strengths to the valuation; for CLAIM-306 it has no account of absence. |
| LIT-1120 | Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors | Andoni et al. (2017), SODA 2017. For c-approximate near-neighbour search (find a point within cr when one lies within r) over n points in Euclidean space, the paper gives a data structure for every pair of exponents with c²√ρq + (c²−1)√ρu ≥ √(2c²−1), using space n^{1+ρu+o(1)} + O(dn) and query time n^{ρq+o(1)} + dn^{o(1)}, on worst-case data (for Hamming and ℓ_1, c√ρq + (c−1)√ρu ≥ √(2c−1)). It proves the curve cannot be beaten by list-of-points structures, the model of all known data-independent hashing schemes, on random instances, and unconditionally in the cell-probe model only at one end (query n^{o(1)}) for one or two probes. So the exact trade between space spent organizing and per-query time is known within the hashing model; outside that model, for more than two probes, it is not, and data-dependent schemes are covered only by conjecture. | Active — read in full 2026-10-15 from the authors’ arXiv full version (arXiv:1608.03580v2, 21 May 2017, 62 pp.), which merges the two preprints behind the SODA 2017 extended abstract; the SODA text itself was not seen, and the quantum-information steps of the two-probe proof were followed, not checked. Worth reading as the exact time–space curve for approximate near-neighbour search: for every c > 1, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) in Euclidean space (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming and ℓ_1), on worst-case data, and optimal for list-of-points (hashing-based) structures on random instances, with unconditional cell-probe matches only for one or two probes at the n^{o(1)}-query end (n^{(c/(c−1))²−o(1)} words). For QUESTION-114 it prices organizing against per-query cost exactly within the hashing model, as replication of each point into n^{ρu} lists against n^{ρq} lists scanned, and its closest-pair remark makes the best point depend on the number of queries; it says nothing about workloads, query mixes, build cost beyond space, exact nearest neighbours or structured data, and data-dependent schemes are covered only by conjecture. For CLAIM-310 it shows overlapping bins are needed for fast similarity queries; for THEORY-340 it gives the similarity analogue of Yao’s lookup floor in the unrestricted cell-probe model at one and two probes; no bearing on THEORY-345 or CLAIM-309. |
| LIT-1127 v2 | Aggregation of Variables in Dynamic Systems | Simon (1961). In nearly decomposable linear systems, short-run dynamics are governed within subsystems and long-run dynamics by interactions among their aggregates, which justifies aggregating variables over the long run. | Unavailable — filed 2026-10-16, not read. On 2026-10-19, at the owner’s request to read it first among the works of the exchange on coarse-to-fine processes, a search for a lawful full text reached none. JSTOR (stable 1909285) returned a block page instead of the PDF. The Econometric Society’s page gives the abstract only and sends full text to JSTOR institutional access. OpenAlex reports it closed with no repository copy, and Semantic Scholar does not index the DOI. The two reprints are not open: Ando, Fisher and Simon, Essays on the Structure of Social Science Models (1963), is on the Internet Archive as a lending-only item, and Simon’s Models of Discovery (1977, ch. 12) is a paywalled Springer chapter. Carnegie Mellon’s digital collections returned 503. A title search found no author-posted copy. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1151 v2 | Optimal compression of approximate inner products and dimension reduction | Alon & Klartag (2017), FOCS 2017, ARXIV-1610.00239. For n points of norm at most 1 in R^k and 1/n^0.49 ≤ ε ≤ 0.1, storing every inner product and squared distance to additive ε takes Θ(n log n/ε²) bits once k ≥ log n/ε², Θ(nk log(2 + log n/(ε²k))) for log n ≤ k ≤ log n/ε², and Θ(nk log(1/ε)) below, with an efficient per-point sketch. Comparing the counts across dimensions re-proves Larsen and Nelson’s bound and extends it: for every ε ≥ 2/√n some n such points need Ω(min{k, log(2 + ε²n)/ε²}) dimensions under any map with additive error ε. A bipartite JL lemma keeps cross inner products in O(log(2 + ε²n)/ε²) dimensions, without bounding norms. | Active — read 2026-10-16 (NOTE-929) in full; worth reading as the paper that turns the Johnson–Lindenstrauss lower bound into a bit count, determining how many bits it takes to store every inner product of n points of the unit ball to additive ε, and so extending Larsen and Nelson’s dimension lower bound to every ε ≥ 2/√n with log(2 + ε²n)/ε² in place of log n/ε². Read it knowing that its dimension statements are for additive error on squared distances, the relative form being stated only for ε ≥ 1/n^0.49; that the step from bits to dimension is a short remark; that constants are unstated; and that its small-ε upper bound is bipartite, keeps only cross inner products and does not bound norms, so the one-set JL lemma at log(2 + ε²n)/ε² stays open. Filed as THEORY-399 and THEORY-393. |
| LIT-1152 v2 | Extending Lipschitz functions via random metric partitions | Lee and Naor (2005), Inventiones Mathematicae 160(1): 59–95. Random partitions of a metric space at every scale, with each point padded away from piece boundaries, give a gentle partition of unity, and averaging over it extends Lipschitz maps into any Banach space and from any superspace. This gives ae(X) ≤ C log n/log log n for n-point X (improving C log n), ≤ C log λ(X) for doubling X, ≤ C r² for K_r-minor-free graph metrics, e(ℓ₂ᵈ, Z) ≤ C√d, and, with the Johnson–Lindenstrauss lemma, eₙ(L₂, Z) ≤ C√(log n). For Hilbert targets it does not improve Johnson and Lindenstrauss’s K√(log n), and whether √(log n) suffices for every target it leaves open. | Active — read 2026-10-16 (NOTE-927) in full; worth reading as the method that turns random metric partitions into Lipschitz extensions: padded random partitions at every scale give a gentle partition of unity, and averaging over it extends maps into any Banach space at a constant depending on the domain alone. Read it knowing that its bound for n-point domains, C log n/log log n, improves Johnson, Lindenstrauss and Schechtman’s C log n for Banach targets but not Johnson and Lindenstrauss’s K√(log n) for Hilbert targets; that it states as open whether √(log n) suffices for every target; that the copy read is the January 2004 preprint; and that it does not discuss separated domains. Filed as THEORY-396 and THEORY-400. |
| LIT-1153 v3 | Extensions of Lipschitz mappings into a Hilbert space | Johnson & Lindenstrauss (1984), Contemporary Mathematics 26: 189–206. Lemma 1: for 1 > τ > 0 there is K(τ) such that any n points of ℓ₂ⁿ map into ℓ₂ᵏ, k = [K log n], with ‖f‖_lip‖f⁻¹‖_lip ≤ (1+τ)/(1−τ), by a Haar-random rank-k orthogonal projection with positive probability; the proof takes K ≥ (10/τ)². The lemma is the tool for Theorem 1, that a Lipschitz map from any n-point subset of a metric space into ℓ₂ extends to the whole space with constant at most C(log n)^{1/2}, verifying a Marcus–Pisier conjecture. A lower bound of order (Log n / Log Log n)^{1/p−1/2} for L_p, 1 ≤ p < 2, and an extension theorem for separated sets complete it. | Active — read 2026-10-14 (NOTE-930) in full; worth reading as the source of the dimension-reduction lemma, in the form its authors gave it: n points of ℓ₂ⁿ go into [K(τ) log n] dimensions with Lipschitz distortion (1+τ)/(1−τ), by a Haar-random orthogonal projection, proved by concentrating one projected norm about its median and taking a union bound over the normalized differences. Read it knowing that the τ⁻² dependence is in the proof (K ≥ (10/τ)²), not the statement; that the paper shows only that log n cannot be improved, not the τ-dependence; and that the lemma is a tool for the paper’s real subject, L(X, n) ≤ C(log n)^{1/2} for extending Lipschitz maps into ℓ₂. |
| LIT-1154 v2 | Optimality of the Johnson-Lindenstrauss Lemma | Larsen & Nelson (2017), FOCS 2017, ARXIV-1609.02094. For integers n, d ≥ 2 and lg^0.5001 n/√min{n, d} < ε < 1, some n points in R^d need m = Ω(ε⁻² lg(ε²n)) dimensions under any map, linear or not, that keeps all squared distances within 1 ± ε. For ε > 1/min{n, d}^0.4999 this is Ω(ε⁻² lg n), so the Johnson–Lindenstrauss bound is optimal up to a constant, and the distributional lemma with a union bound cannot be improved on. The proof is an encoding argument over k-sparse vectors with k = ε⁻²/256, ending in m = Ω(k lg(d/k)). | Active — read 2026-10-14 (NOTE-925) in full; worth reading as the proof that the Johnson–Lindenstrauss order ε⁻² lg n cannot be beaten by any map, linear or not, on a worst-case point set, and as a lower bound that is itself a count of sparse supports. Read it knowing that the bound is Ω(ε⁻² lg(ε²n)) and holds only for lg^0.5001 n/√min{n, d} < ε < 1; that it is tight up to an unstated constant, not a sharp threshold; that the guarantee is on squared distances; and that it says nothing about random features, expansion or typical inputs. |
| LIT-1155 v2 | The geometry of graphs and some of its algorithmic applications | Linial, London & Rabinovich (1995), Combinatorica 15(2): 215–245. Every n-point metric embeds in O(log n)-dimensional Euclidean space with distortion O(log n) (Bourgain’s theorem as Theorem 3.2(1) states it). The paper reproves the distortion by a randomized map to distances from O(log² n) random sets, takes the dimension from Johnson–Lindenstrauss, and computes the least Euclidean distortion of any finite metric by semidefinite programming. Constant-degree expanders need distortion Ω(log n) in ℓ_p, 1 ≤ p ≤ 2, of any dimension (Proposition 4.2), so the bound is tight. The max-flow/min-cut gap for k commodities is O(log k), with a deterministic algorithm (Theorem 4.1). | Active — read 2026-10-16 (NOTE-928) in full; worth reading as the place where Bourgain’s O(log n) embedding of finite metrics into Euclidean space is restated, reproved by distances to random sets of every scale, and shown tight: constant-degree expanders need distortion Ω(log n) in ℓ_p, 1 ≤ p ≤ 2, of any dimension, because their gap between multicommodity flow and cut is Ω(log n). The same embeddings bound that gap by O(log k) for k commodities. Read it knowing that the O(log n) dimension comes from Johnson–Lindenstrauss, not from the construction, which uses O(log² n) coordinates; that the lower bound rests on Leighton and Rao’s gap, cited; that constants are unspecified; and that it says nothing about trees in ℓ₂, for which Sala et al. cite it. Filed as THEORY-394. |
| LIT-1156 v2 | A Simple Proof of the Restricted Isometry Property for Random Matrices | Baraniuk, Davenport, DeVore & Wakin (2008), Constructive Approximation 28(3): 253–263. If an n × N random matrix satisfies the concentration inequality of Johnson–Lindenstrauss proofs, Pr(|‖Φx‖² − ‖x‖²| ≥ ε‖x‖²) ≤ 2e^{−n c₀(ε)}, then a union bound over a (12/δ)^k-point net on each of the (eN/k)^k coordinate k-subspaces, with a bootstrapping step from net to subspace, gives the restricted isometry property for all k ≤ c₁n/log(N/k) with probability at least 1 − 2e^{−c₂n} (Theorem 5.2). Gaussian, ±1 and Achlioptas’s sparse ensembles qualify, also in any fixed orthonormal basis. That this range is the widest possible the paper takes from Gelfand widths. | Active — read 2026-10-14 (NOTE-931) in full; worth reading as the short proof that the restricted isometry property follows from the concentration inequality used in Johnson–Lindenstrauss proofs, by a union bound over nets on the sparse supports. Read it knowing that it proves sufficiency only, with constants left implicit; that the optimality of k ≲ n/(log(N/n) + 1) it cites from Gelfand widths, not from concentration; that its constants are Achlioptas’s; and that the route “through” the JL lemma in §6 is sketched, not written out. |
| LIT-1157 v3 | An elementary proof of a theorem of Johnson and Lindenstrauss | Dasgupta & Gupta (2003), Random Structures & Algorithms 22(1): 60–65. For n points in ℝᵈ and 0 < ε < 1, projecting onto a random k-dimensional subspace and scaling by √(d/k) keeps every squared pairwise distance within 1 ± ε with probability at least 1/n once k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n (Theorem 2.1). Each tail of a random unit vector’s squared length in k coordinates is at most exp((k/2)(1 − β + ln β)) (Lemma 2.2), proved by Markov’s inequality on an exponential of Gaussian squares; a union bound over the pairs finishes it. | Active — read 2026-10-14 (NOTE-932) in full; worth reading as the shortest complete proof of the Johnson–Lindenstrauss lemma with an explicit dimension, k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n, and as a case in which that dimension is visibly the log of the number of pairs, about n², over a squared tolerance. Read it knowing that its map is a scaled projection onto a random subspace, not a Gaussian or ±1 matrix, which §3 only describes; that it keeps squared distances; that one draw succeeds with probability only 1/n; and that it treats k < d only, so it says nothing about maps that raise dimension. |
| LIT-1159 v2 | Database-friendly random projections: Johnson-Lindenstrauss with binary coins | Achlioptas (2003), Journal of Computer and System Sciences 66(4): 671–687. A random projection E = (1/√k)AR keeps all pairwise squared distances of n points within 1 ± ε with probability at least 1 − n^−β when k ≥ (4 + 2β)/(ε²/2 − ε³/3) · log n and R’s entries are independent ±1, or √3 × {+1, 0, −1} with probabilities 1/6, 2/3, 1/6. That k₀ equals the best bound for Gaussian projections. The proof shows the all-equal unit vector has the largest even moments of its projection, and that these are at most the Gaussian ones. | Active — read 2026-10-14 (NOTE-926) in full, from a 2002 journal-submission draft, not the JCSS text; worth reading as the proof that a Johnson–Lindenstrauss embedding needs concentration, not spherical symmetry: independent ±1 entries, or √3·{+1, 0, −1} with probabilities 1/6, 2/3, 1/6, give k₀ = (4 + 2β)/(ε²/2 − ε³/3) · log n, the best Gaussian bound, by showing the all-equal vector is the worst case and its moments are dominated by the Gaussian ones. Read it knowing that the draft omits the proof of the sparse case of Lemma 6.3, that 2/3 zeros is stated without proof as the most that lemma allows, and that its ensembles are signed and zero-mean, which the fly’s {0, 1} matrix is not. |
| LIT-1160 v2 | Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing | Donoho & Tanner (2009), Phil. Trans. R. Soc. A 367: 4273–4293, ARXIV-0906.2530. For Gaussian matrices, recovery of a k-sparse vector from n measurements in dimension N by ℓ1 minimization, or by its nonnegative LP form, succeeds with probability tending to 1 below a curve ρ(δ; Q) in δ = n/N, ρ = k/n and to 0 above it, a result proved earlier and restated here. In 2,948,000 linear programs over eight non-Gaussian ensembles at N up to 1600, the 50% success points fall on the same curves. Exact agreement at finite N is rejected; the differences decay as N^(-1/2), so the paper supports an asymptotic universality, observed and not proved. | Active — read 2026-10-14 (NOTE-933) in full, from the 48-page arXiv copy, not the journal text; worth reading as the measured case for how far the sharp ℓ1 recovery thresholds proved for Gaussian matrices hold for other random matrices. Read it knowing that the transitions sit on the Gaussian curves only in the large-N limit, that finite-N universality is rejected, that the sparse and Hadamard ensembles miss by about one transition width at small N, that the curves belong to (LP) and (P1) and not to other decoders, and that nothing about non-Gaussian matrices is proved. |
| LIT-1161 | On Lipschitz embedding of finite metric spaces in Hilbert space | Bourgain (1985), Israel Journal of Mathematics 52(1–2): 46–52. Every n-point metric space embeds in Hilbert space with distortion O(log n), by a random construction. Unread; known through Linial, London and Rabinovich. | Unavailable — unread, registered 2026-10-16. A lawful copy was sought and not reached: OpenAlex lists only the closed Springer version, and web searches found no author-posted or course-hosted copy. Its theorem is restated and its proof scheme used by Linial, London and Rabinovich (LIT-1155), which is how the record reads it until a copy is supplied. |
| LIT-1202 | Reconsidering unique information: Towards a multivariate information decomposition | Rauh, Bertschinger, Olbrich & Jost (2014), ISIT 2014: 2232–2236, ARXIV-1404.3146. Theorem 2: no redundancy function satisfies the Williams–Beer axioms (symmetry, self-redundancy, monotonicity), local positivity (non-negative partial-information atoms) and the identity axiom. The counterexample is three sources, two independent bits and their XOR, with the target their joint copy: identity forces each atom {i}{jk} to 1 bit and the total is 2 bits, so the atom {12}{13}{23} ≤ −1. Lemmas 3–4: Bertschinger et al.’s unique information decreases as the other source is enlarged and increases as its own source or the target is; their shared information increases with a source but violates left monotonicity, as does their synergy. | Active — read in full 2026-10-17 (NOTE-965); worth reading as the short proof that the identity axiom and a non-negative decomposition on the Williams–Beer lattice cannot both hold once there are three sources: with Y1, Y2 independent bits, Y3 = Y1 ⊕ Y2 and the target their joint copy, identity, the Williams–Beer axioms and invariance under relabelling the target force the atom {12}{13}{23} to be at most −1 bit. So a multivariate extension of Bertschinger et al.’s unique information cannot be a partial information decomposition. Read it knowing that the theorem rests on the identity axiom applied to sources that are sets of variables and on an unstated invariance under relabelling the target, that its other results (monotonicity of unique information, failure of left monotonicity) are about the bivariate measures, and that the multivariate structure it calls for is left open. Filed as THEORY-713. |
| LIT-1212 v2 | Multiple-gradient descent algorithm (MGDA) for multiobjective optimization | Désidéri (2012; first as an INRIA report, 2009). A point is Pareto-stationary when some convex combination of the criteria’s gradients is zero, which every Pareto-optimal point is; when a point is not, minus the minimum-norm element of the gradients’ convex hull is a descent direction common to all smooth criteria, and steepest descent along it (MGDA) has a subsequence converging to a Pareto-stationary point when the criteria are positive and coercive. | Active — read 2026-10-29 (NOTE-1401) in full, from the Comptes Rendus Mathématique’s own open copy of the 2012 note; worth reading as the short published source for Pareto-stationarity and for the minimum-norm point of the gradients’ convex hull as a common descent direction, the constructive half of CLAIM-366’s convex-hull test. Read it knowing that it does not state the converse (zero in the hull excludes common strict descent), that its convergence theorem’s proof is a sketch, and that the direction, unlike its existence, depends on how the gradients are scaled, which the author leaves open. |
| LIT-1215 v2 | Towards a Neuronal Gauge Theory | Sengupta, Tozzi, Cooray, Douglas & Friston (2016). An essay proposing that variational free-energy minimization be cast as a gauge theory. The only connection it names is the Levi-Civita connection of the Fisher metric on the manifold of a belief’s sufficient statistics, so its gauge freedom is reparameterization; it reads attention as precision weighting forced by that manifold’s curvature. No gauge group or invariant Lagrangian is specified. | Active — read 2026-10-29 (NOTE-1350) in full, from the PLOS Biology PDF and its supporting information (prose; the supplements’ equations did not extract); worth reading as the source for the claim that free-energy minimization is a gauge theory, and for what that claim comes to: Riemannian descent on a belief manifold whose connection is the Levi-Civita connection of the Fisher metric. Read it knowing that it is an essay with no gauge group, invariant Lagrangian or derived field, that its gauge freedom is reparameterization, and that it contains errors on curvature and on what a gauge field is. |
| LIT-1222 v2 | A Difficulty in the Concept of Social Welfare | Arrow (1950). No rule that turns every profile of individual orderings into a social ordering can satisfy unrestricted domain, the Pareto condition, independence of irrelevant alternatives and non-dictatorship together: the impossibility theorem. | Unavailable — filed 2026-10-18, not read. A lawful full text was sought on 2026-10-29 and not reached. OpenAlex reports it closed with no repository copy; the Journal of Political Economy’s page (University of Chicago Press) and JSTOR are paywalled; EconPapers lists only the journal; a web search found lecture slides and summaries, no copy of the article. The fuller statement, Social Choice and Individual Values (1951), was not sought. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1231 v2 | Constrained optimization as ecological dynamics with applications to random quadratic programming in high dimensions | Mehta, Cui, Wang & Marsland (2019). A duality between constrained optimization with inequality constraints, quadratic programming included, and consumer-resource models of ecological dynamics; with a cavity solution it gives the statistics of random high-dimensional quadratic programs, in agreement with numerics. | Active — read 2026-10-29 (NOTE-1395) in full, from arXiv 1809.04221v1; worth reading as the plain statement that inequality-constrained optimization is a consumer–resource ecology, constraints as species and multipliers as abundances, and that the Lagrangian dual of quadratic programming is a Lotka–Volterra system with α = CCᵀ. Read it knowing that the arXiv text has several algebraic slips (κ_i in the main text among them), that its “always” reduction of QP to a squared distance is not general, and that the PRE version was not compared. |
| LIT-1233 v2 | Permanence via invasion graphs: incorporating community assembly into modern coexistence theory | Hofbauer & Schreiber (2022). For ecological ODEs whose invasion growth rates have nonzero, well-defined signs on each subcommunity, it defines the invasion graph of communities and invasions and proves that when the graph is acyclic, robust permanence holds if each species has a positive invasion growth rate at every −i community (a community without it that no other missing species can invade), and, for smooth models, only if. With cycles, signs are not enough. | Active — read 2026-10-29 (NOTE-1379) in full, from arXiv 2204.03773v2; worth reading as the theorem that says when signs of invasion growth rates decide coexistence: with an acyclic invasion graph, robust permanence holds exactly when each species can invade each of its −i communities. Read it knowing that it is deterministic, that its sign-parity assumption excludes common non-Lotka–Volterra cases, that cycles are left open, and that its switching-predator example swaps the labels of two invasion conditions. |
| LIT-1239 v2 | Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining | Eyink & Aluie (2009). Builds an exact decomposition of kinetic energy into band-pass contributions from smooth, graded filters on a geometric sequence of scales, with band budgets whose fluxes telescope, and carries over Eyink’s (2005) bounds: if velocity increments scale with exponents strictly between 0 and 1, the mean energy flux across a scale ℓ takes from scales above Δ or below δ at most a power of the scale ratio, 2/3 at K41 scaling. The bounds are sketched here and proved in the 2005 paper. A heuristic decorrelation argument and one 512³ simulation account for Kraichnan’s faster 4/3 decay of the signed contributions. | Active — read 2026-10-20 (NOTE-969) in full, from arXiv:0909.2386v1; worth reading for its exact band-pass decomposition of kinetic energy with smooth filters (a multiscale Germano identity, in which each band’s energy changes by the flux in minus the flux out), for a compact statement of when and why the energy flux across a scale is local in scale (THEORY-754), and for the account of why the rigorous 2/3 exponent is weaker than Kraichnan’s 4/3. Read it knowing that the rigorous bounds are proved in Eyink (2005), LIT-1252, and only sketched here; that they are conditional on a scaling hypothesis for velocity increments (exponents strictly between 0 and 1) and a non-zero mean flux, neither proved for Navier–Stokes; that the 4/3 argument is heuristic and its one simulation brackets rather than confirms it; that the band decomposition’s non-negativity holds only after integration for the kernels it needs; and that locality here is in scale, for one field, and says nothing about units. |
| LIT-1241 v2 | Simon-Ando Decomposability and Fitness Landscapes | Shpak, Stadler, Wagner & Altenberg (2004). States Simon and Ando’s theorem on nearly decomposable linear systems, without proof and without rates, as four time windows separating within-block relaxation, block quasi-equilibrium, aggregate cross-block dynamics and global equilibrium. It applies an entrywise version of the condition to mutation–selection and recombination on fitness landscapes. Point mutation meets it at low mutation rates, trivially, and recombination rarely does. It concludes that Simon–Ando aggregation is one route to a dynamically sufficient aggregate, independent of exact aggregation by symmetry and of character aggregation under linkage equilibrium. | Active — read 2026-10-20 (NOTE-977) in full, from arXiv:q-bio/0403030v1; worth reading for the record’s only open statement of Simon and Ando’s (1961) theorem (Theorems 2.1, 2.2a and 2.2b: block-diagonal A* plus εC, the four time windows T0 < T1 < T2 < T3, and the aggregate Â_IJ = Σ_{i∈I}Σ_{j∈J} A_ij ν*_jJ), for its four-stage account of when a nearly decomposable system is decomposable, aggregable or both, and for its conclusion that quasi-equilibrium, symmetry (equitable partitions) and linkage equilibrium are independent routes to a dynamically sufficient aggregate. Read it knowing that the theorem is quoted without proof and states no rate in ε and no bound on aggregation error; that the landscape “Simon–Ando condition” is an entrywise proxy, met trivially at low mutation rate even on a flat landscape where there is no time-scale separation; that its leading-eigenvector statement fails when block leading eigenvalues coincide, the stochastic case it invokes; and that (6.2b) is wrong for radius κ ≥ 2. Published as Theory in Biosciences 123(2):139–180 (2004), doi:10.1016/j.thbio.2004.04.001. |
| LIT-1242 v2 | Aggregation of Variables and System Decomposition: Applications to Fitness Landscape Analysis | Shpak, Stadler, Wagner & Hermisson (2004). Separates aggregation of variables, a lower-dimensional map whose dynamics close and which may depend on every microvariable, from decomposition of a system into subsets of variables, and treats systems that are both as a special case, with applications to fitness landscapes. | Active — read 2026-10-20 (NOTE-979) in full, from the authors’ manuscript in Leipzig University’s repository; worth reading for the distinction between aggregation (a closed lower-dimensional description) and decomposition (non-interacting subsets), for the exact condition under which a partition’s block sums close in a linear system (column-equitability, THEORY-755), and for its application to fitness landscapes, exact for one or two peaks and not for three. Read it knowing that it treats exact aggregation only, paraphrases Simon–Ando without an error bound, states Result 1.1 in a form that fails for some real 2×2 matrices, and overstates in its discussion that every linear aggregation is an equitable partition. |
| LIT-1245 v2 | Optimal prediction and the Mori–Zwanzig representation of irreversible processes | Chorin, Hald & Kupferman (2000). For any flow with an invariant measure, writes the resolved variables’ exact evolution as Zwanzig’s generalized Langevin equation, with conditional expectation as the projection: a Markovian term E[LA|A], a memory integral and a noise with zero conditional mean. First-order optimal prediction is the Markovian term alone. In a perturbative nonlinear Schrödinger example, keeping the memory recovers the decay of the conditional mean that the Markovian scheme misses. | Active — read 2026-10-20 (NOTE-989) in full, from the authors’ PostScript copy on Chorin’s Berkeley page; worth reading for the generalized Langevin identity of the resolved variables, with conditional expectation under an invariant measure as the projection (Markovian term R(A) = E[LA|A], memory K(s, A) = E[LS⊥ˢF|A], noise S⊥ᵗF with zero conditional mean), for first-order optimal prediction as its Markovian part, and for the renormalized Hamiltonian −log∫e^{−H} in the canonical case. Read it knowing that its claim that no m-equation Markovian model beats first-order optimal prediction in mean square holds only for the instantaneous rate, that it states no fluctuation–dissipation relation and no criterion for when memory may be dropped, and that its one demonstration (m = 0, perturbative, kernels cut off at a time chosen to give a reasonable decay) shows the decay of the mean but not a faithful mean after t ≈ 2. |
| LIT-1246 v3 | Diffusion maps | Coifman & Lafon (2006). From a symmetric non-negative kernel, a reversible random walk; its t-step diffusion distance equals the Euclidean distance between eigenvectors weighted by λ_l^t, and truncating where |λ_l|^t ≤ δ|λ₁|^t gives the diffusion map, whose dimension falls as t grows. An α-normalization by a kernel density estimate gives, as the kernel width ε → 0, a generator with invariant density q^{2(1−α)}: the density-weighted graph Laplacian at α = 0, the Fokker–Planck (Langevin) operator at α = 1/2, and the Laplace–Beltrami operator, free of the density, at α = 1. Symmetric kernels only. | Active — read 2026-10-20 (NOTE-985) in full; worth reading for QUESTION-217 as the model case of a coarse-to-fine order induced by an operator rather than an imposed partition, and for the proofs the PNAS announcement (LIT-981) deferred. The diffusion distance D_t is exactly the Euclidean distance between eigenvector coordinates weighted by λ_l^t. The number of steps t is a scale, and the kernel width ε is a second one. An α-normalization makes the small-ε limit the graph Laplacian with a density term (α = 0), the Langevin generator whose invariant density is the sampling density (α = 1/2), or the Laplace–Beltrami operator (α = 1). Read it knowing that Proposition 1’s “relative accuracy δ” for the truncated map is wrong (the bound is absolute, THEORY-745); that Appendix A’s ψ_l = φ_l/π should be φ_l/√π; that the α = 1/2 display, the circle formula, the discrete kernel of §5 and Criterion 4’s exponent are misprinted or wrong; that the boundary rate and the heat-kernel proof are incomplete (THEORY-744); and that no spectral gap is used anywhere. |
| LIT-1247 v2 | Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff | No, Ingber & Weissman (2015; IEEE TIT 2016). Asks whether successive refinement loses anything at second order, the √n dispersion term, as well as at first. For finite alphabets it gives an achievable rate pair with dispersions (Theorem 5), from which a source is strongly successively refinable if every law in a neighbourhood of it is successively refinable at (D₁, D₂) (Corollary 6). Discrete sources under Hamming distortion and the Gaussian under squared error qualify. It then uses many refinement stages as a complexity device: with L_n stages the encoder searches L_n small codebooks instead of one exponential one, at a rate penalty of order L_n log n / n, shown for binary-Hamming and Gaussian sources. Nothing in it concerns logarithmic loss. | Active — read 2026-10-20 (NOTE-980) in full, from arXiv:1506.03407v2; worth reading for successive refinement at second order (source dispersion): a sufficient condition, Corollary 6, that a source refinable at (D₁, D₂) throughout a neighbourhood of its law is also refinable with both dispersion terms, so that discrete sources under Hamming distortion and the Gaussian under squared error lose nothing even at √n precision; and for layered codes, which use a growing number L_n of refinement stages to trade encoding complexity against rate, with every stage on the rate–distortion curve. Read it knowing that it says nothing about logarithmic loss or the information bottleneck, though NOTE-988 recalled it as the source of a log-loss result (that result is No and Weissman, ISIT 2015, and No, Entropy 2019, and concerns direct log-loss, not the bottleneck’s remote one); that its sufficient condition asks for refinability in a whole neighbourhood of the source, not at the source alone; that the layered-code results cover only binary-Hamming and Gaussian-quadratic sources; and that the abstract’s “O(n^{log n}) algorithm” is an encoding search over codebooks shown to exist, not constructed, with the bound holding as stated only when the rate is below one nat. |
| LIT-1248 v3 | Onsager’s conjecture on the energy conservation for solutions of Euler’s equation | Constantin, E & Titi (1994). Proves the positive half of Onsager’s conjecture: a weak solution of the incompressible Euler equations on the periodic box with u ∈ L³(0,T; B^α_{3,∞}) ∩ C([0,T]; L²) and α > 1/3 conserves energy, because the coarse-grained energy flux across scale ε is O(ε^(3α−1)). Hölder regularity above 1/3 is a special case. It says nothing about solutions below 1/3. | Active — read 2026-10-20 (NOTE-982) in full, from the authors’ manuscript (the url:); worth reading for a complete one-page proof that a weak solution of incompressible Euler with L³-averaged increments scaling as |y|^α, α > 1/3, conserves energy (THEORY-749), and for its mechanism: the energy flux into scales below a mollification scale ε is a commutator of size ε^(3α−1). Read it knowing that it is the positive half of Onsager’s conjecture only and shows nothing about rougher fields: that dissipative solutions exist below 1/3 is Isett (2018) and others, and some rough solutions conserve energy anyway; that its “physical interpretation” (ε̄ = 0) passes to Navier–Stokes and to ensemble averages without proof; that the endpoint 1/3 is not covered; and that the manuscript misprints the exponents in the middle line of its final estimate and integrates (2) over R³ for the box D, neither affecting the result. |
| LIT-1250 v2 | Diffusion maps, spectral clustering and reaction coordinates of dynamical systems | Nadler, Lafon, Coifman & Kevrekidis (2005; ACHA 2006). For data sampled from p = e^{−U}, it expands the α-normalized diffusion-map operators for small kernel width. The normalized graph Laplacian (α = 0) tends to a backward Fokker–Planck generator with potential 2U, equivalently a Schrödinger operator with potential |∇U|² − ΔU. The normalization α = 1/2 gives the generator of the Langevin dynamics ẋ = −∇U + √2ẇ itself. α = 1 gives the Laplace–Beltrami operator, free of the density. So for data from a stochastic system in equilibrium, the leading eigenfunctions approximate the system’s slowest modes, and the paper proposes them as reaction coordinates. Examples: Ornstein–Uhlenbeck processes, where the first eigenfunctions parametrize the slow variable; a double well, where the first eigenfunction is ±1 on the two wells; three wells; and the iris data. The expansions are formal, and the printed generators are twice what the stated kernel and time step give. | Active — read 2026-10-20 (NOTE-986) in full, from arXiv:math/0503445v1; worth reading for the bridge from a diffusion map to the dynamics behind the data. For samples from p = e^{−U}, the ordinary normalized graph Laplacian (α = 0) tends to a backward Fokker–Planck generator with potential 2U, and the anisotropic normalization α = 1/2 to the generator of the Langevin dynamics with potential U itself, whose slow eigenfunctions the paper proposes as reaction coordinates. It also has worked examples: Ornstein–Uhlenbeck (the finite-ε spectrum λ_k = (τ/(τ+ε))^k), a double well whose first eigenfunction is ±1 on the two wells, and a three-well potential with two pathways. Read it knowing that the generators as printed are twice what the paper’s own kernel and time step give (its Ornstein–Uhlenbeck lemma and its equation (32) disagree by that factor, which this reading checked); that “converge” means a formal small-ε expansion of the operator, with the large-sample limit cited and no spectral convergence shown; that its sufficient condition for the first eigenfunctions to depend only on the slow variable is printed the wrong way round; that its double-well example is separable, so the slow coordinate is closed by construction; and that from equilibrium samples the eigenfunctions are recovered well while the slow eigenvalue, the inter-well rate, is not. In what sense slow eigenfunctions are closed coarse variables is the record’s addition (THEORY-762). The published ACHA version was not reached. |
| LIT-1251 v2 | Turbulence: the filtering approach | Germano (1992). Develops algebraic relations among filtered fields and subgrid stresses at different filter widths, the basis of the Germano identity used in dynamic subgrid models. | Unavailable — filed 2026-10-19, not read. On 2026-10-20 a search for a lawful full text reached none. Cambridge Core’s page gives the abstract and offers the article only by purchase or institutional access; Crossref lists only a similarity-checking link and Cambridge’s terms. Unpaywall, OpenAlex and Semantic Scholar report it closed, with no repository copy. Germano’s institutional records hold no copy: Politecnico di Torino’s IRIS finds only a 2021 paper on the filtering approach, a different work, and the UNIMORE catalogue lists the same 2021 paper; Duke Scholars lists the citation with no file. arXiv has nothing by Germano under this title. A title search found no author-posted copy; the nearest was a CFD Online forum thread of readers asking each other for it. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1252 v2 | Locality of turbulent cascades | Eyink (2005). Treats locality of forward and inverse cascades, in three- and two-dimensional turbulence, in one coarse-graining framework, so that locality of transfer and its direction are separate properties. | Unavailable — filed 2026-10-19, not read. On 2026-10-20 a search for a lawful full text reached none. ScienceDirect returned a 403 bot-check page, and Unpaywall, OpenAlex and Semantic Scholar report it closed with no repository copy. Eyink’s Johns Hopkins pages (ams.jhu.edu/~eyink/, including publications.html), where he posts many of his papers, returned a Cloudflare block to both clients tried, and the Internet Archive refused the connection when tried for an archived copy of them. So whether Eyink posts this paper is unknown: the failure there is on access, not availability. An arXiv search of Eyink’s papers found no preprint of it, only later companions (Eyink & Aluie 2009, arXiv:0909.2386 and 0909.2451). A title search found only a ResearchGate record and citing papers. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1253 v3 | Optimal prediction with memory | Chorin, Hald & Kupferman (2002). Writes the resolved variables of an ODE with uncertain unresolved initial data as an exact generalized Langevin equation (Markovian term, memory, noise) under three projections, computes the memory from equilibrium correlation functions through a Volterra equation, and introduces the t-damping (t-model) equation, whose leading-order memory term dissipates the effective energy of separable Hamiltonian systems. On one 4-variable example the memory restores the decay of the conditional mean that the Markovian equation misses. No error bounds. | Active — read 2026-10-20 (NOTE-975) in full, from the published version on Chorin’s Berkeley page; worth reading for the exact generalized Langevin equation under three projections (conditional expectation, linear, finite-rank), for computing the memory kernel from equilibrium correlation functions through a Volterra equation, and for the t-damping (t-model) equation dΦ/dt = R̄(Φ) + tS(Φ) with its proof that, for separable Hamiltonians, it dissipates the effective energy (THEORY-752). Read it knowing that it has one 4-variable example, no error bound for any approximation, no Markovian augmentation by memory coordinates, that its abstract’s “close to the best possible estimate” holds on that example only to t ≈ 8, and that the t-model it calls a short-memory approximation treats the kernel as constant over the whole window. |
| LIT-1255 v2 | A proof of Onsager’s conjecture | Isett (2016; Annals 2018). For every α < 1/3 it constructs a nonzero weak solution of the 3D incompressible Euler equations on R × T³, with velocity in C^α and pressure in C^{2α} in space and time, that is identically zero outside a finite time interval, so its kinetic energy is not constant. This is the negative half of Onsager’s conjecture; the positive half, conservation above 1/3, is Constantin, E and Titi’s. The construction iterates a Main Lemma that adds high-frequency corrections built from Mikado flows, after a new “gluing” step that rearranges the error so that it lives on short, separated time intervals and the corrections never interact. | Active — read 2026-10-20 (NOTE-974), skimmed, from arXiv:1608.08301v2: the introduction, the statement of the theorem, the Main Lemma and its three sublemmas, the gluing construction and the convex-integration ansatz, and the whole proof of the theorem from the Main Lemma; the long estimates of §§9–10 and 17 and the appendix were not checked. Worth reading for the negative half of Onsager’s conjecture: for every α < 1/3 there is a nonzero weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous with exponent α in space and time, that vanishes outside a finite time interval and so does not conserve energy (THEORY-763). With Constantin, E and Titi (LIT-1248) it fixes 1/3 as the threshold. Read it knowing that its solutions gain energy from zero and lose it again, so they are not dissipative (admissible) solutions, which came later (Buckmaster, De Lellis, Székelyhidi and Vicol 2019); that they are built by convex integration from the zero solution and so also show non-uniqueness, not that turbulent or vanishing-viscosity limits are of this kind; that the endpoint 1/3, two dimensions and finite-energy solutions on R³ are left open; and that the arXiv v2 copy read is a 2024 revision, not compared with the Annals text. |
| LIT-1256 v3 | Diffusion wavelets | Coifman & Maggioni (2006). A self-adjoint diffusion operator T defines a multiresolution analysis by thresholding its dyadic powers at a precision ε: V_j is spanned by the eigenvectors with λ^t ≥ ε, so the levels are exactly nested spectral bands set by the operator, not by a partition. The contribution is an algorithm that squares T on each compressed orthonormal basis to build localized scaling functions and wavelets for approximations of these spaces, and compressed dyadic powers that apply the Green’s function fast. Proofs cover the uncompressed construction; the compressed scheme’s error and cost are argued, not proved. | Active — read 2026-10-20 (NOTE-984) in full; worth reading for QUESTION-217 and QUESTION-218 as the multilevel case of an operator-induced coarse-to-fine hierarchy. The levels V_j are spanned by the eigenvectors of a self-adjoint diffusion T with λ^t ≥ ε at dyadic t. They are exactly nested, the wavelet spaces W_j are spectral bands of about one octave, and a localized orthonormal basis of each level is built without computing eigenvectors. Read it knowing that the levels are spectral bands and not spatial blocks (the dyadic cubes only shape the basis); that nesting needs no spectral decay, which buys only compression; that the compressed algorithm as written thresholds T^{2^{j−1}}, not T^{2^j−1} as §5.4 and Fig. 1 say, and the paper’s own Fig. 9 follows the former (THEORY-758); that the indexing of V_j is inconsistent between (4.8), §4.6 and Theorem 27; that only the uncompressed route (Theorems 27 and 29) has proofs, while the compressed scheme’s error, localization and O(n log² n) cost are argued or observed; and that the non-self-adjoint extension is asserted, not carried out. |
| LIT-1259 v2 | Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism | Parish & Duraisamy (2017; arXiv 2016). Derives subgrid closures for large-eddy simulation from the Mori–Zwanzig formalism by Stinis’s finite-memory approximation, which replaces the memory integral by auxiliary relaxation equations for the subgrid force: a finite Markovian augmentation of the resolved modes, with no error bound and a memory length fitted to DNS (Burgers: about 0.2 over the spectral radius of the resolved Jacobian) or preset. On Burgers, isotropic turbulence, the Taylor–Green vortex and channel flow it tracks filtered DNS about as well as dynamic Smagorinsky, and better on the transitional Taylor–Green vortex. | Active — read 2026-10-20 (NOTE-973) in full; worth reading for the clearest working example of a Mori–Zwanzig memory term turned into a finite Markovian augmentation (Stinis’s finite-memory closure: the resolved modes plus a relaxing subgrid-force field), tested against filtered DNS on Burgers and four Navier–Stokes flows. Read it knowing that the augmentation’s error is not bounded, the memory length is fitted or preset, the zero-variance projection makes the formalism a deterministic trajectory closure, “finite support” is exponential decay, and the comparisons are plots without error norms. THEORY-760 (Active) and THEORY-747 (Proposed) are filed from it. |
| LIT-1282 | Random Hyperbolic Graphs: Degree Sequence and Clustering | Gugelmann, Panagiotou and Peter (2012), ICALP. The first rigorous results for the threshold (T = 0) hyperbolic random graph of Krioukov et al., at curvature −1 with radial density exponent α > 1/2 and disk radius R = 2 log n + C: the expected average local clustering coefficient is Θ(1) and concentrated, the fraction of vertices of degree k is given exactly up to 1 + o(1) for k up to a power of n and has a power-law tail with exponent 2α + 1 up to the maximum degree, the average degree is a constant, and the maximum degree is n^(1/(2α) + o(1)). | Active |
| LIT-1283 | Clustering and the hyperbolic geometry of complex networks | Candellero and Fountoulakis (2015), Internet Mathematics. For Krioukov et al.’s hyperbolic random graph with curvature −ζ², density exponent α and 0 < ζ/α < 2, in the soft model with connection probability 1/(exp(β(ζ/2)(d − R)) + 1): when β > 1 (temperature below 1) the global clustering coefficient (three times triangles over paths of length two) converges in probability to an explicit positive constant if ζ/α < 1 and to zero if 1 ≤ ζ/α < 2; when β ≤ 1 it converges to zero. Restricted to vertices of bounded type it stays positive for every ζ/α < 2 when β > 1. | Active |
| LIT-1289 v2 | The L³-based strong Onsager theorem | Giri, Kwon & Novack (2023; Annals of Mathematics 2026). For every β < 1/3 there is a weak solution of the 3D incompressible Euler equations on T³ in C⁰([0, T]; B^β_{3,∞} ∩ L^{1/(1−3β)}) that satisfies the local energy inequality ∂_t(½|u|²) + div((½|u|² + p)u) = −D[u] ≤ 0 and strictly dissipates its total kinetic energy: the L³-based strong Onsager conjecture, previously known only for β < 1/7 (in Hölder spaces, De Lellis and Kwon). In the proof D[u] is exactly a prescribed smooth density, −e(t)e′(t)ρ₀(x₁, x₂)² with e(t) a decaying exponential and ρ₀ a pipe flow along e₃, so dissipation is strictly positive inside a periodic array of pipes and zero outside. The construction is an intermittent, L³-measured convex integration in which successive velocity increments overlap in frequency (“partial wavelet sums”), kept apart in space by multi-scale “Mikado bundles”, with an intermittent pressure that dominates every error pointwise. Section 3’s proofs are in a companion paper (Annals of PDE 10:19, 2024). | Active — read 2026-10-20 (NOTE-1022), skimmed, from arXiv:2305.18509v3: the introduction and its heuristics, the whole of Section 2 with the inductive Proposition 2.11 and the proof of Theorem 1.1 from it, the derivation of the new current errors (§5.1) and the closing of the relaxed local energy identity (§6.3, Lemma 6.10) were read; the estimates of Sections 3–6, the dodging lemmas of Section 4, the parameter choices of Section 7 and the appendix were paged through and not checked, and Section 3’s proofs are in a companion paper (Annals of PDE 2024) that the record does not hold. Worth reading for the local, L³-based version of the negative half of Onsager’s conjecture: for every β < 1/3 there is a weak solution of the 3D incompressible Euler equations on the periodic box in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}) that satisfies the local energy inequality and whose total energy strictly decreases; and, from the proof rather than the theorem’s statement, for the fact that its Duchon–Robert dissipation is exactly the smooth, nonnegative density E(t, x) = −e(t)e′(t)ρ₀(x₁, x₂)² fixed at the iteration’s first step, positive in a periodic array of straight pipes and zero outside them (THEORY-772). So the coarse-grained energy flux into vanishing scales converges, as a distribution in space and time, to a one-signed density: one-way at every place and time in the limit, which Buckmaster, De Lellis, Székelyhidi and Vicol (LIT-1300) showed only for the whole box. Read it knowing that the solutions are not Hölder or even bounded (the strong Hölder version is open; the best Hölder result with the local inequality is β < 1/7, De Lellis and Kwon); that where and how fast energy is dissipated is an input of the construction, carried unchanged through every step, not something the dynamics selects; that the one-signed statement is about the limit ℓ → 0 tested against nonnegative functions, and nothing is said about the sign of the flux at any finite scale; that nothing ties these solutions to vanishing-viscosity limits, though the paper notes that the local inequality is what such limits would satisfy; and that the arXiv v3 copy read is the authors’ final version, not compared with the Annals text. |
| LIT-1292 v3 | A contraction based, singular perturbation approach to near-decomposability in complex systems | Bousquet & Slotine (arXiv 2015, unpublished). Recasts Simon’s near-decomposability as a singularly perturbed system, fast module states x̃ and slow states y with ẏ = O(ε), and bounds the error of the reduced slow system with nonlinear contraction theory. If each module is contracting, the reduced slow system is contracting, and the coupling terms grow at most affinely in |x̃| and |y|, then below an explicit critical separation ε_c, after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε), the fast states stay O(ε) from their quasi-steady values and the slow states O(ε) from the reduced trajectory, for all later time. A worked building example with a negative conductance shows multiple equilibria and instability above ε_c. Deterministic; bounds trajectory error, not dependence; proofs are sketches with a printed inversion in Remark 8. | Active — read 2026-10-20 (NOTE-1025) in full; worth reading for the only rate the record holds for Simon’s near-decomposability: with contracting fast modules (rate β_f) and a contracting reduced slow system (rate β_g), and coupling perturbations bounded affinely in the off-manifold state, Lemma 3 and Theorem 1 keep the fast variables within O(ε) of their quasi-steady manifold and the slow variables within O(ε) of the reduced system, uniformly in time after a transient of (1/β_f + 1/(εβ_g)) log(1/ε), for every ε below an explicit critical timescale separation ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)]; and for Remark 6, that the micro states act on the macro level through their quasi-steady values, “an aggregation in time”, not an ensemble aggregate. Read it knowing that it is a five-page preprint, never published in a journal, whose proofs are sketches; that Remark 8 prints ε_c inverted (the worked example’s value √2/7 is the reciprocal of the printed expression, and matches Hypothesis 2); that Lemma 3’s O(ε) bound on the slow error needs d_g = 0, which the proof’s last line assumes silently while the worked example lists d_g = 1; that it is deterministic throughout and bounds trajectory error only, not any information-theoretic dependence; that the slow coordinates and the partition are given, not found; and that ε_c is sufficient and in linear checks conservative by factors of 3 to 50. |
| LIT-1294 | Near-Decomposability, Partition and Aggregation, and the Relevance of Stability Discussions | Ando & Fisher (International Economic Review 1963). The sequel to Simon & Ando (1961), standardly cited as giving the proofs of the near-decomposability theorems: in a linear dynamic system whose matrix is nearly block-diagonal, the short run is governed within blocks and the long run by the blocks’ aggregates. It also asks when stability analysis of the decomposed parts is relevant to the whole. Not read; this summary is from the title and how the paper is cited. | Unavailable — registered 2026-10-20, not read. A search on 2026-10-20 for a lawful full text reached none. JSTOR (stable 2525455, the DOI’s landing) returned a “Client Challenge” bot-check page instead of the article. Crossref names JSTOR as the publisher and lists no full-text link. OpenAlex, Unpaywall and Semantic Scholar all report it closed with no repository copy. The 1963 book that reprints it, Ando, Fisher and Simon, Essays on the Structure of Social Science Models (MIT Press), is on the Internet Archive in two scans (essaysonstructur0000ando, essaysonstructur0000unse), both access-restricted lending-only items; Google Books shows it with no pages viewable; the MIT Press page returned 403 to this client. No Cowles Foundation discussion paper was found for it: web searches of the Cowles series on Yale’s EliScholar turned up nothing by Ando and Fisher on near-decomposability, and EliScholar’s own search returned 403. CORE returned 429 and BASE a bot check. Title searches, with and without filetype:pdf, found no author-posted copy. It stays Unavailable until a copy is supplied, by library access or the book. No NOTE is filed. |
| LIT-1295 v2 | A Contraction Theory Approach to Singularly Perturbed Systems | Del Vecchio & Slotine (IEEE TAC 2013; conference version CDC-ECC 2011). Revisits singular perturbation with nonlinear contraction theory. For ẋ = f(x, z, t), εż = g(x, z, ε) with g contracting in z, the fast state’s distance from the slow manifold z = γ(x) obeys an explicit global bound, decaying at rate λ_z/ε to a floor εχ_z(d + K)/λ_z (Lemma 2). With a contracting slow subsystem the slow state stays within an explicit O(ε) of the reduced trajectory for all t ≥ 0 (Theorem 3), and the result extends to three time scales and to retroactivity attenuation in a phosphorylation cycle. Theorem 3 as stated assumes only partial contraction of f in x; its proof needs, and a counterexample shows it requires, contraction of the reduced system. Metrics are constant throughout. | Active — read 2026-10-20 (NOTE-1020) in full, from the authors’ revised manuscript of 22 December 2011 on Del Vecchio’s MIT pages, with the CDC-ECC 2011 conference version (MIT DSpace) for comparison; worth reading for the contraction robustness lemma with its one-line proof (Lemma 1: a system contracting at rate λ in a constant metric Θ, disturbed by d with |d| ≤ d̄, stays within χe^{−λt}|Δ(0)| + χd̄/λ of an undisturbed trajectory), for an explicit, global, uniform-in-time bound on the distance of the fast state from the slow manifold (Lemma 2: χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z), and for an explicit O(ε) bound on the error of the quasi-steady-state (Markovian) reduction for all t ≥ 0 (Theorem 3), extended to three time scales and applied to retroactivity in a phosphorylation cycle. Read it knowing that Theorem 3 as stated is false: it assumes only that f is partially contracting in the slow state x, while its proof uses contraction of the reduced system ẋ = f(x, γ(x), t), and a two-variable example satisfying every stated hypothesis has a bistable reduced system and an O(1) error (NOTE-1020); with the reduced system assumed contracting, which the CDC version’s introduction says, the theorem and its proof are correct. Read it also knowing that its metrics are constant (state- and time-dependent metrics are mentioned, not used), that the bound d on |∂γ/∂x · f| is global and cannot hold for a linear system, though the proof needs it only along the trajectory, that the example’s contraction rate λ_z is printed as a max where it is a min, and that the copy read is the revised manuscript, not the version of record. |
| LIT-1300 v2 | Onsager’s conjecture for admissible weak solutions | Buckmaster, De Lellis, Székelyhidi & Vicol (2017; CPAM 2019). For every β < 1/3 and every smooth strictly positive e : [0, T] → R there is a weak solution of the 3D incompressible Euler equations on T³ × [0, T], in C^β in space and time, with ∫|v(x, t)|² dx = e(t) for every t. With e strictly decreasing these are dissipative (admissible in the sense of a non-increasing total energy), which Isett’s solutions were not. An h-principle (Theorem 1.3) adds that any smooth strict subsolution is a weak-* limit of such solutions. The proof modifies Isett’s: a gluing step whose Reynolds stress stays trace-free, then Mikado-flow perturbations placed in disjoint “squiggling” space-time regions that cover every time, so energy can be pumped in even where the stress vanishes. | Active — read 2026-10-20 (NOTE-1013) in full, from arXiv:1701.08678v1: worth reading for the admissible half of Onsager’s negative direction: for every β < 1/3 and every smooth, strictly positive energy profile e(t) on [0, T] there is a weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous with exponent β in space and time, whose energy ∫|v|² equals e(t) at every time; taking e strictly decreasing gives solutions whose kinetic energy strictly decreases, so the energy flux into vanishing scales is one-signed on every time interval (THEORY-776); and for a shorter, more self-contained proof of the 1/3 threshold than Isett’s (LIT-1255), with a trace-free glued stress and perturbations added everywhere in time so as to steer the energy. Read it knowing that “admissible” here means only that the total energy does not increase (the word appears in the title alone), not the local energy inequality, so the transfer is one-way for the whole box and may run backwards in places; that the same theorem gives solutions with increasing energy just as readily, so the direction is chosen by the prescribed profile and is not forced by the equations; that nothing is shown about vanishing-viscosity limits, about uniqueness or non-uniqueness for given initial data, or about the endpoint 1/3; that the h-principle (Theorem 1.3) rests on a result of Daneri and Székelyhidi stated without proof; and that the abstract’s “typical” is looser than the weak-* density that Theorem 1.3 proves. |
| LIT-1304 v2 | Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics | Chorin & Lu (PNAS 2015). Proposes model reduction done entirely in discrete time: the resolved variables are advanced by a fixed numerical scheme plus a discrepancy sequence computed exactly from data, and the discrepancy is fitted as a NARMAX time series (autoregression in itself, lags of functions of the resolved variables, moving average in past innovations). Memory is thereby carried by finitely many lags. The connection to Mori–Zwanzig is drawn by analogy, not derived. On two-scale Lorenz 96 with ε = 0.5, a fit with one autoregressive lag, one or two exogenous lags and at most one moving-average term reproduces the statistics of x₁ better than a polynomial-plus-AR(1) closure, and at observation spacing 0.05 extends the lead time at anomaly correlation 0.6 from 2 to 4. Nothing is proved and no error bound is given. | Active — read 2026-10-20 (NOTE-1028) in full; worth reading for the fully discrete stochastic parametrization: fix a numerical integrator for the resolved equations, take the discrepancy z^{n+1} = (x^{n+1} − x^n)/δ − R_δ(x^n), which the data give without differentiation, and fit it as a NARMAX time series with x as exogenous input, so that memory of past x, past z and past innovations is carried by finitely many lags; for the conditional-likelihood (least-squares) fit; and for a Lorenz 96 test with no time-scale separation (ε = 0.5), where one or two lags reproduce the pdf, autocorrelation and cross-correlation of x₁ and, at δ = 0.05, double the forecast lead time over a polynomial-plus-AR(1) closure (THEORY-818). Read it knowing that the link to Mori–Zwanzig is motivation, not derivation: nothing shows that the fitted Φ approximates the MZ memory or that ξ is the MZ noise, which is orthogonal to the resolved variables but neither white nor independent; that “generalizes the MZ formalism” is an overstatement; that the model structure is found “by trial and error” and changes with δ; that nothing is proved, no error bound is given and no case is shown in which a finite discrete memory suffices exactly; and that the δ = 0.01 fit has a moving-average coefficient 0.9997, at the edge of invertibility. |
| LIT-1305 v2 | Nonuniqueness and existence of continuous, globally dissipative Euler flows | Isett (2017; Archive for Rational Mechanics and Analysis 2022). For every α < 1/15 there are weak solutions of the incompressible Euler equations on T³ (and T^d, d ≥ 3) in C^α_{t,x} that satisfy the local energy inequality ∂_t(|v|²/2) + ∇·((|v|²/2 + p)v) ≤ 0, strictly on an open time interval: the first continuous examples, and a first partial result on the strong Onsager conjecture. From one initial datum there is a Cantor family of them, with no isolated points, positive Hausdorff dimension in C_tL²_x and one common dissipation measure; the same holds for solutions with the local energy equality, and wild data are C⁰ dense. The scheme is a convex integration of “dissipative Euler–Reynolds flows” with one-directional transported plane waves, whose triad interactions cancel the unresolved energy flux current. In the proof the dissipation measure is a function of time alone, switched on only while the construction drains the solution to zero. | Active — read 2026-10-20 (NOTE-1027), skimmed, from arXiv:1710.11186v4: the introduction, Sections 2 and 3 (the dissipative Euler–Reynolds flow and the Main Lemma), Sections 4.1–4.5 (the shape of the construction, the coefficient equations and the energy increment), the opening of 4.10, Section 4.14, Section 5 and Section 6 in full were read; Sections 4.6–4.13, the estimates, were paged through and not checked. Worth reading for the first continuous, indeed C^α_{t,x} with α < 1/15, weak solutions of the 3D incompressible Euler equations that satisfy the local energy inequality with nonzero dissipation (Theorems 1 and 4); for nonuniqueness of such solutions from one initial datum, as a Cantor family with a common dissipation measure and positive Hausdorff dimension in C_tL²_x, and of energy-conserving ones (Theorems 4, 5 and 7); for C⁰ density of wild initial data (Theorem 8); for the notion of a dissipative Euler–Reynolds flow, whose relaxed local energy inequality carries an unresolved flux density and current; and, from the proof rather than any statement, for the fact that the dissipation measure is a function of time alone, equal at each time to the rate of loss of total energy and so spread uniformly over the torus (THEORY-821). Read it knowing that “globally dissipative” means the local energy inequality (not whole-box dissipation, and not strict); that the exponent 1/15 is far below 1/3 (De Lellis and Kwon later reached 1/7 in Hölder spaces, Giri, Kwon and Novack 1/3 in L³-based ones); that the dissipation happens only in a short final interval in which the construction drains the solution to zero, its timing an input; that the shared initial datum is produced by the construction, not prescribed; that the “probabilistic method” is a binary choice per stage, not a random construction; and that the Burgers example of §1.2.1 misstates two constants (−d/dt∫u₀² is 1/3, not 2/3, and the threshold (1/6)^{1/3} is sufficient but not sharp, which is (1/4)^{1/3}), without affecting its point. |
| LIT-1306 v2 | Software in the natural world: A computational approach to hierarchical emergence | Rosas, Geiger, Luppi, Seth, Polani, Gastpar & Mediano (arXiv 2024). Defines three exact closures of a coarse-graining Z of a stationary discrete process X. In informational closure the micro past adds nothing to predicting Z’s future (Barnett–Seth dynamical independence). In causal closure Z’s ε-machine equals its υ-machine, the coarsest partition of micro pasts that predicts Z. In computational closure Z’s ε-machine is a coarse-graining of X’s. It proves informational ⟺ causal closure, and causal ⇒ computational closure for spatial coarse-grainings, not conversely. It ties these to strong and weak lumpability of the micro ε-machine. It proposes the lattice of strongly lumpable coarse-grainings of that ε-machine, modulo computational equivalence, as a system’s “functional architecture”. The case studies are cellular-automaton parity, Ehrenfest, Glauber energy, a modular random walk, agent counts and Hopfield overlaps. Most of them close by a permutation symmetry. | Active — read 2026-10-20 (NOTE-1030) in full; worth reading for three exact notions of a self-contained coarse-graining Z of a stationary discrete process X and how they relate: informational closure, I(X⃗_t; Z⃗ᴸ_{t+1} | Z⃗_t) = 0 for every horizon, which is Barnett and Seth’s dynamical independence without an environment; causal closure, the ε-machine of Z equivalent to its υ-machine (the coarsest partition of micro pasts that predicts Z’s future); and computational closure, Z’s ε-machine a coarse-graining of X’s. Theorem 1 proves informational ⟺ causal closure for any coarse-graining of the past; Theorem 2 proves causal ⇒ computational closure for spatial ones, and Counterexample 4 the converse false. Proposition 4 and Theorem 3 tie closure to Kemeny–Snell lumpability: strong lumpability gives closure, closure with a Markov Z gives strong lumpability, weak lumpability gives only computational closure. Read it knowing that the sublattice claims are asserted and, for informationally closed spatial coarse-grainings, false under common refinement (a three-bit counterexample in the NOTE); that the stated sufficient condition for the Ising energy to close (a permutation-invariant Hamiltonian) fails with a uniform field at coincident energies; that Proposition 3 is a telescoping identity whose closure hypothesis is idle; that (A.11) and Example 2 contain slips; that the modular random walk closes by symmetry of the weights and not by modularity; and that every result is exact, with nothing on approximate closure. |
| LIT-1307 v2 | Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism | Lin & Lu (J. Comput. Phys. 2021). Writes data-driven model reduction for an invertible map with an invariant measure in Koopman-operator terms, derives the discrete Mori–Zwanzig equation from the Dyson formula, and introduces the Wiener projection: orthogonal projection onto the span of the entire past of chosen functions Ψ of the resolved variables. Under it the memory term vanishes, the reduced equation is the causal Wiener filter x_{n+1} = Σ_k Ψ(x_{n−k})h_k + ξ_{n+1}, and the residual is orthogonal to all of Ψ’s past. A rational approximation of the filter’s transfer function, called “uncontrolled” by the authors, turns it into a NARMAX-type recursion. That recursion is fitted by nonlinear least squares with a stability constraint and tested on Kuramoto–Sivashinsky and a stochastically forced Burgers equation. The Wiener projection is proved; the NARMAX derivation is heuristic and no error is bounded. | Active — read 2026-10-20 (NOTE-1029) in full; worth reading for the discrete-time Mori–Zwanzig identity derived from the Dyson formula in a few lines (Eq. 2.5–2.6), valid for any projection; for the Wiener projection, orthogonal projection onto the closed span of the whole past of chosen functions Ψ of the resolved variables, under which the Dyson formula collapses (Eq. 3.5), the memory term disappears into the projection, and the residual is orthogonal to all of Ψ’s past (Eq. 3.6b), not only to its present (THEORY-819); for the skew-product construction that brings randomly forced systems and bounded delays under the same formalism (§3.3); and for the cascade-of-quadratics parametrization that enforces stable memory during fitting (§4.1). Read it knowing that the step from the Wiener projection to NARMAX is a rational approximation the authors themselves call “uncontrolled” and “heuristic”, so no error bound and no convergence statement is proved; that the Wiener projection needs an invertible map and an invariant measure, and its claimed series representation (property (i)) needs more than the linear independence it states; that the KS and Burgers results are numerical, with orders chosen by trial for stability; that Appendix C says the noise term does not appreciably change the KS statistics or forecasts while §5.1 says that without it the models “do considerably worse at all tasks”; and that the record’s check shows the rational form is exact, with orders at most n − 1, for a scalar coarse variable of a stable linear Gaussian VAR(1), where the Wiener filter’s poles are the moving-average zeros, not the eliminated block’s eigenvalues. |
| LIT-1352 v2 | Towards a Mathematical Structure of Global Phenomenal Consciousness | Lee-Youngzie, Tsuchiya, Robinson, Dietz & Monti (2026), Entropy 28(6):615, a hypothesis article. Intra-phenomenal structures model the parts of one experience, and inter-phenomenal structures place an experience among others by similarity judgements. A presheaf maps each part, in a finite topology of parts, to the dissimilarity matrix measured for it. Agreement of narrow qualia on an overlap is read as phenomenal binding, and the presheaf’s being a sheaf as the broad quale’s being fixed by its parts. Worked for a two-hemifield visual field. A sheaf failure is read as disunity for binocular rivalry and as context dependence for the Ebbinghaus illusion. No data. | Active — read in full 2026-10-22 (NOTE-1059) from the version of record in PubMed Central (PMC13298370, CC BY 4.0, via the Europe PMC full-text XML); worth reading as the first worked proposal to join intra-phenomenal (part–whole) and inter-phenomenal (similarity) qualia structures by a presheaf, with phenomenal binding read as agreement of measured narrow qualia on overlaps and a sheaf as a broad quale fixed by its parts. It is a hypothesis article with no data. It does not identify unity with a global section: it assumes unity always holds, and it reads a sheaf failure as disunity for binocular rivalry and as context dependence for the Ebbinghaus illusion without a criterion between them. Its worked topology example is malformed, and by its own definition of gluing the rivalry case is not a sheaf failure. |
| LIT-1435 | Theory of Value: An Axiomatic Analysis of Economic Equilibrium | Debreu (1959), Theory of Value, Cowles Foundation Monograph 17, Wiley. The axiomatic theory of general equilibrium: commodities and prices, producers, consumers, equilibrium, optimum and uncertainty. Read only for §4.6 and its notes. There a continuous complete preorder on a connected separable space is proved to have a continuous utility function. The lexicographic ordering of R² is shown to have none. | Active — read in part 2026-10-23 (NOTE-1112): §4.6, “Continuity assumption on preferences” (pp. 55–59), and the notes to chapter 4 (pp. 72–73), from the PDF the Cowles Foundation posts as Monograph 17. Worth reading for Debreu’s own statement of the utility representation theorem and its limits. A complete preorder on a connected subset of Rˡ (more generally, a connected separable space) with closed upper and lower contour sets has a continuous utility function. The lexicographic ordering of R² has no real-valued representation at all. A utility function is “arbitrary to a large extent”. The rest of the book (commodities, producers, equilibrium, optimum, uncertainty) was not read. |
| LIT-1439 | Statistical ranking and combinatorial Hodge theory | Jiang, Lim, Yao & Ye (2011), Mathematical Programming 127:203–244. HodgeRank: an edge flow of aggregated pairwise comparisons splits orthogonally into a gradient flow, whose potential is the least-squares global ranking (a Borda count generalized to incomplete data), and a cyclic residual, which splits into curl (local) and harmonic (global) inconsistency (Theorems 4.8, 5.1). The residual’s share, the “cyclicity ratio”, certifies the ranking. | Active — read 2026-10-23 (NOTE-1099): in full, from arXiv 0811.1067v2 (42 pp.). HodgeRank. Pairwise comparisons, aggregated into a weighted edge flow on the graph of alternatives, decompose orthogonally into a gradient flow, the l2-best global ranking found by weighted least squares, and a divergence-free residual. The residual splits into curl (local, triangular inconsistency) and harmonic (global inconsistency) parts. Binary win/loss data enter as averaged signs, and on complete balanced binary data the ranking is Borda’s. The decomposition is cardinal: curl-freeness is stronger than transitivity. It is the tool for reading the record’s conflict record. |
| LIT-1442 | Flows and Decompositions of Games: Harmonic and Potential Games | Candogan, Menache, Ozdaglar & Parrilo (2011), Mathematics of Operations Research 36:474–503. The space of finite games is a direct sum P ⊕ H ⊕ N of potential, harmonic and nonstrategic subspaces (Theorem 4.1), pulled back from the Helmholtz decomposition of the game graph’s flow of unilateral payoff differences. Potential games are P ⊕ N (Theorem 5.1). Harmonic games, such as rock–paper–scissors, generically have no pure equilibrium, and the closest potential game is given in closed form under a weighted l2 norm (Theorem 6.2). | Active — read 2026-10-23 (NOTE-1116): in full, from arXiv 1005.2405v2 (45 pp.). Every finite game in strategic form splits uniquely into a potential, a harmonic and a nonstrategic component. The split pulls back the Helmholtz decomposition of the flow of unilateral payoff differences on the game graph. Potential games are exactly the games with no harmonic component. Harmonic games generically have no pure equilibrium. The nearest potential game, in a weighted l2 norm, is given in closed form. It is the paper behind the record’s measure of a potential’s share of a negotiation, and it shows that the share is cardinal. |
| LIT-1454 v2 | Representation of a preference ordering by a numerical function | Debreu (1954), in Thrall, Coombs & Davis (eds), Decision Processes, Wiley, pp. 159–165. Gives conditions under which a complete preference preorder is represented by a real-valued function. On a space satisfying the second axiom of countability, a continuous complete preorder has a continuous representation, with no connectedness needed. Unread: the result as Debreu restates it is read in Theory of Value (LIT-1435). | Unavailable — registered 2026-10-23, not read: no lawful full text could be reached. Looked for on 2026-10-23. The Cowles Foundation’s page for it (cowles.yale.edu/node/148338) lists it as a Discussion Paper, document control number 2083, with no file. The old Cowles Foundation Paper URL cited elsewhere (cowles.econ.yale.edu/P/cp/p00b/p0097.pdf) redirects to the Foundation’s home page. Its archive of Cowles Commission discussion papers lists the economics series only up to Econ 245, and guessed paths for the paper on its file server were not found. The only Crossref record is the 1983 reprint in Debreu’s Mathematical Economics (Cambridge University Press, ch. 6, pp. 105–110), which is paid, and Unpaywall reports no open copy of it. OpenAlex was not consulted (it refused requests without a key that day). Its theorem is read first-hand in Debreu’s Theory of Value, which the Cowles Foundation does post (LIT-1435, §4.6 and ch. 4 note 2). There Debreu proves the connected-separable case, credits that form to Eilenberg, and cites this paper for removing connectedness on second-countable spaces. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-1460 | Semiorders and a Theory of Utility Discrimination | Luce (1956), Econometrica 24(2):178–191. The paper that introduced semiorders: preference relations whose indifference need not be transitive, because differences below a threshold of discrimination go unnoticed. By its abstract, the class it axiomatizes is “substantially equivalent” to a utility theory with just-noticeable-difference functions. Unread: no lawful copy reached. | Unavailable — registered 2026-10-23 from Crossref, not read: no lawful full text could be reached. OpenAlex and Semantic Scholar list no open copy; JSTOR and the Econometric Society give the abstract only (the issue is not in JSTOR’s free early-journal content, which stops well before 1956); a web search found no author, course or repository posting. Filed because it is the paper that introduced the semiorder, the structure the record’s correction to CLAIM-436 turns on: a relation with a discrimination threshold whose indifference is not transitive. It stays Unavailable until a copy is supplied, not on merit. |
| LIT-1462 v3 | Potential Games | Monderer & Shapley (1996), Games and Economic Behavior 14:124–143. The paper defines exact, weighted, ordinal and generalized ordinal potentials for games in strategic form, and it is reported (Lemma 2.5, per Duersch et al.) to show that a finite game has the finite improvement property exactly when it has a generalized ordinal potential. Unread: no lawful copy was reached by this client. | Unfetched — registered 2026-10-23 from Crossref, not read: a lawful copy exists but this client could not fetch it, so this failed on access, not availability. OpenAlex lists the paper as closed, with no repository copy. The publisher (ScienceDirect) is paywalled. Two course-page copies were found: Frieze’s INFONET reading list at CMU, www.math.cmu.edu/~af1p/Teaching/INFONET/Papers/EconomicModels/PotGam.pdf, and a Bowdoin course page, tildesites.bowdoin.edu/~mirfan/files/3210/Monderer_Shapley_1996_Potential%20Games.pdf. Both failed TLS certificate verification from this client on 2026-10-23 and again on 2026-10-30 (curl error 60: the chain each serves ends at emSign Root CA - G1, which this client does not trust; verification was not overridden). A second fetcher got HTTP 503 from both on 2026-10-23. An author upload on ResearchGate (Monderer’s) was not tried, and Unpaywall was skipped because it requires an email address. Its central lemma, that a finite game has the finite improvement property exactly when it has a generalized ordinal potential, is held only as later papers restate it: Duersch, Oechssler and Schipper (arXiv 1003.4274) attribute it to Lemma 2.5, and Candogan et al. (LIT-1442) restate its equilibrium results. The record’s own proof of the finite case is in THEORY-896, which this paper bears on. Filed because the record is renaming “ascendancy” to “potential” in this paper’s sense. It stays Unfetched until a copy is supplied, not on merit; either course copy, fetched by hand, would clear it. Free copies: www.math.cmu.edu/~af1p/Teaching/INFONET/Papers/EconomicModels/PotGam.pdf; tildesites.bowdoin.edu/~mirfan/files/3210/Monderer_Shapley_1996_Potential%20Games.pdf. |
| LIT-1472 v3 | Intransitivity of preferences | Tversky (1969), Psychological Review 76(1):31–48. Shows experimentally that people’s choices can cycle predictably when options vary on two dimensions and differences on the more important one are small. The model is the lexicographic semiorder: the first dimension decides only when its difference exceeds a threshold, and otherwise the next one does. Unread: a lawful course copy exists but could not be fetched. | Unfetched — registered 2026-10-23 from Crossref, not read: a lawful copy exists but this client could not fetch it (“failed on access, not availability”, in ADR-041’s words). The paper is posted for teaching on a UC San Diego faculty page (pages.ucsd.edu/~mckenzie/Tversky1969PsychReview.pdf). This client refused the TLS connection, the server’s certificate chaining to a root not in its trust store; the web fetcher got a 503; the Internet Archive holds no snapshot. OpenAlex and Semantic Scholar list it as closed, APA’s copy is paywalled, a Central European University course copy sits behind a bot check, and a whole-book PDF of Tversky’s selected writings on another UCSD page was not used, its lawfulness being unclear. Filed because its lexicographic semiorder is the structure the correction to CLAIM-436 names. It stays Unfetched until a copy is supplied, not on merit. |
| LIT-1508 | Re-evaluating Evaluation | Balduzzi, Tuyls, Pérolat & Graepel (NeurIPS 2018). Evaluation data are antisymmetric matrices: Elo is their row mean in logit space, exact only when the log-odds are curl-free, and inflatable by copies; the Schur decomposition gives a multidimensional Elo for cyclic interactions. Nash averaging scores agents (and tasks) against the maximum-entropy Nash equilibrium of a zero-sum meta-game, which ignores exact copies when that equilibrium is unique. Applied to Atari, human ties with the best agents. | Active — read in full 2026-10-24 (NOTE-1155) from arXiv v2 (30 October 2018, the authors’ “NIPS 2018, final version”); worth reading as the source of the Hodge and Schur reading of evaluation data and of Nash averaging. Elo is the row mean of the log-odds matrix, exact only when the log-odds have no cyclic part, and copies of an agent move it. Nash averaging scores each agent against the maximum-entropy Nash team of the meta-game. Its invariance to copies is proved only for the Nash set (Lemma 1): when the meta-game has more than one equilibrium, copying an agent moves the maximum-entropy selection and the Nash averages with it, which the reading checked by script. The Atari re-evaluation used an LP solution, not the maximum-entropy one. |
| LIT-1608 v2 | Branching Processes: Variation, Growth, and Extinction of Populations | Haccou, Jagers & Vatutin (2005). A monograph on branching processes as models of populations, written for biologists: growth and extinction probabilities, single- and multi-type processes, and their use in establishing mutants and invading types. No abstract was seen. The summary is what the book is standardly cited for. | Unavailable — filed 2026-10-27, not read. Registered in the second pass on the exchange of 2026-10-27, “Repository coverage assessment”, at the owner’s request to find what it mentioned and did not cite (second curation entry of 2026-10-27). No NOTE is filed, and it stays Unavailable until it is read, not on merit. A lawful full text was looked for on 2026-10-27 and none was found: OpenAlex lists it as closed with no open location, Unpaywall reports it closed with no open locations, and Europe PMC has no free copy. |
| LIT-1632 v2 | The renormalization group: Critical phenomena and the Kondo problem | Wilson (1975). The renormalization group treats problems with many coupled scales by eliminating degrees of freedom scale by scale and following the flow of effective couplings; near fixed points most couplings are irrelevant, which explains universality in critical phenomena, and the method solves the Kondo problem. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: the article is closed on Unpaywall and OpenAlex (Reviews of Modern Physics), with no repository copy. It stays Unavailable until a copy is supplied. |
| LIT-1638 v2 | Geometric singular perturbation theory for ordinary differential equations | Fenichel (1979). For a fast–slow system εẋ = f(x,y), ẏ = g(x,y), a compact normally hyperbolic critical manifold of the fast subsystem persists for small ε as a nearby locally invariant slow manifold, with stable and unstable fibrations, so the reduced slow dynamics on it are a rigorous approximation. | Unfetched — searched 2026-10-27 for a lawful full text; one exists but this client could not fetch it, so it is not read and no NOTE is filed (ADR-053). Unpaywall lists the publisher’s open-archive copy on ScienceDirect, sciencedirect.com/science/article/pii/0022039679901529/pdf; it answered curl with an HTTP 403 Cloudflare challenge, and a headless- browser fetch was not attempted, since it would be working around the challenge. The failure is on access, not availability. A browser fetch of the open-archive copy is the cheapest way to read it. It stays Unfetched until a copy is supplied. |
| LIT-1639 v2 | Interconnection of port-Hamiltonian systems and composition of Dirac structures | Cervera, van der Schaft & Baños (2007). Power-conserving interconnections of port-Hamiltonian systems are described by Dirac structures, and composing Dirac structures gives again a Dirac structure, so interconnected port-Hamiltonian systems stay port-Hamiltonian; the paper characterizes when the composition is well defined. | Unfetched — searched 2026-10-27 for a lawful full text; one exists but this client could not fetch it, so it is not read and no NOTE is filed (ADR-053). Unpaywall lists the submitted version in the University of Groningen repository (research.rug.nl/files/2818803/2007AutomaticaCervera.pdf) and a bronze copy on ScienceDirect; the repository file answered curl with a bot challenge (“Just a moment…”), and a headless-browser fetch was not attempted, since it would be working around the challenge. The failure is on access, not availability. A browser fetch of that copy is the cheapest way to read it. It stays Unfetched until a copy is supplied. |
| LIT-1644 v2 | A Compositional Framework for Passive Linear Networks | Baez & Fong (2018), TAC 33. Open circuits of passive linear two-terminal components (resistors, and inductors and capacitors through the Laplace transform over ℝ(s)) are morphisms of a hypergraph category Circ built from decorated cospans, composed by gluing terminals. The black box functor Circ → LagRel, a hypergraph functor (Thm 1.1, Thm 7.7), sends a circuit to the Lagrangian relation between its terminals’ potentials and currents, the graph of the differential of the power functional left when the interior is eliminated by the principle of minimum power. The external behaviour of a composite is the composite of the parts’ behaviours. Dirichlet forms alone compose by that elimination but lack identities, since an ideal wire has none. | Active — read in full 2026-10-27 (NOTE-1223), from arXiv v6; worth reading as the theorem that black-boxing passive linear circuits is functorial: the Lagrangian relation a circuit imposes on its terminals’ potentials and currents composes as the circuits do. It is also a clean account of eliminating an interior by minimum power, and of why that elimination alone has no identities. Its scope is narrow: passive, linear, two-terminal components, time only through the Laplace transform with zero initial conditions. |
| LIT-1646 v2 | Opinion Dynamics on Discourse Sheaves | Hansen & Ghrist (2020; SIAM J. Appl. Math. 2021). A discourse sheaf gives each agent a private opinion space and each edge a space of discussed topics, with a linear expression map from each agent into each edge, which may be selective, scaled or sign-reversed (“lie”). Sheaf-Laplacian diffusion converges to the projection onto the global sections: agreement in expression, not in held opinion, so stable polarized distributions exist. Stubbornness, control and observability reduce to relative cohomology. Letting the expression maps diffuse converges to the nearest sheaf of which the opinions are a section (“learning to lie”); joint diffusion is gradient descent on the discord, and a conserved quantity keeps the opinions from collapsing to zero when it starts indefinite. Bounded confidence and antagonism are nonlinear Laplacians. | Active — read in full 2026-10-27 (NOTE-1222), from the arXiv preprint; worth reading as the clearest formal case of agreement on what is expressed without identity of what is held, and of correspondence maps revised by interaction. Mathematics with toy examples, no data; its expression learning reaches coherence with nothing outside the loop to check it, which the authors call “learning to lie”. |
| LIT-1649 v2 | Compositional Dynamics in Learning and Mechanics | Spivak (2026). An operad ArrSm of “smooth adaptive arrangements” (a responsive parameter space, a lens of output and input maps between manifolds, and a real potential) with two operad functors into polynomial coalgebras, discrete-time input-output machines, built by lens internalization (Corollary 3.3.4). The functors differ only in the integrator: Φconf stores position and steps down the sharpened gradient, Φphase stores position and momentum and takes a symplectic-Euler step. Φconf gives gradient descent with backpropagation, weight decay and Newton’s method. One arrangement of harmonic particles on a finite directed graph gives the discrete wave equation under Φphase and the discrete heat equation under Φconf, both governed by the symmetrized graph Laplacian, and for harmonic closed systems the phase step preserves the symplectic pairing exactly. | Active — read in full 2026-10-27 (NOTE-1225), from arXiv v2; worth reading as a compositional account in which the coupling data (wiring, potential, response) and the dynamical regime are separate choices: one arrangement of harmonic particles gives a discrete wave equation or a discrete heat equation depending only on whether the integrator stores momentum, and gradient descent with backpropagation is the same construction applied to a parameterized map. Rigorous for its categorical core and its worked examples; many further examples are reported from Claude-written code the author did not audit. |
| LIT-1652 v2 | Counterexample-Guided Abstraction Refinement | Clarke et al. (2000), CAV 2000. Model checking runs on an existential abstraction that merges states, built from the atomic formulas of the program’s conditions and of the property, so a property true of the abstraction is true of the program. An abstract counterexample is checked against the concrete system by forward image computation. If it is spurious, the abstract state where it first has no concrete continuation is split, to separate the states reachable along it from those with the transition onward, and checking runs again. The coarsest split is stated to be NP-hard (proof omitted), so a polynomial non-optimal one is used. For finite-state programs and ACTL* properties whose counterexamples are paths or loops, the loop ends in a real counterexample or a proof. The property is never revised. | Active — read in full 2026-10-27 (NOTE-1231), from the authors’ manuscript of the CAV paper in CMU’s repository; worth reading as the founding paper of counterexample-guided abstraction refinement and as an exact case of a failure under a fixed specification read as a missing distinction, located and repaired by a split. The specification itself is never revised. Two of its four theorems have no proof in this text. |
| LIT-1654 v2 | Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system | Ashwin, Wieczorek, Vitolo and Cox (2012), arXiv 1103.0169, Phil. Trans. R. Soc. A 370. Tipping in an open system ẋ = f(x, λ(t)) is split into B-tipping (a bifurcation of the quasi-static attractor), N-tipping (noise carries the system out of its neighbourhood) and R-tipping, in which “the system fails to track a continuously changing quasi-static attractor”. In a linear model with a fixed stable M and a tipping radius R, steady drift r of the equilibrium tips the system if ‖M‖⁻¹|r| > R and cannot if ‖M⁻¹‖|r| < R, with no bifurcation and no noise; in a co-moving frame it is the ability to track that bifurcates. Examples include a unique, globally stable equilibrium whose threshold is no basin boundary, and a toy energy balance model shows each type alone. Neither N- nor R-tipping needs a change of stability, so critical-slowing warnings have no reason to apply. | Active — read in full 2026-10-27 (NOTE-1215), from the corrected arXiv v5; worth reading as the paper that names rate-induced tipping beside bifurcation and noise, and for its plainest form: a linear model with a fixed stable matrix, so no bifurcation and no noise, in which an attractor that keeps its stability is lost when it moves faster than the system relaxes, relative to the distance to the threshold. A toy climate model shows each type alone. Mathematics and illustration, not evidence about any real system. |
| LIT-1655 v2 | Angular synchronization by eigenvectors and semidefinite programming | Singer (2011). Estimates n angles, up to one common phase, from noisy measurements of their pairwise offsets when an unknown share are outliers uniform on the circle, by the top eigenvector of the Hermitian matrix of e^{iδᵢⱼ} or by a complex Goemans–Williamson SDP. A random-matrix argument puts the eigenvector’s threshold at a share p > 1/√n of good offsets on the complete graph, and simulations recover 400 angles with 85% outliers at correlation 0.97. Fano’s inequality shows no algorithm can succeed below a threshold of order √(n/m), and the eigenvector method is within a factor √((L − 1)/(2 log₂ L)) of it, where exhaustive maximum likelihood costs O(mLⁿ). Extensions to other compact groups, cryo-EM and clock synchronization are sketched. | Active — read in full 2026-10-27 (NOTE-1224), from arXiv v2; worth reading as the paper that posed angular synchronization with outliers and sketched its extension to other groups, and as a clean case that separates what pairwise evidence determines from what a cheap method can recover: an information bound that binds every algorithm, and a spectral relaxation whose threshold sits within an L-dependent factor of it. The spectral threshold is a careful heuristic confirmed by simulation, not a theorem. |
| LIT-1658 v2 | Sheaves are the canonical data structure for sensor integration | Robinson (2017), Information Fusion 36. An expository paper on sensor integration with sheaves. Sensor models become restriction maps between spaces of observations over sets of entities; the consistency radius of an assignment is its largest disagreement between an observation and the restriction of another, and with K-Lipschitz restrictions it bounds the distance to every global section below by ε/(1 + K). Fusion is the search for the nearest global section; in one simulated search-and-rescue scenario it cut the crash-location error 2–8×, and the pattern of discrepancies singled out a biased sensor. Cohomology, for linear sheaves, flags sensor and processing configurations whose data can be locally but not globally consistent. “Canonical” rests on axioms, not a theorem. | Active — read in full 2026-10-27 (NOTE-1221), from arXiv v3; worth reading as the source of the consistency radius and of sheaf-based data fusion as the nearest global section, with a clear worked example. It is expository: “canonical” is argued by positing axioms whose models are sheaves, not proved, and the paper calls sheaves one of “many co-universal information representations”. The demonstration is one simulated scenario; its uniqueness result for linear fusion (Proposition 25) does not follow from its proof. |
| LIT-1659 v2 | Toward a spectral theory of cellular sheaves | Hansen & Ghrist (2019), J. Appl. Comput. Topology 3. Defines the Hodge Laplacians of a cellular sheaf of inner-product spaces; by discrete Hodge theory the degree-k kernel is the cohomology H^k, in degree 0 the global sections, and the flow ẋ = −Δ^k x converges exponentially to the orthogonal projection onto it (Prop. 8.1). Lifts spectral graph theory’s basics with short proofs: harmonic extension, a bound k + 2 for normalized sheaves, interlacing, pushforward, covering and product spectra, effective resistance and spectral sparsification. The Laplacian does not determine the sheaf, Kron reduction fails in general because an interior vertex can impose a non-pairwise boundary constraint, and no Cheeger inequality is proved. Consensus, flocking, opinion dynamics, communication compression and synchronization are sketched as applications. | Active — read in full 2026-10-27 (NOTE-1233), from arXiv v2; worth reading as the standard introduction to sheaf Laplacians: the kernel is the global sections, the heat flow projects onto them, and a set of spectral graph theory’s basics lift with short proofs. Exploratory by its own account: no Cheeger inequality, and the applications are sketches. |
| LIT-1660 v2 | The Behavioral Approach to Open and Interconnected Systems | Willems (2007). A dynamical system is its behavior, the set of trajectories its laws allow; inputs and outputs are not assumed; systems are interconnected by sharing variables and imposing constraints on them, and modelling proceeds by tearing a system into parts, zooming into each and linking them again. | Unavailable — filed 2026-10-27, not read. A reading was attempted on 2026-10-27 and no lawful full text was reached: Unpaywall and OpenAlex both give the article as closed; OpenAlex lists a CiteSeerX cache, not used, and a University of Groningen record, whose page offered no file. It stays Unavailable until a copy is supplied. |
| LIT-1661 v3 | Institutions: abstract model theory for specification and programming | Goguen & Burstall (1992). An institution is a category of signatures with sentences, models and satisfaction, subject to the Satisfaction Condition: a translated sentence holds in a model exactly when the original holds in the model’s reduct. From it, theories and model classes form a Galois connection; theories have colimits whenever signatures do, so specifications can be glued; constraints forcing free or initial interpretation can be added as sentences without losing this; and institution morphisms translate between logics, letting one logic’s prover be used on another’s theories when every model is reached. | Active — read in full 2026-10-29 (NOTE-1348) from a copy of the published article supplied by the owner; worth reading as the source of the Satisfaction Condition, the one axiom that makes truth invariant under change of notation, and of what follows from it for every logic at once: theories glue wherever signatures do (Theorem 11), constraints for abstract data types glue too, and a translation between logics that meets the condition and reaches every model lets one logic’s prover decide another’s entailments (Proposition 35). For this record it gives a precise standard of correct translation, relative to a translation of models, distinct from structural resemblance and from route agreement. Proofs are omitted for the Satisfaction Condition of its own examples; natural language is named as an application and not developed. Source of THEORY-1150 and THEORY-1142. |
| LIT-1668 v2 | Cycle holonomy captures higher-order compatibility constraints in remote synchronization | Torres-Hugas et al. (2026), Communications Physics. For a phase model whose edges carry antisymmetric phase lags, the twisted Laplacian Lα = δα*δα has a zero mode exactly when every cycle holonomy is 0 mod 2π; otherwise its smallest eigenvalue measures the residual incompatibility, largest on an isolated cycle at holonomy π. The symmetric Sakaguchi–Kuramoto model is a different model, but on cactus graphs its lag gives each ℓ-cycle a path mismatch (ℓ − 2)α that an associated connection is built to carry as holonomy. In the triangle–pentagon remote-synchronization motif the pentagon’s mismatch reaches π at α = π/3, where the locked branch is an exact two-group state; the branch has already lost transverse stability at αc ≃ 1.0164, so the holonomy marks geometry, not stability. | Active — read in full 2026-10-27 (NOTE-1218), from the arXiv preprint v2; worth reading for a clean statement of when edge-wise phase relations can be met around cycles, and for its own caution that the point of maximal cycle frustration is not where remote synchronization loses stability. The spectral criterion is standard mathematics; the synchronization result is one seven-node motif. |
| LIT-1669 v2 | Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life | Rosen (1991). Organisms are relational systems; in the metabolism–repair ((M,R)) model an organism is closed to efficient causation, every efficient cause of its constitutive processes being produced within it, which Rosen argues no mechanism (a system with a computable model) can be. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: the book has no DOI to look up on Unpaywall, OpenAlex’s title search was out of its daily budget for this client, and the publisher sells it; no open copy is known. It stays Unavailable until a copy is supplied. |
| LIT-1671 v2 | Slaving principle revisited | Haken (1996). Near an instability the few modes whose damping goes to zero become slow order parameters, and the many stable modes are enslaved to them, expressible as functions of the order parameters, so the dynamics reduce to order-parameter equations; the paper restates the principle and its derivation. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: the article is closed on Unpaywall and OpenAlex (Physica D, Elsevier), with no repository copy. It stays Unavailable until a copy is supplied. |
| LIT-1674 v2 | Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks | Stewart, Golubitsky & Pivato (2003), SIAM J. Appl. Dyn. Syst. 2(4). In a coupled cell network, a typed directed graph whose admissible dynamics give each cell an arbitrary smooth function of its input set, the same for cells with isomorphic inputs, a synchrony pattern is invariant under every admissible vector field iff its equivalence relation is balanced: equivalent cells receive the same number of inputs from each class along each arrow type (Theorem 6.5). Symmetry gives balance but is not needed, so robust synchrony occurs in networks with no symmetry and among cells with no arrows between them. The dynamics on a balanced subspace are admissible for a quotient network (Theorem 9.2), which can have symmetry the original lacks; not every quotient field lifts. The paper studies “existence, not stability”. | Active — read in full 2026-10-27 (NOTE-1217), from the accepted version in the Warwick repository; worth reading as the theorem behind robust synchrony in networks: an architecture forces a synchrony pattern for every admissible dynamics exactly when the pattern is balanced, which symmetry implies but does not exhaust, and the dynamics on it are a quotient network’s. It treats existence only, not stability. |
| LIT-1797 v2 | Gromov-Wasserstein Averaging of Kernel and Distance Matrices | Peyré, Cuturi & Solomon (2016). An entropic Gromov–Wasserstein discrepancy between similarity matrices of any size and loss, computed by an iteration in which each step is a Sinkhorn problem, and a barycenter of such matrices as a Fréchet mean of user-chosen size, histogram and weights. Block coordinate relaxation alternates coupling updates with a closed-form matrix update: for square loss the weighted average of the inputs realigned by their couplings, for KL a weighted geometric average, so PSD and infinitely divisible kernels stay in their classes. Convergence at the step size used is observed, not proved, and only stationary points are sought. Applications: shape barycenters, point-cloud k-means and molecular energy regression. | Active — read in full 2026-10-29 (NOTE-1343), from the PMLR PDF; worth reading for the explicit form of a Gromov–Wasserstein barycenter, a weighted average of realigned inputs whose size, weights and regularization are chosen in advance, and for the authors’ statement of what its algorithm does not guarantee. |
| LIT-1801 v2 | Word Embeddings as Metric Recovery in Semantic Spaces | Hashimoto, Alvarez-Melis & Jaakkola (2016). If text were a random walk whose steps depend on distance in a latent space, negative log co-occurrence over a long window, less per-word terms, would converge to squared latent (geodesic) distance whatever the one-step link. Under that toy model, which the authors say is not a model of language, GloVe, softmax word2vec and PMI-SVD have the true embedding among their optima, up to isometry; a negative-binomial “metric regression” matches them on analogy, series and classification tasks, and word embedding of graph random walks does manifold learning on MNIST. Evidence that real co-occurrences form a Euclidean space is a correlation of 0.75 with human similarity and two nearest-neighbour statistics. | Active — read in full 2026-10-29 (NOTE-1329) from the ACL Anthology PDF; worth reading for the proof that, under a metric random walk, log co-occurrence over long windows converges to squared latent distance whatever the one-step link, so embeddings identify a metric and coordinates only up to isometry; its evidence that real corpora have a Euclidean semantic space is weaker than its theorem. |
| LIT-1803 | Relational Composition as Parallel Transport: What Per-Entity Frames Force in Attention | Sarkar, Mukherjee & Sato (2026), ARXIV-2610.34945. Under per-entity O(d) frames, an invariant attention score depends on two entities only through norms and ⟨xᵢ, Tᵢⱼxⱼ⟩, and a transport computed from one vector per endpoint has rank at most one, so relations must be supplied. On parsed CLUTRR unconstrained transports score 0.848 out of distribution, a transformer 0.436, and orthogonal ones 0.202; on StepGame’s commutative algebra rotations help. The paper calls the premise “a chosen inductive bias, not a consequence of fresh names”. | Active — skimmed 2026-10-29 (NOTE-1326) as the companion LIT-1820 cites for its premise. Worth keeping for its own statement that per-entity frame freedom is an inductive bias, and for the parsed-CLUTRR control in which forcing orthogonal transports costs 64.6 points out of distribution. |
| LIT-1804 v2 | CO-Optimal Transport | Redko, Vayer, Flamary & Courty (2020). Co-optimal transport matches two data matrices of different shapes by fitting a coupling of rows (samples) and a coupling of columns (features) together, under an entry-wise bilinear cost. With equal sizes and uniform weights it is zero exactly when the matrices agree up to separate row and column permutations, and it satisfies the triangle inequality; it proves no uniqueness of the plans, which are fixed only up to joint row–column symmetries. For symmetric matrices it lower-bounds Gromov–Wasserstein, with equality and shared optimal couplings for squared Euclidean distances. Exact solution is NP-hard in general; block coordinate descent reaches block-wise optima. It reports the best accuracy among the baselines on heterogeneous domain adaptation and co-clustering. | Active — read in full 2026-10-29 (NOTE-1335), from arXiv v3; worth reading for the exact relation between co-optimal transport and Gromov–Wasserstein (a lower bound, equal for squared Euclidean distances) and for what its zero theorem implies: joint sample and feature plans are fixed only up to the matrix’s joint row–column symmetries. |
| LIT-1805 v2 | Towards Optimal Transport with Global Invariances | Alvarez-Melis, Jegelka & Jaakkola (2019). Optimal transport between two point sets whose spaces differ by one unknown global linear map, drawn from a Schatten ℓp ball, posed as a joint optimisation over the coupling and the map. When the maps preserve angles or the target is whitened, the map can be eliminated in closed form: orthogonal maps leave the nuclear norm of the transported cross-covariance XΓYᵀ to maximise, and Frobenius-sphere maps its Frobenius norm, which is exactly Gromov–Wasserstein with cosine similarity. Alternating SVD and annealed Sinkhorn steps solve it without adversarial initialisation; on word translation it is level with GW and behind self-learning. | Active — read in full 2026-10-29 (NOTE-1333); worth reading for the exact statement of what a global-invariance alignment treats as arbitrary, one transformation from a stated class acting on the whole space, and for Lemma 4.3, which shows that for one class this is the same problem as Gromov–Wasserstein (THEORY-1135). |
| LIT-1806 v2 | Causal Optimal Transport of Abstractions | Felekis et al. (2023; CLeaR 2024). COTA learns a stochastic map from a low-level causal model’s states to a high-level one’s, given both DAGs, the map between their interventions and samples of each paired interventional distribution, without the structural equations. It solves one entropic optimal-transport problem per intervention, with a cost that rewards states compatible with the same paired interventions and a term meant to impose do-calculus relations between the plans. As written that term depends only on the plans’ marginals, which the transport constraints fix, so it is constant on the feasible set. The reported gains over non-causal transport come from the compatibility cost, and reverse when the intervention set is small. | Active — read in full 2026-10-29 (NOTE-1342) from arXiv v1; worth reading as the first method to learn a causal abstraction map from paired interventional samples without the structural equations, and for what it shows unintentionally: its do-calculus term is constant on the feasible set as written, so its gains come from an intervention-compatibility cost. |
| LIT-1807 v2 | AvAtar: Learning to Align via Active Optimal Transport | Yu et al. (2026). Active learning for alignment by entropic optimal transport. The oracle answers one fixed question, which target matches a given source object, and an answer sets that pair’s cost to zero. AvAtar ranks source objects by the derivative of a proxy utility of the transport plan (its squared norm, entropy, or graph consistency) with respect to supervising each candidate match, averaged over matches by the current plan. The derivative is exact, from one singular positive semidefinite (n + m) linear system solved by conjugate gradients. With 20% of true pairs given and 20% more chosen in ten rounds, it beats uncertainty, centrality and matching-sampling rules on network alignment and image–text tasks by up to 2.8 MRR points, without reported variances; weighting by the current plan is essential. | Active — read in full 2026-10-29 (NOTE-1327), from the arXiv v2; worth reading as a precise case of budgeted question choice within a fixed vocabulary: which object’s correspondence to ask about, scored by an exact derivative through entropic transport of a coherence proxy, and evaluated on accuracy. |
| LIT-1808 v2 | Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces | Alvarez-Melis, Mroueh & Jaakkola (2020). Matches two hierarchies, such as WordNets in two languages or two ontologies, from their Poincaré embeddings alone. Embeddings of one hierarchy from different seeds place branches differently and change pairwise distances, so Euclidean aligners and hyperbolic OT with an orthogonal map fail (P@10 at most 11%). Registering one space with a hyperbolic neural network trained through a Sinkhorn divergence reaches P@10 of 38–60% across languages, but P@1 of only 22% when a hierarchy is matched to a second embedding of itself, and fails without a pretraining step. GW is dismissed, not tested. | Active — read in full 2026-10-29 (NOTE-1323); worth reading as a case where the arbitrariness between two learned representations is not a group action at all: embeddings of one hierarchy from different seeds differ non-isometrically, and only a non-linear registration recovers part of the correspondence (THEORY-1136). Its matching uses a symmetric distance and does not preserve the hierarchy’s direction. |
| LIT-1809 v2 | POT: Python Optimal Transport | Flamary et al. (2021). A software paper for POT, an MIT-licensed Python library of optimal transport solvers as of 2021: exact transport by a C++ solver, entropic transport by Sinkhorn and stabilized variants, Wasserstein barycenters, partial and unbalanced transport, regularized transport with generic solvers, Gromov–Wasserstein, sliced Wasserstein and domain adaptation, tested to 92% line coverage. It compares features with OTT, OTJulia and GeomLoss and reports no benchmarks. Fused and partial Gromov–Wasserstein and co-optimal transport are not in it. | Active — read in full 2026-10-29 (NOTE-1325); the citable description of the POT library as of 2021. Worth citing for the library, with the version used, but not for fused or partial Gromov–Wasserstein or co-optimal transport, which this paper does not describe. |
| LIT-1811 v2 | Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs | Bodnar et al. (2022). In the time limit, diffusion with a sheaf Laplacian projects node features onto its harmonic space, and the class of restriction maps decides which labellings that space can linearly separate. Weighted graph Laplacians cannot separate the sides of a balanced bipartite graph; signed one-dimensional maps separate any two classes and no one-dimensional sheaf separates three; diagonal maps with d ≥ C stalks and orthogonal bundles with C ≤ 2d separate any number. Sheaf convolutions with non-symmetric maps can raise the Dirichlet energy that graph convolutions must lower. A model that learns the maps from node features at each layer is competitive on small heterophilic benchmarks. The separating sheaves are built from the labels, and the learned ones are inspected only on synthetic data. | Active — read in full 2026-10-29 (NOTE-1332) from arXiv v4; worth reading for the cleanest statement of what a sheaf’s restriction maps decide about diffusion: which labellings its harmonic space can separate, class by class of maps, with proofs. The separating sheaves are all built from the labels, and the orthogonal construction works at d = 8, which the paper denies. |
| LIT-1813 v2 | Optimal Transport for structured data with application on graphs | Vayer, Chapel, Flamary, Tavenard & Courty (2019). Fused Gromov–Wasserstein compares two attributed graphs with one coupling whose cost mixes, with weight α, a Wasserstein term on node features and a Gromov–Wasserstein term on structure. It tends to W as α → 0 and to GW as α → 1, and for q = 1, interior α and distance-matrix structures it is a metric that vanishes exactly when a bijection preserves weights, features and structure (a 2^{q−1}-relaxed semi-metric for q > 1). A conditional-gradient solver reaches stationary points, and a block coordinate barycenter supports graph k-means. On graph classification it is competitive with kernels and graph CNNs, though the supplement’s Wasserstein-only scores match or beat it on several datasets. | Active — read in full 2026-10-29 (NOTE-1328), from arXiv v3; worth reading for the metric and interpolation theorems of fused Gromov–Wasserstein, from which it follows that anchors as features break exactly the structural symmetries that move them. |
| LIT-1814 v2 | Hierarchical Multi-Marginal Optimal Transport for Network Alignment | Zeng et al. (2024). HOT aligns several networks with one multi-marginal fused Gromov–Wasserstein coupling tensor. A fused GW barycenter first co-clusters the networks, and nodes are then aligned jointly within aligned clusters, with a cost from random-walk-with-restart scores relative to anchor node sets and a KL proximal point solver. Under square loss the joint objective at a tensor is the sum of the pairwise objectives at its marginals, so it is never below the sum of pairwise optima. On three-network benchmarks with 10% anchors, HOT beats 13 baselines in pairwise and whole-tuple hits and scales far better than a dense tensor; the ground truth is consistent by construction, and no ablation isolates joint fitting from the new cost and the clustering. | Active — read in full 2026-10-29 (NOTE-1334), from the arXiv v2; worth reading for the identity behind joint alignment, that a multi-marginal objective is the sum of pairwise objectives at its marginals, so joint fitting buys consistency at a measurable price; its experiments do not test whether that consistency costs true local distinctions. |
| LIT-1815 v2 | Structured Optimal Transport | Alvarez-Melis, Jaakkola & Jegelka (2017). Optimal transport whose cost is the Lovász extension of a submodular set function on source–target assignments, so that assignments made together, such as one labelled group sent to one region, can cost less than the sum of their parts. The problem stays convex and is a zero-sum game against cost matrices in the base polytope; it is ordinary OT when the set function is modular, and is solved by saddle-point mirror prox with Sinkhorn and base-polytope projections. The groups are supplied in advance. Single-run experiments on point matching, colour transfer, 100-sample digit adaptation and sentence similarity favour the structured couplings. | Active — read in full 2026-10-29 (NOTE-1324), from arXiv v1; worth reading as a convex way to make a transport cost depend on sets of assignments rather than single ones, and for its limits: the structure is supplied, only submodular interactions are tractable, and its own threshold cost is not submodular. |
| LIT-1818 v2 | Gromov-Wasserstein Alignment of Word Embedding Spaces | Alvarez-Melis & Jaakkola (2018). Casts unsupervised word translation as a Gromov–Wasserstein problem between the two languages’ within-language cosine-similarity matrices, solved by entropic projected gradient with no anchors, and reads the optimal coupling as the translation. It matches adversarial-plus-refinement methods on related languages trained on comparable Wikipedia corpora (P@1 72–82) at about 4% of their CPU time, and does markedly worse on English–Russian (45) and English–Finnish (18). The objective sees only within-space relations, so it is unchanged by a rotation of either space and cannot distinguish couplings that differ by a symmetry of the data. | Active — read in full 2026-10-29 (NOTE-1336); worth reading as the cleanest statement of correspondence recovered from within-space relations alone, and for what that costs: the objective cannot tell a correspondence from one composed with a symmetry of either space (THEORY-1133), which the paper does not say. |
| LIT-1820 | Attention as Lattice Gauge Theory: Relations Interact Only Through Wilson Loops | Sarkar, Mukherjee & Sato (2026), ARXIV-2610.37656. If every entity’s coordinate frame may be rotated independently, so that relation transports Tᵢⱼ ∈ O(d) change as gᵢTᵢⱼgⱼ⁻¹, then relations with no cycle between them cannot interact (in the pure-gauge sector), a triangle’s only invariants are class functions of its holonomy, and an equivariant edge update has the form Ψ(H)Tᵢⱼ with Ψ conjugation-equivariant. An untrained instance on S4 synchronization with outliers gains 26–45 points over a single path by composition, and 0.6–7.8 more by Wilson-loop routing; it does not beat spectral synchronization. | Active — read in full 2026-10-29 (NOTE-1331), after the owner brought it to the exchange “Paper assessment” asking whether it is “on to something or if this is AI slop nonsense”. Worth keeping for its conditional theorem, which is sound and is QUESTION-130’s construction carried further, and for its measured caution about its own experiments. Its framing outruns both. |
| LIT-1822 v2 | The Gromov–Wasserstein distance between networks and stable network invariants | Chowdhury & Mémoli (2019). Extends the Gromov–Wasserstein distance to measure networks: a Polish node set with a probability measure and any bounded, possibly asymmetric, edge-weight function. Proves it is a pseudometric whose zero set is weak isomorphism, so nodes with identical in- and out-weights can be merged or split freely. Gives in- and out-going versions of the first, second and third lower bounds and proves each equals a transport problem on the real line, so the third is computed in closed form. Adds sublevel size functions, stable in an interleaving sense, which raise the known lower bound between the circle and the 2-sphere to 0.17. Illustrates the third lower bound on simulated block-model networks and World Bank migration networks. | Active — read in full 2026-10-29 (NOTE-1330), from the arXiv v2; worth reading as the clean statement of what a Gromov–Wasserstein comparison of directed weighted networks can and cannot see: it is a pseudometric on any bounded weights, zero exactly at weak isomorphism, and its cheap invariants are lower bounds that can all vanish for networks apart. |
| LIT-1824 v2 | Tractable models of ecological assembly | Serván & Allesina (2021), Ecology Letters 24: 1029–1037. Represents assembly by rare, low-density, well-separated invasions as a directed graph on feasible sub-communities, and proves that sequential (bottom-up) and all-at-once (top-down) assembly reach the same end-points when the graph is acyclic and attractors are exactly the uninvadable sub-communities. For competitive Lotka–Volterra with A = D(v)BD(w), B symmetric with positive entries, MacArthur’s Lyapunov function orders the sub-communities, so both hold: with B positive definite there is one end-point, the sub-community of greatest total biomass when r = v, reachable by invasions without extinctions; with B indefinite there are several, and order of arrival matters. A consumer–resource model whose consumers can exclude resources can have cycles. | Active — read 2026-10-29 (NOTE-1422) in full, from its first author’s dissertation in the University of Chicago’s repository (chapter 3, appendix included); worth reading as the result that, for symmetric competitive Lotka–Volterra communities, MacArthur’s Lyapunov function orders every sub-community, so the assembly graph is acyclic, every feasible sub-community can be built by invasions without extinctions, and sequential and all-at-once assembly reach the same end-points. Read it knowing that the copy read is the dissertation chapter of August 2020, not the Ecology Letters version, which was not compared; that order-independence (a single end-point) needs the interaction matrix positive definite, and an indefinite one still has priority effects; and that the step from the Lyapunov function to acyclicity is asserted, not written out. |
| LIT-1826 | MacArthur’s consumer-resource model | Chesson (1990), Theoretical Population Biology 37: 26–38. By the abstract: reviews MacArthur’s consumer-resource model and develops statistical measures of association between species’ utilization functions that express its coexistence conditions without strong symmetry assumptions. Not read. | Unfetched — filed 2026-10-29 from the triage of the day, not read. A lawful free copy is located: the author’s own reprint on his University of Arizona faculty pages, at http://www.eebweb.arizona.edu/Faculty/chesson/Peter/Reprints/1990_MacArthur’s%20consumer-resource%20model.pdf (given by Semantic Scholar as the paper’s open copy). On 2026-10-29 the host returned 503 over http and reset the connection over https, to curl and to WebFetch, and web.archive.org reset every connection, so no Wayback capture could be checked. An academia.edu upload (https://www.academia.edu/113408254/MacArthurs_consumer_resource_model, uploader not seen) returned a Cloudflare 403 challenge. OpenAlex reports the Elsevier copy closed. No NOTE is filed. |
| LIT-1827 | Multispecies Competition in Variable Environments | Chesson (1994), Theoretical Population Biology 45: 227–276. By the abstract: a general model of several species competing in a fluctuating environment, analysed by one method that sorts mechanisms of coexistence and exclusion into three categories: those that do not need fluctuations, the storage effect, and fluctuating competition with species-specific nonlinear responses to it; illustrated with a Lotka–Volterra model, nonlinear resource consumption and two recruitment models. Not read. | Unavailable — filed 2026-10-29 from the triage of the day, not read. A lawful full text was sought on 2026-10-29 and not reached. OpenAlex reports it closed with the Elsevier DOI as its only location, and Semantic Scholar reports no open copy. The author’s other reprints are on his old University of Arizona faculty pages (www.eebweb.arizona.edu/Faculty/chesson/Peter/Reprints/), but that host returned 503 or reset the connection on 2026-10-29, its publication list could not be opened, and no address for this paper’s reprint was found by search; the lab’s current pages (eeb.arizona.edu/chesson-lab, /chesson-publications) list it without a link. web.archive.org reset every connection, so no capture of a reprint could be checked. It stays Unavailable until a copy is supplied or the faculty pages answer. No NOTE is filed. |
| LIT-1837 | Comparison between different methods of level identification | Pfante, Bertschinger, Olbrich, Ay & Jost (2014), Advances in Complex Systems 17(2), 1450007. Compares criteria for a level of a discrete dynamical system, a coarse-graining that admits a closed description in its own terms: informational closure, commutativity of aggregation with the dynamics (lumpability), Markovianity of the coarse process and related measures. By the readings that cite it, it shows that for Markov spatial coarse-grainings informational closure is equivalent to commutativity (NOTE-1030), and that closure is stronger than the sufficiency Krakauer et al. use (NOTE-1400). Not read. | Unavailable — registered 2026-10-29 from the triage of the day, not read. A lawful full text was sought on 2026-10-29 and not reached. OpenAlex reports it closed, with no repository full text; it lists CiteSeerX (10.1.1.401.4652; the connection was reset), MPG’s edoc record 708568 (it redirects to pure.mpg.de, which answered 403 and a bot check, and no file was seen) and TUHH’s repository (hdl 11420/14500, a metadata record whose PDF link is a placeholder). World Scientific offers only paid access. No arXiv deposit was found. The MPI MIS publication list for Olbrich gives the journal article and no MiS preprint for it. A web search names a Santa Fe Institute working paper, 13-11-035, at https://www.santafe.edu/media/workingpapers/13-11-035.pdf; that address returned 404 on 2026-10-29, and the working paper was not found elsewhere on the SFI site or at EconPapers. It stays Unavailable until a copy is supplied. Its results on informational closure, commutativity (lumpability) and Markovianity are known to the record second-hand, through NOTE-1030 and NOTE-1400. |
| LIT-1866 | Potential Games | Apt (2013), lecture slides, CWI. Best-response dynamics; exact, ordinal and generalized ordinal potentials after Monderer and Shapley (1996); the finite improvement property (every improvement path is finite); the theorem, credited to Monderer and Shapley, that a finite game has a generalized ordinal potential if and only if it has the FIP, proved one way and sketched the other by counting the improvement sequences that end at each profile; a game with the FIP and no ordinal potential; Rosenthal’s theorem that congestion games are potential games; the Braess paradox; and price-of-stability bounds. | Active — read 2026-10-30 (NOTE-1427) in full, all 47 slides, from the author’s course page at CWI (homepages.cwi.nl/~apt/stra13/potential13-sli.pdf). Worth reading as a clear secondary statement of the potential-game definitions and of Monderer and Shapley’s theorem that a finite game has a generalized ordinal potential exactly when it has the finite improvement property, with a two-by-two game that has the property but no ordinal potential. Read it knowing that it is lecture slides: the theorem is attributed to “Monderer and Shapley ‘96” with no lemma number or page, and the converse is a sketch. |
| LIT-1880 v2 | When Can Safe Controllers Adapt? Information before Commitment | Saligrama (2026). A safe adaptive controller must stay safe under every model not yet ruled out. Its first action that leaves no safe continuation under an alternative is a commitment, and the observation that action produces comes too late to justify it. With the information before commitment measured as a KL divergence between stopped records, every uniformly safe policy forfeits at least about ½e^{−I} of the value reachable only by committing. Acting on an exclusion at risk α needs about log(1/(2α)) nats beforehand. Bounded information and a linear gap give linear regret, shown for a headroom-limited device and for constrained linear regulation. Finite information can still certify a smaller candidate list. Safe recovery at O(√(T log T)) holds when a safe experiment is repeatable, informative and returns to a common state. | Active — read in full 2026-10-31 (NOTE-1466), with the main proofs; worth reading for the precise sense in which evidence must precede commitment: the observation a committing action produces cannot justify it, and bounded information before commitment leaves a fixed fraction of the oracle gap to every uniformly safe controller (THEORY-1254). It also proves the positive case, safe recovery from a repeatable informative experiment, and shows certainty equivalence failing at an active constraint (THEORY-1256). |
| LIT-1881 v2 | Bridging Freidlin-Wentzell large deviations theory and stochastic thermodynamics | Santolin, Freitas, Esposito & Falasco (2025). For overdamped diffusions with weak noise and a nonconservative force, the Freidlin–Wentzell quasi-potential is expanded in powers of the force about the detailed-balance free energy, each order computed along detailed-balanced relaxation; the first order saturates the emergent second law. Splitting the drift into a quasi-potential gradient and an orthogonal circulation, the escape rate from an attractor is bracketed between minus the entropy produced in relaxing into it and the entropy produced along the most likely escape path. The bracket closes at detailed balance; a sheared double well illustrates it. | Active — read in full 2026-10-31 (NOTE-1465); worth reading as the short, self-contained proof, for overdamped diffusions, that a nonequilibrium escape rate e^{−ΩΔI} is bracketed by the entropy produced in relaxing into the attractor and in escaping it (THEORY-1264), with a recursive expansion of the quasi-potential about the free energy. The bracket closes only at detailed balance. |
| LIT-1893 v2 | Input-to-State Safety With Control Barrier Functions | Kolathaya & Ames (2019). Defines input-to-state safety for a safe set C = {h ≥ 0}: under an input disturbance bounded by ‖d‖∞, the larger set {h + γ(‖d‖∞) ≥ 0} stays forward invariant, for some class-K γ. A barrier inequality with a disturbance penalty, ḣ ≥ −α(h) − ι(‖d‖), implies it. Adding a damping term εL_g hᵀ to any barrier controller, or the matching constraint to a CLF–CBF quadratic program, gives ι = ‖d‖²/(4ε). The paper’s examples show that nominal barrier controllers, including the standard quadratic program, can leave the safe set without bound under small disturbances. | Active — read in full 2026-10-31 (NOTE-1477); worth reading as the short statement of graded robust safety: under a bounded input disturbance a barrier-function controller keeps the state not in the safe set but in a set enlarged by a class-K function of the disturbance, and a damping term along the barrier’s input direction delivers this with an enlargement of ‖d‖²/(4ε) (THEORY-1267). It is not the robust box invariance CASE-190 certifies, which is THEORY-134’s. |
| LIT-1897 v2 | Geometry of Nonequilibrium Reaction Networks | Dal Cengio, Lecomte & Polettini (2023). Extends Hill–Schnakenberg cycle analysis from linear reaction networks to interacting ones, which are hypergraphs, by row-reducing the stoichiometric matrix: cycles span its kernel and carry steady currents and the non-conservative part of the affinity, cocycles span the orthogonal complement and carry transient currents and conservative forces, and an integrator recovers chemical potentials. One chemostat cannot drive a network; suppressing a cocycle creates a new conservation law. Near equilibrium the relaxation and response Onsager matrices share their spectrum, and the decomposition gives a linear-regime reconstruction of metabolic fluxes that satisfies Kirchhoff’s laws. | Active — read 2026-10-31 (NOTE-1476; appendices skimmed); worth reading for the cycle–cocycle geometry of reaction hypergraphs and as the exact form of the exchange’s “a productive cycle need not be a stoichiometric cycle”: catalysts cancel from the stoichiometric matrix, so mutual catalytic production is not a cycle of it (THEORY-1248). |
| LIT-1975 v2 | Adjustable robust solutions of uncertain linear programs | Ben-Tal, Goryashko, Guslitzer and Nemirovski (2004), Math. Program. 99: 351–376. For uncertain LPs with here-and-now and wait-and-see variables, the Adjustable Robust Counterpart (∀ data ∃ adjustment) is never more conservative than the usual Robust Counterpart (∃ one solution ∀ data), coincides with it under constraint-wise uncertainty, and is usually NP-hard. Restricting adjustments to affine functions of the data gives a tractable counterpart for fixed recourse and tight semidefinite approximations otherwise. Each decision may have an information basis, the part of the data it may depend on; in an inventory model, poorer bases raise cost and the non-adjustable counterpart is infeasible. | Active — read 2026-11-03 (NOTE-1503), from the published text on A. Nemirovski’s faculty page; worth reading as the standard distinction between robust (∃ solution ∀ data) and adjustable robust (∀ data ∃ adjustment) decisions, with information bases between them, which CLAIM-537’s conditions I and II instantiate; its own intermediate case is a restriction on the rule’s form (affine), not on what is read. |
| LIT-1981 v2 | The geometric minimum action method: A least action principle on the space of curves | Heymann and Vanden-Eijnden (2008), CPAM 61: 1052–1117. The Freidlin–Wentzell quasi-potential, a minimization over paths and their durations that has no minimizer between equilibria, is rewritten as a minimization of a parametrization-free action over curves, which has one; for a diffusion it is 2 inf ∫ |b| sin²(θ/2) ds. The minimizer is the maximum-likelihood transition curve in the small-noise limit. The geometric minimum action method computes it for nongradient SDEs, SPDEs and Markov jump processes. With gradient drift and identity noise transition paths run parallel to the drift; in the Maier–Stein model they leave the heteroclinic orbit for β > 4. | Active — read in part 2026-11-03 (NOTE-1504, Skimmed: §§1, 2.1–2.2, 4.1 and 5), from the published text on a Peking University course page; worth reading as the standard computation of most likely transition paths in small-noise systems, which confirms QUESTION-071’s note of 2026-11-02 for gradient drift with identity noise and shows, in the Maier–Stein model, that the noise path stays the reversed relaxation path until the drift is far from gradient. |
| LIT-1986 v2 | A Survey of Viability Theory | Aubin (1990), doi:10.1137/0328044. Registered unread from the exchange “Assess SDT Coverage” of 2026-11-02 (A21). Aubin’s survey. THEORY-134 holds the viability kernel from the 1999 notes (LIT-714); the robust, discriminating kernel A21 uses is not held. | Unavailable — searched 2026-11-03 for a lawful full text and none was reached, so it is not read and no NOTE is filed. OpenAlex and Unpaywall (noreply address) report it closed; epubs.siam.org returned 403 to this client. Web searches found citations and Aubin’s later book (Viability Theory, Birkhäuser), which is not this work, and no free copy. THEORY-134 holds the viability kernel from LIT-714; the robust and discriminating kernels A21 uses remain unread. It stays Unavailable until a copy is supplied. |