{ "schemaVersion": "1.6", "slug": "entropy-concavity-four-sevenths", "title": "The critical signal fraction for local entropy concavity", "shortTitle": "How much noise restores entropy concavity? Exactly enough to push the signal below 4/7", "url": "https://evidencepress.org/releases/entropy-concavity-four-sevenths/", "oneLine": "Mix two noisy copies of a near-Gaussian symmetric signal: the entropy of the blend is guaranteed concave in the mixing weight exactly when the noise variance exceeds three quarters of the signal variance (signal fraction below 4/7, signal-to-noise ratio 4/3); whether 4/7 itself qualifies is open.", "abstract": "Take two independent copies of a random signal, add Gaussian noise to each, and mix the noisy copies with weights √t and √(1−t). Ball, Nayar and Tkocz conjectured that, for log-concave signals, the entropy of the mixture is concave in t. Luo (2026) disproved this and showed that symmetric near-Gaussian counterexamples survive added noise whenever the signal accounts for more than 4/7 of the total variance; Schubr (2026) proved local concavity below 1/2. This candidate shows that 4/7 is the true local threshold for symmetric signals. For every signal fraction q < 4/7 there is an explicit neighbourhood of the Gaussian, defined by a uniform bound on the relative density f/φ − 1, in which every symmetric variance-one input gives a strictly concave entropy curve (Theorem A); with Luo's counterexamples, 4/7 is the supremum of such signal fractions, a signal-to-noise ratio of 4/3. Whether q = 4/7 itself qualifies is open. For every q in (4/7, 1], one sequence of smooth strongly log-concave inputs makes equal weighting a strict local entropy minimum, with exact curvature constant γ(q) = 21√5 q/(49q − 4) and a Gaussian-shaped local profile (Theorem B); the sign on (4/7, 2 − √2], Luo's Problem 10.3, was also proposed in an earlier public, unrefereed GitHub argument. The threshold is the fixed point q = b(q/2) of an Ornstein–Uhlenbeck smoothing inequality whose range is proved sharp, with Luo's family extremal within the Hermite test functions (Proposition S); at q = 4/7 Luo's family is strictly concave near equal weighting (Proposition E); the proof's radius shrinks like (4/7 − Q)^(9/4) (Remark R). The effects are minute (dips of order 10⁻¹³ nats); the results are structural. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-03", "dateModified": "2026-11-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23910629", "doiUrl": "https://doi.org/10.5281/zenodo.23910629", "conceptDoi": "10.5281/zenodo.23910628", "pdfUrl": "https://github.com/ipitchford/entropy-concavity-four-sevenths/releases/download/v0.1.0-candidate/entropy-concavity-four-sevenths-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23910629", "repoUrl": "https://github.com/ipitchford/entropy-concavity-four-sevenths", "releaseUrl": "https://github.com/ipitchford/entropy-concavity-four-sevenths/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/entropy-concavity-four-sevenths/index.md", "bibtexUrl": "https://evidencepress.org/releases/entropy-concavity-four-sevenths/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/entropy-concavity-four-sevenths.mp3", "imageUrl": "https://evidencepress.org/assets/og/entropy-concavity-four-sevenths.png?v=8ee3662a4f", "coverArtUrl": "https://evidencepress.org/assets/art/entropy-concavity-four-sevenths.svg?v=9a296b3c22", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/entropy-concavity-four-sevenths.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/entropy-concavity-four-sevenths.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematics candidate. The theorems have written analytic proofs; the shipped checks test exact identities, closed-form constants, the radius table and finite-n asymptotics in finite precision, not interval certificates. Internal AI-assisted review in five roles, a separate-model adversarial review and audit of the new proofs, a separate proof audit, a clean replay, one revision round and a re-review. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23910629", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "All four checkers were rerun with Python -I -B and -I -B -O, and the negative-control harness rerun, from a fresh copy of the package before freezing and again from the frozen archive; every JSON output was identical to the shipped one apart from timing and interpreter fields. GitHub CI repeats every checker and the negative controls on Linux." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "During internal review, reviewers recomputed the load-bearing constants (the 4/7 condition, γ(q), the radius table, the Lemma II.5 matrix, Proposition S and E constants) with separately written high-precision scripts. This is producer-side diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Python standard library only (CPython 3.12 tested on macOS and Linux CI); exact and fixed-precision Decimal arithmetic, no random seeds; not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, separate-model adversarial review and proof audit, proof audit, clean replay, re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (revision of 3 November 2026 with DOI insertion, 32 pages), its checkers and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 3 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "differential entropy", "entropy concavity", "Ball-Nayar-Tkocz conjecture", "Gaussian smoothing", "Ornstein-Uhlenbeck semigroup", "Hermite polynomials", "signal-to-noise ratio", "blind source separation" ], "keyResults": [ "Theorem A (proof): for every Q < 4/7 there is an explicit δ_Q > 0 such that every admissible even g with ‖g‖∞ ≤ δ_Q gives F'' ≤ −(1/16)‖N(T_√(qt) g + T_√(q(1−t)) g)‖² < 0 for all q ≤ Q and t in [0, 1]; δ_0.5 ≈ 1.7 × 10⁻⁵.", "Corollary (proof, with Luo 2026): (0, 4/7) ⊆ C ⊆ (0, 4/7], so sup C = 4/7, a noise variance of 3/4 of the signal variance (signal-to-noise ratio 4/3); whether 4/7 ∈ C is open.", "Theorem B (proof): for every q in (4/7, 1], one sequence of smooth strongly log-concave inputs gives F''(1/2) > 0 eventually, with γ(q) = 21√5 q/(49q − 4) and local profile 1 − exp(−4γ(q)s²); the sign on (4/7, 2 − √2] was also proposed in an earlier public GitHub argument.", "Proposition S (proof): the bounded-input smoothing inequality ‖T_√s g‖² ≤ J‖g‖∞‖T_√r g‖ holds for s < b(r) = √(2r(1+r)) − r and fails beyond; 4/7 is the fixed point q = b(q/2), and Luo's parameter a = 7/(3√5) is extremal within the Hermite test functions.", "Proposition E and Remark R (proof): at q = 4/7 Luo's family is strictly concave near t = 1/2, so it gives no boundary counterexample; the proof's radius decays like C_δ(4/7 − Q)^(9/4), C_δ ≈ 5.93 × 10⁻³." ], "reviews": [], "evidencePackage": "Written analytic proofs of Theorems A and B, the Corollary, Propositions S and E and Remark R (32-page manuscript); five standard-library Python checkers with retained outputs in normal and optimized mode: Part II coefficient asymptotics against −8γ(q) and the exact recurrence (233 checks), the Part I radius recipe and table in 100-digit arithmetic (165), the Part II Hermite identities and two-weight inner products to about 10⁻⁷⁸ (178), the Proposition S envelope, Proposition E diagnostics and Remark R constant (315, with 6 internal negative controls), and a negative-control harness requiring 16 deliberately broken checker copies to fail under both normal and -O runs; clean replay from a fresh copy of the package with every output identical apart from timing fields.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Ball, Nayar and Tkocz (2016), A reverse entropy power inequality for log-concave random vectors. Studia Mathematica. States the conjecture (Conjecture 2) that the entropy of √t X + √(1−t) Y is concave in t for log-concave inputs.", "url": "https://doi.org/10.4064/sm8418-6-2016", "doi": "10.4064/sm8418-6-2016" }, { "citation": "Luo (2026), Near-Gaussian counterexamples to the Ball-Nayar-Tkocz entropy concavity conjecture. arXiv:2609.11418 v2, unrefereed. Disproves the conjecture, gives the symmetric counterexamples for every q > 4/7, the Gaussian-kernel interpolation used in Lemma I.2, and Problems 10.1–10.3.", "url": "https://arxiv.org/abs/2609.11418v2", "doi": "10.48550/arXiv.2609.11418" }, { "citation": "Schubr (2026), Mixing-weight entropy concavity near the Gaussian below one-half signal. Luminary Research Archive, public edition 3, not peer reviewed. Proves local concavity for q < 1/2; its entropy-energy argument is used in Part I.", "url": "https://research.luminaryoddities.org/local-and-eventual-entropy-concavity-under-gaussian-smoothing/v/3", "doi": null }, { "citation": "the-omega-institute/trureturing (2026), Open problem (preregistration): prove Congyi Luo Problem 10.3 (arXiv:2609.11418v2): positive midpoint entropy curvature for every 4/7 < q <= 2-sqrt2. GitHub issue 25275, 1 November 2026; proposed argument, not refereed. Earlier public proposal of the sign in Theorem B on (4/7, 2−√2].", "url": "https://github.com/the-omega-institute/trureturing/issues/25275", "doi": null }, { "citation": "Abu-Ajamieh (2026), Noise-Induced Restoration of Entropy Concavity Near Gaussian Equilibrium. arXiv:2610.22386 v1, unrefereed. Concavity up to q = 5/8 for a single fixed He₄ direction, reconciled with the 4/7 threshold in Section 1.3.", "url": "https://arxiv.org/abs/2610.22386v1", "doi": "10.48550/arXiv.2610.22386" }, { "citation": "Artstein, Ball, Barthe and Naor (2004), Solution of Shannon’s problem on the monotonicity of entropy. Journal of the American Mathematical Society. Mixing makes sums more Gaussian.", "url": "https://doi.org/10.1090/S0894-0347-04-00459-X", "doi": "10.1090/S0894-0347-04-00459-X" }, { "citation": "Lieb (1978), Proof of an entropy conjecture of Wehrl. Communications in Mathematical Physics. The form of the entropy power inequality giving the chord inequality that concavity would strengthen.", "url": "https://doi.org/10.1007/BF01940328", "doi": "10.1007/BF01940328" }, { "citation": "Eskenazis, Nayar and Tkocz (2018), Gaussian mixtures: Entropy and geometric inequalities. The Annals of Probability. Entropy comparisons for Gaussian mixtures.", "url": "https://doi.org/10.1214/17-AOP1242", "doi": "10.1214/17-AOP1242" }, { "citation": "Eskenazis and Gavalakis (2024), On the entropy and information of Gaussian mixtures. Mathematika. Proves the concavity conjecture for Gaussian mixtures.", "url": "https://doi.org/10.1112/mtk.12246", "doi": "10.1112/mtk.12246" }, { "citation": "Le (2014), Normal and isometric weighted composition operators on the Fock space. Bulletin of the London Mathematical Society. Fock-space operator theory behind the Gaussian-kernel comparison.", "url": "https://doi.org/10.1112/blms/bdu046", "doi": "10.1112/blms/bdu046" }, { "citation": "Carroll and Gilmore (2021), Weighted composition operators on Fock spaces and their dynamics. Journal of Mathematical Analysis and Applications. Fock-space operator theory behind the Gaussian-kernel comparison.", "url": "https://doi.org/10.1016/j.jmaa.2021.125234", "doi": "10.1016/j.jmaa.2021.125234" }, { "citation": "Abbe and Zheng (2012), A Coordinate System for Gaussian Networks. IEEE Transactions on Information Theory. Hermite expansions of entropy near the Gaussian.", "url": "https://doi.org/10.1109/TIT.2011.2169536", "doi": "10.1109/TIT.2011.2169536" }, { "citation": "Costa (1985), A new entropy power inequality. IEEE Transactions on Information Theory. Concavity of entropy power along the noise direction, contrasted in Discussion D.4.", "url": "https://doi.org/10.1109/TIT.1985.1057105", "doi": "10.1109/TIT.1985.1057105" }, { "citation": "Villani (2000), A short proof of the \"concavity of entropy power\". IEEE Transactions on Information Theory. Concavity along the noise direction.", "url": "https://doi.org/10.1109/18.850718", "doi": "10.1109/18.850718" }, { "citation": "Guo, Shamai and Verdu (2005), Mutual Information and Minimum Mean-Square Error in Gaussian Channels. IEEE Transactions on Information Theory. The I-MMSE relation, contrasted in Discussion D.4.", "url": "https://doi.org/10.1109/TIT.2005.844072", "doi": "10.1109/TIT.2005.844072" }, { "citation": "Comon (1994), Independent component analysis, A new concept? Signal Processing. The entropy contrast behind the blind-source-separation reading (Discussion D.3).", "url": "https://doi.org/10.1016/0165-1684(94)90029-9", "doi": "10.1016/0165-1684(94)90029-9" }, { "citation": "Vrins, Pham and Verleysen (2007), Mixing and Non-Mixing Local Minima of the Entropy Contrast for Blind Source Separation. IEEE Transactions on Information Theory. Spurious mixing minima for strongly multimodal sources.", "url": "https://doi.org/10.1109/TIT.2006.890716", "doi": "10.1109/TIT.2006.890716" }, { "citation": "Courtade, Fathi and Pananjady (2018), Quantitative Stability of the Entropy Power Inequality. IEEE Transactions on Information Theory. 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