{ "schemaVersion": "1.6", "slug": "gaussian-cosine-local-stability", "title": "Local stability of cosine configurations for Gaussian minima", "shortTitle": "Cosine configurations resist every small tilt", "url": "https://evidencepress.org/releases/gaussian-cosine-local-stability/", "oneLine": "Equally spaced planar directions are strict local minima for Gaussian minimum moments, even against rank-increasing perturbations, for every n ≥ 3 and p > 1.", "abstract": "An explicit second-variation formula proves strict local stability of the cosine correlation configuration for every number of coordinates n ≥ 3 and every moment order p > 1. Angular changes yield a cycle Laplacian; normal changes yield a shifted signed-cycle matrix whose kernel consists precisely of rotations. The analytic argument includes covariance-rank increases and the integrable singularity for 1 < p < 2. The neighbourhood radius is existential. Four-coordinate supporting comparisons, finite evaluators and retained optimisation records are separate from the universal proof. The global Gaussian-minimum inequality and stronger stochastic conjecture remain unresolved here.", "datePublished": "2026-10-18", "dateModified": "2026-10-18", "version": "1.0.0", "doi": "10.5281/zenodo.23453252", "doiUrl": "https://doi.org/10.5281/zenodo.23453252", "conceptDoi": "10.5281/zenodo.23453251", "pdfUrl": "https://github.com/ipitchford/gaussian-cosine-local-stability/releases/download/v1.0.0-candidate/gaussian-cosine-local-stability.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23453252", "repoUrl": "https://github.com/ipitchford/gaussian-cosine-local-stability", "releaseUrl": "https://github.com/ipitchford/gaussian-cosine-local-stability/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/gaussian-cosine-local-stability/index.md", "bibtexUrl": "https://evidencepress.org/releases/gaussian-cosine-local-stability/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/gaussian-cosine-local-stability.mp3", "imageUrl": "https://evidencepress.org/assets/og/gaussian-cosine-local-stability.png?v=ecafa39c4a", "coverArtUrl": "https://evidencepress.org/assets/art/gaussian-cosine-local-stability.svg?v=5653c0b916", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/gaussian-cosine-local-stability.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/gaussian-cosine-local-stability.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted candidate. Local stability, not a global solution; no external peer review or formal verification 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/23453252", "note": "Public GitHub v1.0.0-candidate and Zenodo record 23453252: full downloads of PDF, ZIP and checksum manifest matched the frozen local assets on 18 October 2026. DOI issuance and byte parity do not establish scientific validity." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Clean-package producer replay passed ordinary and optimised Python: exact four-cycle algebra, all 200 saved endpoints including 22 unsuccessful flags, 200 sampled-frame comparisons, eight local paths, four Sobol sanity comparisons and seven semantic regression methods. Original optimisation trajectories were not regenerated. Finite diagnostics do not prove the analytic all-n, p>1 theorem or global optimality." }, { "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 established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Producer-workflow implementation diversity is not independent reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant verification established." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "No external specialist review established." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Internal AI-assisted assessment is not external peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed", "note": "Cross-version numerical validation does not reproduce the exact historical environment." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted research workflow", "OpenAI Codex: mathematical extension, checking, revision and publication" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted assessment, not external peer review." }, "attributionBoundary": null, "problem": { "name": "Gaussian-minimum cosine conjecture", "url": "https://arxiv.org/abs/2605.02023v1" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Version 1.0.0 manuscript, claim map and clean-package replay checked on 18 October 2026. Public GitHub and Zenodo PDF, ZIP and checksum assets were downloaded in full and matched against the frozen local bytes. Availability does not establish mathematical correctness or external review.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Gaussian minima", "cosine configuration", "local stability", "second variation", "signed cycle", "correlation matrices" ], "keyResults": [ "Strict local minimality in the full correlation domain for every fixed n ≥ 3 and p > 1, including rank-increasing perturbations.", "An explicit angular Laplacian and normal signed-cycle form isolate the rotational zero modes and give quadratic growth near the orbit.", "For four coordinates, independence is a nonglobal strict local minimum; an independent-noise bound excludes a specified interior region, not the full domain." ], "reviews": [], "evidencePackage": "Analytic all-n, p>1 local theorem; exact finite algebra and floating-point four-row diagnostics have narrower scope.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "D. Kunisky (2026), A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture. Version 1.", "url": "https://arxiv.org/abs/2605.02023v1", "doi": "10.48550/arXiv.2605.02023" }, { "citation": "Y. Long (2025), Hyperbolic embedding of infinite-dimensional convex bodies. Known intrinsic-volume identity.", "url": "https://doi.org/10.1007/s10711-024-00957-5", "doi": "10.1007/s10711-024-00957-5" }, { "citation": "Y.-F. Zhong, T. Ma and Z.-C. Hu (2020), A note on the Gaussian minimum conjecture.", "url": "https://arxiv.org/abs/2008.06211v1", "doi": "10.48550/arXiv.2008.06211" }, { "citation": "cyclemath (2026), Convex Hulls of Great Subspheres and Gaussian Minima. 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