{ "schemaVersion": "1.6", "slug": "perron-minima", "title": "Exact Perron optimisation with factorised pair interactions", "shortTitle": "When every local minimum is global", "url": "https://evidencepress.org/releases/perron-minima/", "oneLine": "Factorised pair interactions make every local Perron minimum globally optimal; an arbitrarily small departure can create a trap.", "abstract": "For positive factorised pair weights, a nonnegative diagonal and a bias parameter strictly between zero and one, this candidate classifies all local Perron minima on the skew box: they are precisely saturated transitive orders, and all share one characteristic polynomial and global optimum. It derives a common-optimiser robust design rule, characterises factorisation through order-independent spectra, and exhibits strict nonglobal minima under arbitrarily small nonfactorised perturbations. A separate even-order tournament result gives constructive recovery with a spectral deficit bound.", "datePublished": "2026-09-28", "dateModified": "2026-09-28", "version": "1.0.1-candidate", "doi": "10.5281/zenodo.23014818", "doiUrl": "https://doi.org/10.5281/zenodo.23014818", "conceptDoi": "10.5281/zenodo.23014817", "pdfUrl": "https://github.com/ipitchford/perron-minima/releases/download/v1.0.1-candidate/perron_minima.pdf", "altPdfUrl": "https://zenodo.org/records/23014818/files/perron_minima.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23014818", "repoUrl": "https://github.com/ipitchford/perron-minima", "releaseUrl": "https://github.com/ipitchford/perron-minima/releases/tag/v1.0.1-candidate", "markdownUrl": "https://evidencepress.org/releases/perron-minima/index.md", "bibtexUrl": "https://evidencepress.org/releases/perron-minima/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/perron-minima.mp3?v=b5f75b4416", "imageUrl": "https://evidencepress.org/assets/og/perron-minima.png?v=fd5ca818f1", "coverArtUrl": "https://evidencepress.org/assets/art/perron-minima.svg?v=956e10e340", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/perron-minima.mp3", "name": "Plain-English audio briefing", "description": "OpenAI synthetic voice; explanation, not scientific evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/perron-minima.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate. Producer-side replay is not formal verification or unaffiliated reproduction. Numerical checks are tolerance-based; the general arguments remain written proofs. No applied validation, arbitrary-switching stability or exhaustive priority claim." }, "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/23014818", "note": "All six release assets fully downloaded from GitHub and Zenodo and matched to the frozen inventory." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Eleven full local jobs pass: exact/symbolic checks, semantic controls, historical and supplied referee replay. Numerical equality uses a tested tolerance contract." }, { "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 is evidenced; coordinated agents and supplied replay records do not establish independence." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Auxiliary symbolic and exact checks are implementation diversity within the producing workflow." }, { "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 of these written arguments." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The supplied favourable review is retained as a document, not an authenticated external specialist endorsement." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No journal or external venue peer review is claimed." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Normal and optimised pinned Linux replay pass at the tagged commit; clean-extraction macOS replay also passes. Numerical outputs use explicit tolerances, not byte identity. PDF build is not hermetic.", "evidenceUrl": "https://github.com/ipitchford/perron-minima/actions/runs/36421901694" } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted research; originating model identities not authenticated", "Supplied review and checking code", "Codex publication intake, targeted repair and replay" ], "humanRole": "Publication authority and supplied materials; no human mathematical contribution is inferred. Scholarly creator remains Anonymous.", "disclosure": "The supplied review, internal checks and publication-intake replay belong to a producer-coordinated workflow. They do not establish unaffiliated reproduction, authenticated specialist review or formal verification." }, "attributionBoundary": null, "problem": { "name": "Local and global Perron minima under fixed factorised pair sums", "url": "https://github.com/ipitchford/perron-minima" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied v1.0 retained by input hash. Publication revision clarifies fixed-orientation existence and replaces a nonportable numerical byte-equality gate with an explicit tolerance contract and corruption controls. Public asset identities and bytes checked on 28 September 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Perron eigenvalue", "spectral optimisation", "positive matrices", "robust stability", "tournament matrices", "factorised interactions" ], "keyResults": [ "Every local minimum is saturated and transitive; all n! orders attain the same global minimum under the stated factorisation.", "One fixed transitive order optimises every admissible parameter instance, giving an exact rectangular-uncertainty worst corner.", "For n≥4, factorisation is characterised by transitive-order cospectrality for every nonnegative diagonal.", "Arbitrarily small nonfactorised perturbations can create strict nonglobal local minima; multiplicative near-factorisation still gives a κ² approximation bound.", "A separate constructive even-order tournament recovery bound need not return the nearest extremiser." ], "reviews": [], "evidencePackage": "Written proofs, exact and symbolic checks, interval certificates, 40 semantic controls and six numerical-corruption controls; replay of historical and supplied referee code.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Psarrakos and Tsatsomeros (2003): rank-one skew perturbations and the Levinger connection. The current diagonal-plus-rank-one hypotheses must be compared explicitly, not identified with the earlier setting.", "url": "https://doi.org/10.1016/S0024-3795(02)00439-1", "doi": "10.1016/S0024-3795(02)00439-1" }, { "citation": "Psarrakos and Tsatsomeros (2006): variance comparisons and restricted Brualdi–Li results, notably Theorem 14, Corollary 15 and Section 6.", "url": "https://doi.org/10.1016/j.laa.2005.12.018", "doi": "10.1016/j.laa.2005.12.018" }, { "citation": "Kirkland (1995): spectral radii of tournament matrices related by an arc reversal; related local transformations in a distinct problem.", "url": "https://doi.org/10.1016/0024-3795(94)00160-F", "doi": "10.1016/0024-3795(94)00160-F" }, { "citation": "Engel and Sergeev: independent row-permutation optimisation, Definition 1.1 and Theorems 3.3–3.4; a different admissible domain.", "url": "https://arxiv.org/abs/2209.01991", "doi": "10.48550/arXiv.2209.01991" }, { "citation": "Drton, Sturmfels and Sullivant: classical one-factor tetrad relations, Theorem 16. The algebraic criterion is prior work; the spectral equivalence is the claimed contribution here.", "url": "https://arxiv.org/abs/math/0509390", "doi": null }, { "citation": "Drury: author exposition of the Brualdi–Li endpoint argument using a rank-one/skew decomposition.", "url": "https://www.math.mcgill.ca/drury/research/brualdili/", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:perron-minima", "attemptIds": [ "ep-attempt:perron-minima-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "stability-margin", "description": "Exact minimum Perron root and fixed-orientation robust stability criterion.", "scope": "Factorised positive pair interactions, nonnegative diagonal, 0