{ "schemaVersion": "1.6", "slug": "reversible-negative-eigenfunction-moments", "title": "Sharp moment bounds for reversible negative eigenfunctions and spectral graph homomorphisms", "shortTitle": "Balanced averages, unequal moments", "url": "https://evidencepress.org/releases/reversible-negative-eigenfunction-moments/", "oneLine": "A sharp limit on positive–negative moment imbalance extends a known spectral graph-mapping obstruction beyond square energies.", "abstract": "A reversible process can have a balanced average while its positive and negative higher moments differ. This candidate proves a sharp comparison for exchangeable pairs with exact negative linear regression, valid for every real exponent at least two, and determines the universal positive-exponent range. A spectral-projector reduction transfers the result to positive unital self-adjoint maps and strict vector colourings. It extends the known square-energy homomorphism bound to all real exponents above two for edge-transitive targets. The paper includes explicit extremisers, reversible-chain counterexamples below two, residual-error estimates and an inherited 1-walk-regular target corollary. The stronger ordinary-vector conjecture and the graph exponent classification below two remain unresolved here.", "datePublished": "2026-11-06", "dateModified": "2026-11-06", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23996890", "doiUrl": "https://doi.org/10.5281/zenodo.23996890", "conceptDoi": "10.5281/zenodo.23996889", "pdfUrl": "https://github.com/ipitchford/reversible-negative-eigenfunction-moments/releases/download/v0.1.0-candidate/reversible-negative-eigenfunction-moments-v0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23996890", "repoUrl": "https://github.com/ipitchford/reversible-negative-eigenfunction-moments", "releaseUrl": "https://github.com/ipitchford/reversible-negative-eigenfunction-moments/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/reversible-negative-eigenfunction-moments/index.md", "bibtexUrl": "https://evidencepress.org/releases/reversible-negative-eigenfunction-moments/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/reversible-negative-eigenfunction-moments.mp3", "imageUrl": "https://evidencepress.org/assets/og/reversible-negative-eigenfunction-moments.png?v=85377e2035", "coverArtUrl": "https://evidencepress.org/assets/art/reversible-negative-eigenfunction-moments.svg?v=9d03839f97", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/reversible-negative-eigenfunction-moments.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/reversible-negative-eigenfunction-moments.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted research candidate. Internal AI-assisted checking is not external specialist review, independent reproduction or formal verification. Priority is subject to a bounded source search." }, "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/23996890", "note": "Public GitHub and Zenodo assets verified by full download against the frozen ZIP, PDF and checksum ledger on 6 November 2026." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Nine producer-side frozen-package gates passed, including normal and optimized checker runs and negative controls. Finite examples and numerical grids do not prove the universal theorem." }, { "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": "No unaffiliated independent reimplementation established." }, { "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 claimed." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted checking 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 established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Final package includes source, checker, controls, environment record and checksums; clean producer-side replay is retained. No unaffiliated reproduction is claimed." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical development and revision", "Internal AI-assisted source and proof checking", "Codex publication preparation" ], "humanRole": "Publication authority only; no human mathematical contribution inferred.", "reviewStatus": "Internal AI-assisted review is not unaffiliated 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 and its bibliography on 6 November 2026. GitHub and Zenodo publication were verified against the frozen package, including full downloads of all three assets from both providers. Producer-side clean-package checks and their scope are retained separately from public availability.", "grandfatheredAtSchemaVersion": null, "keywords": [ "reversible Markov chain", "exchangeable pair", "negative eigenfunction", "spectral graph theory", "strict vector colouring", "positive map", "moment inequality" ], "keyResults": [ "For exact negative regression E[Y|X]=−rX, the positive and negative p-moments differ by at most r^(1−p), sharply for every p≥2.", "The universal positive-exponent range for all admissible reversible pairs is {1} union [2,infinity); this is not a graph exponent classification.", "Positive-map reduction yields the strict-vector/theta bound for p≥2; p>2 is the graph extension beyond known square-energy results.", "Edge-transitive homomorphism obstructions extend to all p≥2, with an elementary 1-walk-regular target corollary using an established colouring construction.", "Complex Hermitian and controlled residual formulations are elementary consequences; the ordinary-vector conjecture remains open here." ], "reviews": [], "evidencePackage": "Written analytic proofs with a sharp two-state extremiser, an exact rational five-state counterexample, a supplementary floating-point scalar grid and targeted checker controls. Finite checks do not prove the universal theorem.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Elphick, Tang and Zhang (2026). A spectral lower bound on chromatic numbers using p-energy. European Journal of Combinatorics 132, 104252. Question numbering refers to arXiv v5.", "url": "https://doi.org/10.1016/j.ejc.2025.104252", "doi": "10.1016/j.ejc.2025.104252" }, { "citation": "Guo and Spiro (2024). New eigenvalue bound for the fractional chromatic number. Journal of Graph Theory 106(1), 167–181. The square-energy homomorphism predecessor.", "url": "https://doi.org/10.1002/jgt.23071", "doi": "10.1002/jgt.23071" }, { "citation": "Godsil, Roberson, Rooney, Šámal and Varvitsiotis (2017). Universal completability, least eigenvalue frameworks, and vector colorings. Discrete & Computational Geometry 58(2), 265–292. The established target-colouring construction.", "url": "https://doi.org/10.1007/s00454-017-9899-2", "doi": "10.1007/s00454-017-9899-2" }, { "citation": "Coutinho, Spier and Zhang (2024). Conic programming to understand sums of squares of eigenvalues of graphs. arXiv:2411.08184v1. The stronger ordinary-vector square-energy predecessor.", "url": "https://arxiv.org/abs/2411.08184v1", "doi": "10.48550/arXiv.2411.08184" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:reversible-negative-eigenfunction-moments", "attemptIds": [ "ep-attempt:reversible-negative-eigenfunction-moments-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "Sharp one-sided moment bounds for reversible negative eigenfunctions, with scoped positive-map and strict-vector-colouring reductions.", "scope": "Real exponents p at least two under the stated reversibility and negative-eigenfunction hypotheses; positive-map and graph consequences retain their separate assumptions." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "A reversible pair yields the scalar moment bound; finite trace-state compression transfers it to self-adjoint positive unital maps, and strict vector colourings supply graph eigenfunctions.", "remainingRisks": [ "Bounded prior-art search is not exhaustive priority clearance.", "Positivity does not assert complete positivity or a physical quantum-channel interpretation.", "Strict vector colouring does not establish the ordinary-vector-colouring conjecture; direct corollaries are not separate priority claims." ] }, "humanJudgmentGates": [ "Publication authority is not scientific endorsement.", "External specialist validation remains open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "priorityAssessment" ], "nextAction": "Invite external checking of the stated proof and scoped reductions, with explicit separation from producer-side finite checks.", "claimCeiling": "Written proof and scoped producer checks; no external validation, formal verification or absolute originality guarantee." }, "impactClaims": [ { "id": "reversible-negative-eigenfunction-moments-no-impact", "aim": "science", "outcome": "Potential reuse of a sharp moment comparison in the stated operator and graph settings.", "setting": "Probability, positive maps and spectral graph theory", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched research-workflow comparison.", "estimand": "No acceleration or downstream impact effect estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. 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