{ "schemaVersion": "1.6", "slug": "free-parafermion-converse", "title": "A full-spectrum converse for free parafermions", "shortTitle": "When the whole spectrum fits one pattern", "url": "https://evidencepress.org/releases/free-parafermion-converse/", "oneLine": "For every order N ≥ 3, a graph criterion exactly characterises generic canonical free spectra—with a crucial distinction between the whole spectrum and its separate sectors.", "abstract": "This candidate proves an all-order converse for independently coupled universal Weyl graph algebras whose commutation phases are 1, q and q inverse. Canonical full-spectrum freeness is equivalent to the oriented graph being an oriented indifference graph. Outside this class the free-coupling locus is a proper complex algebraic set. The package also provides a cubic-time switching recogniser, an injective-label physical-realisation criterion and an exact census through six Hamiltonian terms. Arbitrary phases, dependent labels and sector-specific mode lists are outside the theorem.", "datePublished": "2026-09-28", "dateModified": "2026-09-28", "version": "1.0.1-candidate", "doi": "10.5281/zenodo.23019437", "doiUrl": "https://doi.org/10.5281/zenodo.23019437", "conceptDoi": "10.5281/zenodo.23019436", "pdfUrl": "https://github.com/ipitchford/free-parafermion-converse/releases/download/v1.0.1-candidate/paper.pdf", "altPdfUrl": "https://zenodo.org/records/23019437/files/paper.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23019437", "repoUrl": "https://github.com/ipitchford/free-parafermion-converse", "releaseUrl": "https://github.com/ipitchford/free-parafermion-converse/releases/tag/v1.0.1-candidate", "markdownUrl": "https://evidencepress.org/releases/free-parafermion-converse/index.md", "bibtexUrl": "https://evidencepress.org/releases/free-parafermion-converse/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/free-parafermion-converse.mp3?v=a885e4ca1a", "imageUrl": "https://evidencepress.org/assets/og/free-parafermion-converse.png?v=0a12bae9be", "coverArtUrl": "https://evidencepress.org/assets/art/free-parafermion-converse.svg?v=9a80a991bc", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/free-parafermion-converse.mp3", "name": "Plain-English audio briefing", "description": "OpenAI synthetic voice; explanation, not scientific evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/free-parafermion-converse.txt" }, { "type": "video", "url": "https://youtu.be/550V8QChXzE", "name": "A Full-Spectrum Converse for Free Parafermions", "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate. All local finite replay and supplied reviewer checks passed. The universal proof is not formally verified; supplied AI-assisted review and separate implementations are not unaffiliated reproduction or external human peer review. No exhaustive priority clearance or applied validation is claimed." }, "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/23019437", "note": "All six assets fully downloaded from GitHub and Zenodo and hash-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": "Complete finite replay and supplied v1 reviewer code passed; 400 held-out switching cases and corruption/refusal controls passed." }, { "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": "A separately implemented review checker was replayed, but coordinated model-based work is not evidence of unaffiliated 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 of the analytic claims." }, { "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": "Clean-extraction macOS and exact-commit pinned Linux full replay passed. Optimization refusal tested, not an optimized mathematical replay. PDF build is not hermetic.", "evidenceUrl": "https://github.com/ipitchford/free-parafermion-converse/actions/runs/36448091118" } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted research; exact originating model identities not authenticated", "Supplied AI-assisted review and checking code", "Codex publication intake, revisions and replay" ], "humanRole": "Publication authority and supplied materials only; no human scientific contribution is inferred. Scholarly creator remains Anonymous.", "disclosure": "Received reviews, internal replay and publication checks are producer-coordinated evidence, not authenticated external review or independent reproduction." }, "attributionBoundary": null, "problem": { "name": "Generic canonical full-spectrum free parafermions at every order N ≥ 3", "url": "https://arxiv.org/abs/2408.09684v2" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied v1 package and separate reviewer audit preserved by hash. The public revision clarifies imported sufficiency, even-order phases, the fork calculation, physical applicability and related-work boundaries without changing the main classification. Public assets checked on 28 September 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "free parafermions", "oriented indifference graphs", "Weyl graph algebra", "canonical spectrum", "central sectors", "switching recognition" ], "keyResults": [ "For every N ≥ 3, generic and all-coupling canonical freeness are equivalent to the graph being oriented indifference.", "Outside this graph class, free couplings form a proper complex algebraic exceptional set.", "The three-term fork is free exactly when at least one coupling is zero; directed cycles have too many generic distinct eigenvalues.", "O(n³) switching recognition returns an explicit certificate; switching changes the Hamiltonian.", "Exact census through six terms: 22,114 isomorphism classes, 120 free classes and 1,322 switchable classes." ], "reviews": [], "evidencePackage": "Written universal proof; exact cyclotomic and finite-field checks; exhaustive graph census; preserved reviewer implementations; semantic corruption and optimization-refusal controls.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Mann, Elman, Wood and Chapman (2025), A graph-theoretic framework for free-parafermion solvability: oriented-indifference sufficiency is the imported baseline. The candidate supplies necessity for its stricter canonical full-spectrum formulation.", "url": "https://doi.org/10.1098/rspa.2024.0671", "doi": "10.1098/rspa.2024.0671" }, { "citation": "Fendley (2014), Free parafermions: foundational free-parafermion models and spectral structure.", "url": "https://doi.org/10.1088/1751-8113/47/7/075001", "doi": "10.1088/1751-8113/47/7/075001" }, { "citation": "Baricz and Singh (2018), Zeros of some special entire functions: positive-parameter hypergeometric zero-location input to the cumulant argument.", "url": "https://arxiv.org/abs/1702.00626v2", "doi": "10.48550/arXiv.1702.00626" }, { "citation": "Rutter, Strash, Stumpf and Vollmer (2025), Simultaneous representation of proper and unit interval graphs: straight-enumeration uniqueness, Proposition 2.", "url": "https://doi.org/10.1007/s00453-025-01296-x", "doi": "10.1007/s00453-025-01296-x" }, { "citation": "Ruh and Elman (2026), Expanding the class of free fermions via twin-collapse methods: sector reductions and representation equivalence, not the same global-grid converse; qudit false-twin orientations differ from the same-direction fork here.", "url": "https://doi.org/10.1103/svfv-vdh5", "doi": "10.1103/svfv-vdh5" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:free-parafermion-converse", "attemptIds": [ "ep-attempt:free-parafermion-converse-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "certificate-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "certificate", "description": "Graph classification of generic canonical full-spectrum freeness, with switching certificates.", "scope": "Universal independent-generator Weyl algebra; N ≥ 3; phases 1,q,q inverse; independent couplings." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Canonical trace and multiplicity constraints connect the graph algebra to one global occupation-mode spectrum; injective labels transfer the result to physical representations.", "remainingRisks": [ "Written proof not formally verified.", "Dependent-label quotients and unequal sector weights are outside the theorem.", "Finite tests and switching certificates have narrower scopes than the universal claim." ] }, "humanJudgmentGates": [ "Publication authorisation only; no human scientific contribution or additional review stage is inferred." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://arxiv.org/abs/2408.09684v2", "inheritedClaim": "Oriented-indifference sufficiency under the source hypotheses.", "inheritedAssuranceCeiling": "Imported published mathematical theorem, extended to canonical multiplicities and complex couplings by explicit arguments." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "formalVerification", "specialistReview", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Unaffiliated audit of the all-order fork and cycle arguments, canonical sufficiency and physical transfer; independent replay.", "claimCeiling": "Unrefereed candidate with producer-side replay." }, "impactClaims": [ { "id": "free-parafermion-converse-no-impact", "aim": "science", "outcome": "Reusable all-order spectral criterion and checkable graph decisions", "setting": "Producer-coordinated publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None.", "estimand": "No impact effect estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/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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