{ "schemaVersion": "1.6", "slug": "sun-circular-primitive-permutation", "title": "A primitive circle—with one exception", "shortTitle": "A primitive circle, except once", "url": "https://evidencepress.org/releases/sun-circular-primitive-permutation/", "oneLine": "A ten-vertex obstruction corrects Sun’s circular shifted-product conjecture; an exact sieve and finite replay establish the candidate classification.", "abstract": "For every prime power q>7 and every parameter a in Fq, the simple graph on nonzero field elements with adjacency a+xy primitive is Hamiltonian exactly outside (q,a)=(11,2). The exceptional case has a forced four-cycle proof. The positive classification combines Luo–She’s inherited infinite range and zero-parameter branch with a classical sieve, a refined independent-set bound and regenerated cycle witnesses for 510 residual fields. The revised manuscript adds a sufficient vertex-and-edge deletion criterion. Internal checking does not establish external peer review, formal verification or four-star significance.", "datePublished": "2026-10-12", "dateModified": "2026-10-12", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23307915", "doiUrl": "https://doi.org/10.5281/zenodo.23307915", "conceptDoi": "10.5281/zenodo.23307914", "pdfUrl": "https://github.com/ipitchford/sun-circular-primitive-permutation/releases/download/v1.1.0-candidate/sun-circular-primitive-permutation-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23307915/files/sun-circular-primitive-permutation-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23307915", "repoUrl": "https://github.com/ipitchford/sun-circular-primitive-permutation", "releaseUrl": "https://github.com/ipitchford/sun-circular-primitive-permutation/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/sun-circular-primitive-permutation/index.md", "bibtexUrl": "https://evidencepress.org/releases/sun-circular-primitive-permutation/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/sun-circular-primitive-permutation.mp3", "imageUrl": "https://evidencepress.org/assets/og/sun-circular-primitive-permutation.png?v=4c3277e8b1", "coverArtUrl": "https://evidencepress.org/assets/art/sun-circular-primitive-permutation.svg?v=6821660532", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/sun-circular-primitive-permutation.mp3", "name": "Audio briefing — A primitive circle—with one exception", "description": "Synthetic plain-English briefing, not additional scientific evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/sun-circular-primitive-permutation.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed computer-assisted candidate. Internal replay and targeted internal checking are not external peer review, formal verification or unaffiliated reproduction." }, "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/23307915", "note": "The frozen ZIP, PDF and checksum file were fully downloaded from both GitHub and Zenodo and matched the local publication assets. Byte parity is not 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": "Fresh full-cycle workspace replay passed for all 510 fields and 3,472,147 cycles. Clean-extraction source/receipt binding, full residual enumeration, selected regeneration and 12 normal/optimized negative controls passed. Linux CI is selected-field regression, not full replay.", "evidenceUrl": "https://github.com/ipitchford/sun-circular-primitive-permutation/blob/v1.1.0-candidate/REPLAY_RECEIPT.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Not established by internal checking or publication." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Not established by internal checking or publication." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Not established by internal checking or publication." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Not established by internal checking or publication." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Not established by internal checking or publication." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "evidenceUrl": "https://github.com/ipitchford/sun-circular-primitive-permutation/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "Pinned Python dependency, source and field representations supplied. Local and Linux internal replay are not unaffiliated reproduction." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted research workflow", "Codex: revisions, bounded extension, internal checking and publication" ], "humanRole": "Publication authority only; no human research contribution asserted.", "reviewStatus": "Unrefereed computer-assisted candidate. Internal replay and targeted internal checking are not external peer review, formal verification or unaffiliated reproduction." }, "attributionBoundary": null, "problem": { "name": "Sun’s circular shifted-product conjecture, Conjecture 3.8", "url": "https://arxiv.org/abs/1309.1679v9" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "User-supplied candidate and referee audit, revised 12 October 2026; supplied partial replay and publication replay remain distinct.", "grandfatheredAtSchemaVersion": null, "keywords": [ "finite fields", "Hamiltonian cycles", "primitive elements", "circular permutations", "computer-assisted proof" ], "keyResults": [ "The graph on Fq* with adjacency a+xy primitive is Hamiltonian for q>7 exactly outside (11,2).", "A forced four-cycle proves the exception; exact finite reduction leaves 510 fields.", "A sufficient discrepancy-and-degree condition certifies Hamiltonicity after bounded vertex and edge deletions." ], "reviews": [], "evidencePackage": "Written proof, exact rational residual calculations and regenerated Hamilton cycles checked with separate field arithmetic; historical, reviewer and publication receipts are distinguished.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Sun, Some new problems in additive combinatorics, arXiv v9, Conjecture 3.8: exact source question.", "url": "https://arxiv.org/abs/1309.1679v9", "doi": "10.48550/arXiv.1309.1679" }, { "citation": "Luo and She, A conjecture on circular permutations over finite fields, v1: inherited large-field and zero-parameter branches.", "url": "https://arxiv.org/abs/2610.06289v1", "doi": "10.48550/arXiv.2610.06289" }, { "citation": "Cohen and Prešern, Primitive polynomials with prescribed second coefficient, Lemma 2.3: classical core sieve.", "url": "https://doi.org/10.1017/S0017089506003077", "doi": "10.1017/S0017089506003077" }, { "citation": "Cohen and Huczynska, The strong primitive normal basis theorem, Section 4: complementary-divisor sieve.", "url": "https://doi.org/10.4064/aa143-4-1", "doi": "10.4064/aa143-4-1" }, { "citation": "Chvátal and Erdős, A note on Hamiltonian circuits: connectivity versus independence.", "url": "https://doi.org/10.1016/0012-365X(72)90079-9", "doi": "10.1016/0012-365X(72)90079-9" }, { "citation": "Evidence Press: the distinct Sun–Hou adjacent-sum graph on all field elements.", "url": "https://evidencepress.org/releases/sun-hou-primitive-sum-circle/", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:sun-circular-primitive-permutation", "attemptIds": [ "ep-attempt:sun-circular-primitive-permutation-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "certificate", "description": "Exact shifted-product classification with one elementary obstruction.", "scope": "All prime powers q>7 and all field parameters, with inherited ranges." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Vertices are nonzero field elements; adjacency means a+xy has multiplicative order q−1. The verifier checks fieldhood, cyclic permutations, every edge and all required parameters.", "remainingRisks": [ "Written and imported results are not formally verified.", "No unaffiliated full replay or comprehensive priority search." ] }, "humanJudgmentGates": [ "External specialist scrutiny remains open; no human research work inferred." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentReimplementation" ], "nextAction": "Independently reimplement the graph and full finite replay, and inspect the imported infinite-range argument.", "claimCeiling": "Unrefereed computer-assisted candidate. 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