{
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  "slug": "fano-plane-spectrum",
  "title": "The homogeneous adjacency spectrum of the Fano plane",
  "shortTitle": "Seven points. Nineteen roots.",
  "url": "https://evidencepress.org/releases/fano-plane-spectrum/",
  "oneLine": "An exact tensor spectrum, a real eigenvalue without real eigenvectors, and a hidden genus-three curve.",
  "abstract": "The seven-point Fano plane has a homogeneous adjacency characteristic polynomial of degree 448. This computational candidate gives its complete factorization into ten irreducible factors and nineteen distinct roots, explicit eigenvectors, the exact H-spectrum, and a characteristic-zero certificate for a smooth degree-eight genus-three eigencurve. Producer replay, saved computation artifacts and internal review remain distinct from unaffiliated validation and historical priority.",
  "datePublished": "2026-09-19",
  "dateModified": "2026-09-19",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22850084",
  "doiUrl": "https://doi.org/10.5281/zenodo.22850084",
  "conceptDoi": "10.5281/zenodo.22850083",
  "pdfUrl": "https://github.com/ipitchford/fano-plane-spectrum/releases/download/v0.1.0-candidate/fano_spectrum.pdf",
  "altPdfUrl": "https://zenodo.org/records/22850084/files/fano_spectrum.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22850084",
  "repoUrl": "https://github.com/ipitchford/fano-plane-spectrum",
  "releaseUrl": "https://github.com/ipitchford/fano-plane-spectrum/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/fano-plane-spectrum/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/fano-plane-spectrum/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/fano-plane-spectrum.mp3?v=1867a28117",
  "imageUrl": "https://evidencepress.org/assets/og/fano-plane-spectrum.png?v=0823a92ea2",
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      "url": "https://evidencepress.org/assets/audio/fano-plane-spectrum.mp3",
      "name": "Audio briefing",
      "transcriptUrl": "https://evidencepress.org/assets/audio/fano-plane-spectrum.txt",
      "description": "AI-generated communication, not additional mathematical evidence."
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
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    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed computational theorem candidate. Default replay does not freshly repeat the complete thirty-prime computation or every finite number-field decomposition. No unaffiliated validation, formal proof, exhaustive priority or impact claim."
  },
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      "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/22850084",
      "note": "Public archive readback establishes availability, not correctness."
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    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Saved residues and scoped fresh computations, ordinary/optimized/clean extraction and negative controls; not full fresh thirty-prime reconstruction.",
      "evidenceUrl": "https://github.com/ipitchford/fano-plane-spectrum/blob/v0.1.0-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": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal model review and producer-coordinated checks do not establish this external assurance dimension."
    },
    {
      "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/fano-plane-spectrum/blob/v0.1.0-candidate/requirements.txt",
      "note": "Python packages pinned and Singular version recorded; environment not hermetic."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python, SymPy, python-flint and Singular exact computation",
      "Pandoc and Tectonic"
    ],
    "humanRole": "Research direction, supplied candidate and review, and full publication authorization; scholarly creator Anonymous.",
    "disclosure": "Internal model reports and producer replay do not establish unaffiliated mathematical validation."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Homogeneous adjacency spectrum of the Fano plane (AMR-030-0030)",
    "url": "https://people.math.sc.edu/cooper/combprob.html"
  },
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  "corrections": [],
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  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Supplied candidate and review; scoped producer reconstruction, revision and internal editorial gate precede public archive and deployment readback.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Fano plane",
    "tensor eigenvalues",
    "hypergraph spectrum",
    "resultants",
    "eigenschemes"
  ],
  "keyResults": [
    "Exact degree-448 resultant factorization with nineteen distinct eigenvalues.",
    "H-spectrum consists of 0, 1, 3 and the unique real root of lambda cubed plus twice lambda squared plus twice lambda minus two.",
    "The real eigenvalue 2 has no real eigenvector.",
    "Two quadratic eigenvalues have smooth geometrically irreducible degree-eight genus-three eigencurves."
  ],
  "reviews": [],
  "evidencePackage": "Exact modular resultant artifacts and deterministic CRT bounds; explicit witnesses; characteristic-zero saturation, Jacobian and resolution; rational real obstruction; normal/optimized clean replay; semantic negative controls; five internal role reports and focused confirmation.",
  "openProblems": [
    "Unaffiliated full resultant reconstruction and semantic audit.",
    "Specialist audit of finite eigenscheme classification and curve geometry.",
    "Conceptual explanation of multiplicity 52 and possible classical curve identification.",
    "Broader primary-source work on historical priority."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Cooper and Dutle (2012), Spectra of uniform hypergraphs. Normalization and spectral framework.",
      "url": "https://doi.org/10.1016/j.laa.2011.11.018",
      "doi": "10.1016/j.laa.2011.11.018"
    },
    {
      "citation": "Cooper and Dutle (2015), Computing hypermatrix spectra with the Poisson product formula. Established algorithmic precedent.",
      "url": "https://doi.org/10.1080/03081087.2014.910207",
      "doi": "10.1080/03081087.2014.910207"
    },
    {
      "citation": "Clark and Cooper, Applications of the Harary–Sachs theorem for hypergraphs. Inspected author version: leading Fano coefficients and complete Rowling polynomial.",
      "url": "https://arxiv.org/abs/2107.10781v1",
      "doi": "10.48550/arXiv.2107.10781"
    },
    {
      "citation": "Doğan, Tsigaridas and Zafeirakopoulos (2026), Resultant multiplicity via projective degrees and applications to tensor eigenvalues. Recent preprint, contextual bound.",
      "url": "https://arxiv.org/abs/2609.01268v1",
      "doi": "10.48550/arXiv.2609.01268"
    }
  ],
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    "decisionObject": {
      "type": "certificate",
      "description": "Exact Fano homogeneous resultant and eigencurve certificates.",
      "scope": "One specified order-three normalized tensor in characteristic zero; no general multiplicity or priority theorem."
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    "semanticBridge": {
      "state": "explicit",
      "description": "Seven-line incidence data determines quadratic eigen-equations; Poisson recursion, CRT and characteristic-zero algebra support the declared claims.",
      "remainingRisks": [
        "CAS and source-to-equations correctness.",
        "Saved versus fresh computational provenance.",
        "Finite decomposition and historical-priority coverage."
      ]
    },
    "humanJudgmentGates": [
      "Audit mathematical arguments and equation encoding.",
      "Assess prior art and priority separately.",
      "Retain rights and assurance boundaries."
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    "assuranceTarget": {
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        "independentRerun",
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      "nextAction": "Unaffiliated full computation and specialist proof audit.",
      "claimCeiling": "Unrefereed computational theorem candidate. Default replay does not freshly repeat the complete thirty-prime computation or every finite number-field decomposition. No unaffiliated validation, formal proof, exhaustive priority or impact claim."
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        "id": "fano-plane-spectrum-no-impact",
        "aim": "science",
        "outcome": "Inspectable exact spectral and eigenscheme example",
        "setting": "Producer-coordinated mathematical publication",
        "status": "NO_IMPACT_EVIDENCE",
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        "comparator": "None.",
        "estimand": "No acceleration or impact effect estimated.",
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          "measurementScope": "assurance-through-publication",
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            "positiveSignalObserved": true,
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