{
  "schemaVersion": "1.6",
  "slug": "quartic-inverse-coefficients",
  "title": "Strict interlacing and exact intersections for quartic inverse coefficients",
  "shortTitle": "Roots that never meet",
  "url": "https://evidencepress.org/releases/quartic-inverse-coefficients/",
  "oneLine": "Uniform quartic boundary interlacing, with an exact affine point count using external rigidity.",
  "abstract": "The candidate proves strict positive-root interlacing and coprimality for adjacent quartic inverse-coefficient boundary factors. A common Jacobi convolution factor and a positive no-collision determinant treat both parity pairings uniformly. Combining this boundary theorem with external R3 rigidity and credited differential identities gives finite etale affine intersections of degree floor(d(d+1)/6), with Jacobian and next coefficient units.",
  "datePublished": "2026-09-12",
  "dateModified": "2026-09-12",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22727740",
  "doiUrl": "https://doi.org/10.5281/zenodo.22727740",
  "conceptDoi": "10.5281/zenodo.22727739",
  "pdfUrl": "https://github.com/ipitchford/quartic-inverse-coefficients/releases/download/v0.1.0-candidate/manuscript.pdf",
  "altPdfUrl": "https://zenodo.org/records/22727740/files/manuscript.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22727740",
  "repoUrl": "https://github.com/ipitchford/quartic-inverse-coefficients",
  "releaseUrl": "https://github.com/ipitchford/quartic-inverse-coefficients/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/quartic-inverse-coefficients/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/quartic-inverse-coefficients/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/quartic-inverse-coefficients.mp3?v=2907014a29",
  "imageUrl": "https://evidencepress.org/assets/og/quartic-inverse-coefficients.png?v=a1cd9fbeb4",
  "coverArtUrl": "https://evidencepress.org/assets/art/quartic-inverse-coefficients.svg?v=50054a98c7",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/quartic-inverse-coefficients.mp3",
      "name": "Audio briefing",
      "transcriptUrl": "https://evidencepress.org/assets/audio/quartic-inverse-coefficients.txt",
      "description": "AI-generated communication, not additional mathematical evidence."
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate. Uniform boundary interlacing is independent of rigidity; affine reducedness, exact geometric count and units use external R3. No coordinate/residue-field classification, external peer review, formal verification, exhaustive priority or impact 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/22727740",
      "note": "Public archive bytes must pass readback before deployment; availability is not correctness."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Exact scoped symbolic and finite checks, normal and optimized modes; not a proof assistant.",
      "evidenceUrl": "https://github.com/ipitchford/quartic-inverse-coefficients/blob/v0.1.0-candidate/verification.txt"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Internal model reports and supplied evidence 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 reports and supplied evidence 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 reports and supplied evidence 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 reports and supplied evidence 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 reports and supplied evidence 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/quartic-inverse-coefficients/blob/v0.1.0-candidate/requirements.txt",
      "note": "SymPy pinned and Linux replay; TeX environment not hermetic."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI Codex under human direction",
      "Python and SymPy exact arithmetic",
      "Pandoc and pdfLaTeX"
    ],
    "humanRole": "Research direction, supplied reviews and full publication authority; scholarly creator Anonymous.",
    "disclosure": "Internal model review and supplied-review checks do not establish unaffiliated mathematical validation."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Uniform quartic boundary coprimality and structural affine intersections",
    "url": "https://doi.org/10.5281/zenodo.21864574"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Source repairs and internal assessment are distinct from subsequent operational publication and public readback.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "polynomial inversion",
    "strict interlacing",
    "finite free convolution",
    "Jacobi polynomials",
    "affine intersections"
  ],
  "keyResults": [
    "Uniform adjacent positive-root strict interlacing and coprimality, independent of R3.",
    "Using external R3: finite etale affine algebra of degree floor(d(d+1)/6).",
    "Jacobian and next inverse coefficient are units; geometric points need not be rational.",
    "Exact symbolic determinant certificates and bounded diagnostics remain distinct from uniform proofs."
  ],
  "reviews": [],
  "evidencePackage": "Written proof; three symbolic determinant identities; 96 finite coefficient identities; 59 root diagnostics including 53 directed-order checks; finite affine checks; normal/optimized/fresh replay; five internal editorial reports.",
  "openProblems": [
    "Independent specialist scrutiny and reproduction of the proof and source bridge.",
    "Arithmetic structure and residue fields of affine points.",
    "Higher-degree analogues require new arguments.",
    "Formalization of the strictness and determinant proof."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "liqsweep (2026), Polynomial composition rigidity via critical values. External R3 input, pinned research manuscript.",
      "url": "https://github.com/blueberryvertigo/polynomial-composition-rigidity/blob/b17b6b9f7440b12fd85df2c2db9c98209ecc174d/rigidity.typ",
      "doi": null
    },
    {
      "citation": "Martinez-Finkelshtein, Morales and Perales (2024), Real roots of hypergeometric polynomials via finite free convolution.",
      "url": "https://doi.org/10.1093/imrn/rnae120",
      "doi": "10.1093/imrn/rnae120"
    },
    {
      "citation": "Lewis, Perry and Straub (2019), An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms.",
      "url": "https://doi.org/10.1016/j.jpaa.2019.04.002",
      "doi": "10.1016/j.jpaa.2019.04.002"
    },
    {
      "citation": "Anonymous (2026), Smooth-point certificates for Polydegree containments. Motivating affine conjecture.",
      "url": "https://doi.org/10.5281/zenodo.21864574",
      "doi": "10.5281/zenodo.21864574"
    },
    {
      "citation": "Anonymous (2026), Complementary-minor duality and fixed-width reductions for the Polydegree Ideal Conjecture. Credited differential identities.",
      "url": "https://doi.org/10.5281/zenodo.22100350",
      "doi": "10.5281/zenodo.22100350"
    },
    {
      "citation": "Anonymous (2026), A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture. Existential containment candidate, not all-point classification.",
      "url": "https://github.com/ipitchford/polydegree-full-e3-column/blob/v0.1.0-candidate/paper/MANUSCRIPT.md",
      "doi": null
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:quartic-inverse-coefficients",
    "attemptIds": [
      "ep-attempt:quartic-inverse-coefficients-assurance-publication"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "communication"
    ],
    "lineageId": null,
    "parentLinks": [
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": "smooth-point-certificates-polydegree-containments",
        "externalUrl": null,
        "inheritedClaim": "Boundary family and motivating affine conjecture; earlier structural interpretation.",
        "inheritedAssuranceCeiling": "Unrefereed parent; reuse is not independent corroboration."
      },
      {
        "relation": "reuses-method",
        "workId": "ep-work:lps-structural-reductions",
        "legacyReleaseSlug": null,
        "externalUrl": null,
        "inheritedClaim": "Quasismoothness and raising identities, reproduced with attribution.",
        "inheritedAssuranceCeiling": "Unrefereed structural parent; no external validation inherited."
      },
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://github.com/blueberryvertigo/polynomial-composition-rigidity/blob/b17b6b9f7440b12fd85df2c2db9c98209ecc174d/rigidity.typ",
        "inheritedClaim": "R3 excludes a three-coefficient zero block, source m=3,n=d-1.",
        "inheritedAssuranceCeiling": "Public research manuscript; not re-proved or represented as journal-refereed."
      }
    ],
    "accelerationPrimitives": [
      "structural-compression",
      "regime-stitching",
      "research-lineage-reuse",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "reusable-method",
      "description": "Adjacent-boundary strict interlacing and structural affine count.",
      "scope": "Normalized quartic inverse coefficients in characteristic zero, d>=2; affine theorem uses R3."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "Common Jacobi convolution factors and a no-collision determinant prove boundary ordering; R3 and differential rank justify reducedness before Hilbert-series counting.",
      "remainingRisks": [
        "External rigidity validity is inherited, not established by replay.",
        "Finite diagnostics do not prove universal statements.",
        "Bounded prior-work comparison is not exhaustive priority."
      ]
    },
    "humanJudgmentGates": [
      "Audit the proof and exact source-to-R3 mapping.",
      "Assess novelty and priority separately.",
      "Retain rights, dependency and assurance boundaries."
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Independent proof scrutiny and reproduction; preserve the external rigidity dependency.",
      "claimCeiling": "Unrefereed candidate. Uniform boundary interlacing is independent of rigidity; affine reducedness, exact geometric count and units use external R3. No coordinate/residue-field classification, external peer review, formal verification, exhaustive priority or impact claim."
    },
    "impactClaims": [
      {
        "id": "quartic-inverse-coefficients-no-impact",
        "aim": "science",
        "outcome": "Inspectable quartic boundary theorem and conditional structural affine classification",
        "setting": "Producer-coordinated mathematical publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "None.",
        "estimand": "No acceleration or 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. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.",
    "attempts": [
      {
        "attemptId": "ep-attempt:quartic-inverse-coefficients-assurance-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "Prospective remaining assurance and publication only. Original discovery, submitted review and initial source/test edits precede registration and are excluded; no discovery clock is reconstructed.",
          "forecast": {
            "frozenAt": "2026-09-12T16:06:30.879Z",
            "procedureClass": "reviewed-mathematics-publication",
            "targetOutcome": "Complete review revisions and publish GitHub/Zenodo assets and guarded Evidence Press page, media and readback.",
            "expectedActiveMinutes": 150,
            "plausibleLowMinutes": 90,
            "plausibleHighMinutes": 240,
            "expectedUnattendedWaitMinutes": 15,
            "referenceClass": {
              "label": "Completed proof with minor revisions",
              "sampleSize": 0,
              "basis": "Procedural prior, not measured discovery."
            },
            "fermiComponents": [
              {
                "component": "Source repairs and internal editorial gate",
                "count": 1,
                "lowMinutesPerUnit": 30,
                "centralMinutesPerUnit": 50,
                "highMinutesPerUnit": 80,
                "basis": "Established publication route; not an empirical speed comparison."
              },
              {
                "component": "Immutable archives and communication assets",
                "count": 1,
                "lowMinutesPerUnit": 30,
                "centralMinutesPerUnit": 50,
                "highMinutesPerUnit": 80,
                "basis": "Established publication route; not an empirical speed comparison."
              },
              {
                "component": "Composite CI and two deployment cycles",
                "count": 1,
                "lowMinutesPerUnit": 30,
                "centralMinutesPerUnit": 50,
                "highMinutesPerUnit": 80,
                "basis": "Established publication route; not an empirical speed comparison."
              }
            ],
            "probabilityPositiveSignal": 0.95,
            "probabilityTargetClosure": 0.85,
            "probabilityHorizonMinutes": 240,
            "assumptions": [
              "No fatal integrity defect.",
              "Providers available."
            ],
            "reforecastTriggers": [
              "Provider outage or integrity failure."
            ],
            "stopRule": "Timing is telemetry, not a cap; continue unless integrity or provider access blocks publication."
          },
          "outcome": {
            "measuredAt": "2026-09-12T17:40:08.456Z",
            "calendarElapsedMinutes": 94,
            "activeAgentMinutes": 22,
            "activeHumanMinutes": null,
            "computeMinutes": null,
            "unattendedWaitMinutes": 2,
            "blockedMinutes": 0,
            "reworkMinutes": 1,
            "agentRuns": 6,
            "maxParallelAgents": 4,
            "modelTurns": 2,
            "deduplicatedModelTokens": null,
            "uncachedInputTokens": null,
            "researchCycles": {
              "positive": 0,
              "negative": 0,
              "inconclusive": 0
            },
            "falsificationGatesRun": 1,
            "candidateArchitecturesTested": 0,
            "candidateArchitecturesRejected": 0,
            "substantiveReviewRounds": 1,
            "p0Findings": 0,
            "p1Findings": 0,
            "prepublicationClaimCorrections": 0,
            "resultState": "target-closed",
            "resultSummary": "Public GitHub/Zenodo and first guarded canonical site publication completed. Clocks are instrumented lower bounds, not total effort: 18 minutes in the registered initial coordinator window plus four in the resumed observed window; two minutes of CI wait and one minute of tooling repair separately observed. The interrupted permission discussion and other uninstrumented intervals are not assigned to active effort or retrospectively reconstructed; blockedMinutes=0 means no separately timed blocked interval, not absence of a blocker. A speech-API approval interruption occurred and was resolved by explicit user clarification. Final ledger deployment is outside this outcome boundary. modelTurns counts the two publication execution turns, not intervening clarification messages or API inference rounds. No discovery-time, causal acceleration or independent validation claim.",
            "positiveSignalObserved": true,
            "targetReached": true,
            "forecastErrorMinutes": -128,
            "forecastRatio": 0.14666666666666667,
            "withinForecastInterval": false,
            "varianceReason": "Partial lower-bound coordinator timing and an interrupted run are not comparable to the complete-route forecast. Do not use this observation as a full-effort speed estimate.",
            "missingFields": [
              {
                "field": "activeHumanMinutes",
                "reason": "Human effort was not instrumented."
              },
              {
                "field": "computeMinutes",
                "reason": "Hardware inference time was not instrumented."
              },
              {
                "field": "deduplicatedModelTokens",
                "reason": "No task-local fork-aware cumulative counter available."
              },
              {
                "field": "uncachedInputTokens",
                "reason": "No task-local uncached-input counter available."
              }
            ]
          }
        },
        "missingnessReason": null
      }
    ]
  }
}