{
  "schemaVersion": "1.6",
  "slug": "hilbert-series-ternary-septic-octic-nonic",
  "title": "The Hilbert series of the invariants of ternary septics, octics and nonics",
  "shortTitle": "Invariants of plane curves of degree 7, 8 and 9",
  "url": "https://evidencepress.org/releases/hilbert-series-ternary-septic-octic-nonic/",
  "oneLine": "Exact Hilbert series for the invariants of plane curves of degrees 7, 8 and 9, beyond the previous record of degree 6, by known methods made fully auditable.",
  "abstract": "This unrefereed candidate determines the Hilbert series of the rings of SL3(C)-invariants of ternary forms of degrees 7, 8 and 9 (plane septics, octics and nonics). Within the literature we found, the largest degree previously completed was 6 (Bedratyuk–Xin, 2011). The septic series equals the candidate of the earlier Evidence Press release, where it was proved only through degree 760; the octic and nonic series have least denominators of degrees 2331 and 1393 with palindromic numerators. The method is known: pole-order bounds from torus data (Derksen's theory of universal denominators, proved directly for the case used) and the functional equation reduce each series to finitely many coefficients, as in Makam's work on matrix semi-invariants. The release contributes exhaustive pole-bound computations reproduced by a separately written program; correctness statements for two coefficient algorithms that share no code and give the same exact integers over the whole range needed; certificates with archived per-prime data, required out-of-sample predictions and negative controls, validated by re-deriving the known degree-5 and degree-6 series; and structural consequences: exact numbers of minimal generators in low degree and constraints on every homogeneous system of parameters. It also records two attribution corrections to the earlier release. The work was produced by an AI research agent and has not been checked by a human specialist.",
  "datePublished": "2026-09-26",
  "dateModified": "2026-09-26",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22978726",
  "doiUrl": "https://doi.org/10.5281/zenodo.22978726",
  "conceptDoi": "10.5281/zenodo.22978725",
  "pdfUrl": "https://github.com/ipitchford/hilbert-series-ternary-forms/releases/download/v0.2.0-candidate/hilbert-series-ternary-septic-octic-nonic-v0.2.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22978726/files/hilbert-series-ternary-septic-octic-nonic-v0.2.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22978726",
  "repoUrl": "https://github.com/ipitchford/hilbert-series-ternary-forms",
  "releaseUrl": "https://github.com/ipitchford/hilbert-series-ternary-forms/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/hilbert-series-ternary-septic-octic-nonic/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/hilbert-series-ternary-septic-octic-nonic/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/hilbert-series-ternary-septic-octic-nonic.mp3?v=924d3c21c6",
  "imageUrl": "https://evidencepress.org/assets/og/hilbert-series-ternary-septic-octic-nonic.png?v=e41277e995",
  "coverArtUrl": "https://evidencepress.org/assets/art/hilbert-series-ternary-septic-octic-nonic.svg?v=8ba0fb5bf0",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/hilbert-series-ternary-septic-octic-nonic.mp3",
      "name": "Audio briefing",
      "description": "AI-generated communication, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/hilbert-series-ternary-septic-octic-nonic.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate produced by an AI research agent. Computer-assisted theorems resting on published theorems (Knop, Stanley, Herbig–Schwarz, Hochster–Roberts) and on pole-order bounds proved in the paper following Derksen; the method is not new. No formal verification, independent human reproduction, or specialist or journal review; novelty is established only within a bounded literature 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/22978726",
      "note": "Public GitHub prerelease v0.2.0-candidate and published Zenodo record (3 files each); every file matched the SHA-256 sums by server checksum and by independent download."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Replay from a fresh extraction in the archived and fresh modes; the receipts record code hashes and bind to the shipped manifest. A full regeneration mode is documented but was not receipted in this version.",
      "evidenceUrl": "https://github.com/ipitchford/hilbert-series-ternary-forms/blob/v0.2.0-candidate/REPLAY_RECEIPT_fresh.json"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Model reviewers re-ran the replay and small computations inside one producer workflow; not an unaffiliated rerun."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Separately written pole-bound program (blind agent) and a second coefficient algorithm (weight counting) giving the same exact integers, within one producer workflow: implementation diversity, not independent 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; the C engines and Python checkers are trusted."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Producer-coordinated editorial gate and external model reviews 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": "No journal or venue review."
    },
    {
      "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/hilbert-series-ternary-forms/actions",
      "note": "Linux CI runs the archived and fresh modes; versions recorded."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "Claude (Anthropic) research agent under user direction",
      "C with OpenMP (GCC); Python with python-flint (FLINT), SymPy, NumPy and SciPy",
      "pdfTeX and pandoc"
    ],
    "humanRole": "The user set the task, supplied an external review and authorised publication; scholarly creator Anonymous. No human mathematical contribution is claimed.",
    "disclosure": "AI-produced research, computation and drafting. Separate AI agents wrote the independent checks (the blind pole-bound program; earlier the septic weight counting); cross-vendor adversarial reviews (GPT-5.6, two rounds on the septic argument); one publisher-supplied external model review (major revisions, actioned); a producer-coordinated five-role editorial gate (major revision, actioned). None of these is specialist or journal peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Hilbert series of invariants of ternary forms beyond degree 6",
    "url": "https://doi.org/10.1080/03081087.2010.536763"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Research, the prior-art check, the external model review and its repairs, the internal editorial gate and its repairs, and publication are recorded separately.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "invariant theory",
    "Hilbert series",
    "Poincaré series",
    "ternary forms",
    "plane curves",
    "universal denominator",
    "computer-assisted proof",
    "systems of parameters"
  ],
  "keyResults": [
    "The Hilbert series of SL3-invariants of plane septics equals the candidate of the earlier release, now unconditionally (least denominator of degree 1386).",
    "The octic and nonic series are determined: least denominators of degrees 2331 and 1393, palindromic numerators of degrees 2286 and 1338.",
    "Two algorithms sharing no code give the same exact coefficients over the whole range needed (to degrees 1194 and 690).",
    "Exact minimal-generator counts in low degree, e.g. 45, 118, 606 and 2009 new generators in degrees 8 to 11 for nonics.",
    "Every homogeneous system of parameters has a member of degree divisible by 108, 147 and 64 for degrees 7, 8, 9 respectively; solver-computed minimum degree sums are 1428, 2391 and 1457."
  ],
  "reviews": [],
  "evidencePackage": "Ten-page paper with self-contained pole-order lemmas and correctness statements for both coefficient algorithms; exact certificates for d = 7, 8, 9 with required out-of-sample predictions; exact agreement of two algorithms that share no code over the whole range needed; archived raw per-prime residues re-lifted on every replay; negative controls including undersized denominators; replay in labelled modes with archived- and fresh-mode receipts that bind code hashes to the shipped manifest; validation on the known d = 5, 6 cases; research-gates record; editorial-gate reports and response matrix.",
  "openProblems": [
    "Explain the recurring excess of exactly 2 in the pole bounds: centralisers explain it only at r = 2 (Remark 9).",
    "Degree 10 (66 weights): needs an engine change and roughly 100 CPU-hours with the present bound.",
    "Construct explicit systems of parameters and generators consistent with the constraints of Section 6, and certify the degree-sum minima exactly.",
    "Specialist checking of Propositions 3-4 and Lemmas 5-8, and an unaffiliated rerun of the full regeneration mode."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Evidence Press (2026), earlier release: matrix-invariant Hilbert series and a candidate septic series (version 0.1.0-candidate).",
      "url": "https://doi.org/10.5281/zenodo.22974308",
      "doi": "10.5281/zenodo.22974308"
    },
    {
      "citation": "Derksen (2005): universal denominators of Hilbert series; the pole bound and torus reduction used here.",
      "url": "https://doi.org/10.1016/j.jalgebra.2004.10.029",
      "doi": "10.1016/j.jalgebra.2004.10.029"
    },
    {
      "citation": "Makam (2016): Hilbert series and degree bounds for matrix (semi-)invariants; the same finite-determination strategy.",
      "url": "https://doi.org/10.1016/j.jalgebra.2016.01.032",
      "doi": "10.1016/j.jalgebra.2016.01.032"
    },
    {
      "citation": "Bedratyuk and Xin (2011): Poincaré series of invariants of ternary forms of degrees 5 and 6.",
      "url": "https://doi.org/10.1080/03081087.2010.536763",
      "doi": "10.1080/03081087.2010.536763"
    },
    {
      "citation": "Bedratyuk (2009): analogue of the Sylvester–Cayley formula for ternary forms; septic coefficients printed to degree 21.",
      "url": "https://arxiv.org/abs/0806.1920",
      "doi": "10.48550/arXiv.0806.1920"
    },
    {
      "citation": "Herbig and Schwarz: the Koszul complex of a moment map (2-largeness, Theorem 3.7).",
      "url": "https://arxiv.org/abs/1205.4608",
      "doi": "10.48550/arXiv.1205.4608"
    },
    {
      "citation": "Brouwer, Draisma and Popoviciu (2015): the degrees of a system of parameters of the ring of invariants of a binary form; the method behind the hsop constraints.",
      "url": "https://doi.org/10.1007/s00031-015-9335-8",
      "doi": "10.1007/s00031-015-9335-8"
    },
    {
      "citation": "Benanti, Boumova, Drensky, Genov and Koev (2012): survey of computing Hilbert series of invariants, including the ternary frontier.",
      "url": "https://arxiv.org/abs/1201.4448",
      "doi": "10.48550/arXiv.1201.4448"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:hilbert-series-ternary-septic-octic-nonic",
    "attemptIds": [
      "ep-attempt:hilbert-series-ternary-septic-octic-nonic-publication"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "communication"
    ],
    "lineageId": null,
    "parentLinks": [],
    "accelerationPrimitives": [
      "structural-compression",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "certificate",
      "description": "Exact Hilbert series of C[S^d C^3]^SL3 for d = 7, 8, 9 via pole-order bounds from torus data (Lemmas 5-8), the functional equation and exact coefficients.",
      "scope": "Unrefined total-degree grading; d = 7, 8, 9 only."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "Propositions 3-4 (coefficient algorithms) supply exact coefficients; Lemmas 5-8 prove and compute the pole bounds; Proposition 2 turns finitely many coefficients into the full series; certify scripts implement it, with a required out-of-sample prediction.",
      "remainingRisks": [
        "Engines and checkers are trusted; no formal verification.",
        "Knop's criterion used via Makam's statement.",
        "The two coefficient algorithms are independent in code but both rest on Weyl's formulas."
      ]
    },
    "humanJudgmentGates": [
      "Assess Propositions 3-4 and Lemmas 5-8.",
      "Preserve candidate status and the bounded novelty statement.",
      "Publication is authorised; external review is a separate dimension."
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Specialist checking of Propositions 3-4 and Lemmas 5-8; unaffiliated rerun of replay mode full.",
      "claimCeiling": "Unrefereed candidate; computer-assisted theorems for d = 7, 8, 9 by known methods; no formal verification or external review."
    },
    "impactClaims": [
      {
        "id": "hilbert-series-ternary-septic-octic-nonic-no-impact",
        "aim": "science",
        "outcome": "Inspectable computer-assisted Hilbert series",
        "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:hilbert-series-ternary-septic-octic-nonic-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "Prospective publication-scope measurement from this registration: completion of the recomputation with raw residues, the internal five-role editorial gate with one repair batch, research repository and immutable GitHub/Zenodo identity, Evidence Press page, media and canonical readback. The procedural forecast was frozen in the local publication state at registration. The unconditional septic argument, the octic/nonic computations, the external review and its repairs before registration are left-censored; research-phase forecasts and outcomes are in the package's PRE-REGISTERED.md. No acceleration comparison.",
          "forecast": {
            "frozenAt": "2026-09-26T14:18:59Z",
            "procedureClass": "mathematical-candidate-publication-small-archive",
            "targetOutcome": "Internal editorial acceptance, immutable public research assets (GitHub release and published Zenodo DOI 10.5281/zenodo.22978726 with byte-identical assets) and canonical Evidence Press availability.",
            "expectedActiveMinutes": 120,
            "plausibleLowMinutes": 80,
            "plausibleHighMinutes": 200,
            "expectedUnattendedWaitMinutes": 60,
            "referenceClass": {
              "label": "Reviewed mathematical candidate publication with a sub-megabyte archive",
              "sampleSize": 3,
              "basis": "Seymour, Davies-Levitin and Hilbert v0.1.0 releases; procedural prior only; no acceleration comparison."
            },
            "fermiComponents": [
              {
                "component": "Deterministic readiness baseline and package assembly",
                "count": 1,
                "lowMinutesPerUnit": 10,
                "centralMinutesPerUnit": 15,
                "highMinutesPerUnit": 30,
                "basis": "Procedural prior; no comparative acceleration claim."
              },
              {
                "component": "Five-role internal editorial gate and one repair batch",
                "count": 1,
                "lowMinutesPerUnit": 30,
                "centralMinutesPerUnit": 45,
                "highMinutesPerUnit": 70,
                "basis": "Procedural prior; no comparative acceleration claim."
              },
              {
                "component": "GitHub repository, tag, release and Zenodo publication",
                "count": 1,
                "lowMinutesPerUnit": 10,
                "centralMinutesPerUnit": 15,
                "highMinutesPerUnit": 30,
                "basis": "Procedural prior; no comparative acceleration claim."
              },
              {
                "component": "Evidence Press page, explanatory art, OG card, thumbnail and audio",
                "count": 1,
                "lowMinutesPerUnit": 20,
                "centralMinutesPerUnit": 25,
                "highMinutesPerUnit": 40,
                "basis": "Procedural prior; no comparative acceleration claim."
              },
              {
                "component": "Two-commit seals, CI, guarded deployment and two readback cycles",
                "count": 1,
                "lowMinutesPerUnit": 10,
                "centralMinutesPerUnit": 20,
                "highMinutesPerUnit": 30,
                "basis": "Procedural prior; no comparative acceleration claim."
              }
            ],
            "probabilityPositiveSignal": 0.9,
            "probabilityTargetClosure": 0.8,
            "probabilityHorizonMinutes": 300,
            "assumptions": [
              "No fatal integrity defect surfaces in the editorial gate.",
              "GitHub, Zenodo, Cloudflare and OpenAI providers remain available."
            ],
            "reforecastTriggers": [
              "Provider outage, credential expiry or integrity failure.",
              "Editorial decision of Major Revision or Reject."
            ],
            "stopRule": "Timing is telemetry, not a cap; continue unless integrity, rights or provider access blocks publication."
          },
          "outcome": {
            "measuredAt": "2026-09-26T19:55:58Z",
            "calendarElapsedMinutes": 337,
            "activeAgentMinutes": 225,
            "activeHumanMinutes": null,
            "computeMinutes": null,
            "unattendedWaitMinutes": 52,
            "blockedMinutes": 60,
            "reworkMinutes": 60,
            "agentRuns": 7,
            "maxParallelAgents": 5,
            "modelTurns": 12,
            "deduplicatedModelTokens": null,
            "uncachedInputTokens": null,
            "researchCycles": {
              "positive": 2,
              "negative": 1,
              "inconclusive": 0
            },
            "falsificationGatesRun": 9,
            "candidateArchitecturesTested": 1,
            "candidateArchitecturesRejected": 0,
            "substantiveReviewRounds": 1,
            "p0Findings": 0,
            "p1Findings": 6,
            "prepublicationClaimCorrections": 5,
            "resultState": "target-closed",
            "resultSummary": "External model review (Major) and five-role internal gate (Major Revision; no mathematical error) actioned in one repair batch: self-contained pole-order lemmas, the out-of-sample prediction made part of the certificate with undersized-denominator controls, second-algorithm exact agreement over the whole determining range (packed weight counting), sharper generator counts, a centraliser-bound hypothesis tested and largely refuted, and a cross-reference gate. Published after deterministic checks; the full replay mode is documented but not receipted.",
            "positiveSignalObserved": true,
            "targetReached": true,
            "forecastErrorMinutes": 105,
            "forecastRatio": 1.875,
            "withinForecastInterval": false,
            "varianceReason": "Active minutes estimated from session timestamps. Overrun came from the editorial gate's material repairs (second algorithm, certificate change), a replay-receipt/manifest binding redo, an accessibility overflow fix, a missing method-registry changeLog entry, and a confirmation round and a 3.5-hour full replay that are not part of the Evidence Press process and were stopped at the publisher's direction; about 60 minutes were blocked by a usage limit. modelTurns (12) is an estimate of user-facing turns since registration; the count is not instrumented across the context compaction.",
            "missingFields": [
              {
                "field": "activeHumanMinutes",
                "reason": "Human effort not instrumented."
              },
              {
                "field": "computeMinutes",
                "reason": "No separately instrumented total."
              },
              {
                "field": "deduplicatedModelTokens",
                "reason": "Fork-aware runtime counter unavailable."
              },
              {
                "field": "uncachedInputTokens",
                "reason": "Runtime token/cache accounting unavailable."
              }
            ]
          }
        },
        "missingnessReason": null
      }
    ]
  }
}