{ "schemaVersion": "1.6", "slug": "hilbert-series-matrix-septic-invariants", "title": "Hilbert series of the invariants of tuples of 4×4 and 5×5 matrices, and a candidate series for plane septics", "shortTitle": "Matrix invariants: nine new Hilbert series", "url": "https://evidencepress.org/releases/hilbert-series-matrix-septic-invariants/", "oneLine": "Exact counts of the invariants of three to ten 4×4 matrices and of three 5×5 matrices, confirming Berele's denominator conjecture in these nine cases, plus a plane-septic Hilbert series proved to degree 760.", "abstract": "This unrefereed candidate determines the one-variable Hilbert series of the ring of invariants of k generic n×n matrices under simultaneous conjugation for n = 4 with 3 ≤ k ≤ 10 and for (n, k) = (5, 3). Within the literature we found, series with three or more matrices were previously known only for n ≤ 3. Each series is a computer-assisted theorem: Van den Bergh's denominator theorem, the Cohen–Macaulay property and Formanek's functional equation reduce it to finitely many coefficients, which are computed exactly modulo several primes by evaluating the Molien–Weyl integral on grids of roots of unity. In all nine cases the least denominator is the one conjectured by Berele (J. Algebra, 2023), which therefore holds for these pairs. The leading constants of the series fix the number of secondary invariants for any choice of primary degrees. For plane septics (SL3 acting on ternary forms of degree 7) the release gives an explicit candidate rational function R(t) with a least denominator of degree 1386, proves that the Hilbert series agrees with it in every coefficient up to degree 760, and proves full equality conditionally on the existence of a homogeneous system of parameters with compatible degrees, which is unknown. An a priori denominator whose pole set coincides with that of R reduces an unconditional proof to exact coefficients up to degree 10902. Agreement modulo single primes extends to degree 4300 as evidence, not proof. 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": "0.1.0-candidate", "doi": "10.5281/zenodo.22974308", "doiUrl": "https://doi.org/10.5281/zenodo.22974308", "conceptDoi": "10.5281/zenodo.22974307", "pdfUrl": "https://github.com/ipitchford/hilbert-series-matrix-septic-invariants/releases/download/v0.1.0-candidate/hilbert-series-matrix-septic-invariants-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22974308/files/hilbert-series-matrix-septic-invariants-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22974308", "repoUrl": "https://github.com/ipitchford/hilbert-series-matrix-septic-invariants", "releaseUrl": "https://github.com/ipitchford/hilbert-series-matrix-septic-invariants/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/hilbert-series-matrix-septic-invariants/index.md", "bibtexUrl": "https://evidencepress.org/releases/hilbert-series-matrix-septic-invariants/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/hilbert-series-matrix-septic-invariants.mp3?v=7dee03073e", "imageUrl": "https://evidencepress.org/assets/og/hilbert-series-matrix-septic-invariants.png?v=03fa0deef8", "coverArtUrl": "https://evidencepress.org/assets/art/hilbert-series-matrix-septic-invariants.svg?v=5a1b8330d7", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/hilbert-series-matrix-septic-invariants.mp3", "name": "Audio briefing", "description": "AI-generated communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/hilbert-series-matrix-septic-invariants.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. The matrix theorems are computer-assisted proofs resting on published theorems and exact computation; the septic identity is proved only through degree 760 and conditionally beyond. There is 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/22974308", "note": "Public GitHub prerelease v0.1.0-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": "Fresh-extraction FULL replay of the final candidate: manifest before and after; engine validation; ten matrix certificates; the blind second engine over every certificate range; exact septic regeneration to degree 760; character check to degree 32; septic grid, pole-set and Theorem 2 checks; 19 negative controls in normal and optimised Python. Linux CI runs the quick replay.", "evidenceUrl": "https://github.com/ipitchford/hilbert-series-matrix-septic-invariants/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": "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 code inside one producer workflow (blind septic weight counting; a second matrix grid engine; a character formula; reviewers' own grids): 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 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-matrix-septic-invariants/actions", "note": "Linux CI runs the quick replay; versions recorded. Recomputing the stored grid residues takes CPU-hours and is documented, not replayed." } ], "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 and supplied publication authority and an external review; scholarly creator Anonymous. No human mathematical contribution is claimed.", "disclosure": "AI-produced research, computation and drafting. Separate AI agents wrote the independent septic checks and ran a literature search; one cross-vendor adversarial review (GPT-5.6, Major Revision), one publisher-supplied external model review (Major Revision, no fatal defect) and a producer-coordinated five-role AI editorial gate. None of these is specialist or journal peer review." }, "attributionBoundary": null, "problem": { "name": "Least denominators of one-variable Poincaré series of generic matrices (Berele, Conjecture 1)", "url": "https://doi.org/10.1016/j.jalgebra.2022.11.023" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Research, the responses to a GPT-5.6 adversarial review and to a publisher-supplied external model review, the internal editorial gate, and publication are recorded separately.", "grandfatheredAtSchemaVersion": null, "keywords": [ "invariant theory", "Hilbert series", "Poincaré series", "generic matrices", "trace identities", "Berele conjecture", "ternary forms", "plane septics", "Molien–Weyl formula", "computer-assisted proof" ], "keyResults": [ "Exact one-variable Hilbert series of the pure trace ring C̄(4,k) for 3 ≤ k ≤ 10 and of C̄(5,3); for n = 4 the least denominator is (1−t)^{3k−3}(1−t²)^{4k−4}(1−t³)^{5k−5}(1−t⁴)^{4k−3} (Theorem 1).", "Berele's Conjecture 1 on the least denominator holds in all nine cases (exact gcd and denominator-equality checks).", "The leading constants c_{n,k} fix the number of secondary invariants for any primary degrees; for example c_{4,3} = 521/(2^25·3^8), so the product of any primary degrees of C̄(4,3) is divisible by 2^25·3^8.", "Plane septics: the Hilbert series of SL3-invariants of ternary septics agrees with an explicit candidate R(t) in every coefficient up to degree 760 (Theorem 2(1)); full equality holds if an hsop of compatible degrees with sum at most 1557 exists (Theorem 2(2)); whether one exists is unknown.", "An a priori septic denominator of degree 21840 from the 1761 extreme rays of the weight cone has exactly the pole set of R, reducing an unconditional proof to exact coefficients up to degree 10902 (Section 7).", "An independent symmetric-group character formula reproduces every matrix series through degree 32; 19 negative controls are rejected." ], "reviews": [ { "schemaVersion": "1.0", "reviewId": "hilbert-series-matrix-septic-internal-editorial-v1", "reviewClass": "editorial-assessment", "status": "published", "subject": { "canonicalUrl": "https://evidencepress.org/releases/hilbert-series-matrix-septic-invariants/", "doi": "10.5281/zenodo.22974308", "version": "0.1.0-candidate", "releaseTag": "v0.1.0-candidate", "title": "Hilbert series of the invariants of tuples of 4×4 and 5×5 matrices, and a candidate series for plane septics" }, "reviewer": { "name": "Five internal model editorial roles and two confirmation reviewers", "independence": "producer-coordinated" }, "assessment": { "title": "Internal five-role editorial assessment with confirmation round", "date": "2026-09-26", "recommendation": "Minor Revision" }, "assuranceImpact": { "changesVerificationStatus": false, "changesPeerReviewedFlag": false, "changesIndependentReproductionFlag": false, "changesFormalVerificationFlag": false, "explanation": "Producer-coordinated model review does not establish external peer review, reproduction or formal proof." }, "publication": { "publicUrl": "https://github.com/ipitchford/hilbert-series-matrix-septic-invariants/blob/v0.1.0-candidate/review/editorial-gate/EDITORIAL_DECISION.md", "publicSummary": "Five roles reviewed one frozen target; all recommended Minor Revision and none found a mathematical error. Several re-computed load-bearing results with their own code. The repair batch fixed an artefact defect (a negative control had overwritten a shipped evidence record), reframed the septic result as a candidate series with an explicitly unknown hsop hypothesis, added an a priori septic denominator that reduces an unconditional proof to a finite computation, stated Berele's conjecture explicitly, corrected the prior-art table, and added a second matrix engine written blind by a separate agent that agrees over every certificate range. Two confirmation reviewers (mathematics, computation) accepted the repairs with minor edits, which were applied; both independently reproduced the new a priori pole-set computation." } } ], "evidencePackage": "Eleven-page paper; ten matrix certificates recomputed from stored exact residues on every replay; exact septic integers to degree 760 regenerable from scratch; a separate grid computation exact to degree 700; modular agreement to degrees 2100 and 4300; an independent character-formula check; 19 semantic negative controls; a machine-readable verification summary labelling each check exact or modular; model-based review records.", "openProblems": [ "Obtain specialist checking of the certificate argument (Proposition 5) and an independent rerun or reimplementation of the matrix grid engine.", "Prove the septic identity unconditionally: compute exact coefficients up to degree 10902 (about 3000 CPU-hours with the present engine), sharpen the a priori pole-order bounds, or construct a homogeneous system of parameters.", "Extend to C̄(5,k) for k ≥ 4, C̄(6,3), ternary octics and quaternary quartics.", "Construct explicit primary invariants for C̄(4,3) compatible with the divisibility constraint from c_{4,3}." ], "mathObjects": [], "relatedWorks": [ { "citation": "Berele (2023): denominators for one-variable Poincaré series of generic matrices; Conjecture 1 (least denominator).", "url": "https://doi.org/10.1016/j.jalgebra.2022.11.023", "doi": "10.1016/j.jalgebra.2022.11.023" }, { "citation": "de Mello Koch and Jevicki (2025): structure of loop space at finite N; Hilbert series for two matrices up to N = 7 and Hironaka decompositions.", "url": "https://doi.org/10.1007/JHEP06(2025)011", "doi": "10.1007/JHEP06(2025)011" }, { "citation": "Kristensson and Wilhelm (2020): partition functions of N = 4 SYM theory at finite N, including two-matrix series and mixed su(2|3) partition functions.", "url": "https://arxiv.org/abs/2005.06480", "doi": "10.48550/arXiv.2005.06480" }, { "citation": "Djoković (2007): Poincaré series of pure and mixed trace algebras of two generic matrices.", "url": "https://arxiv.org/abs/math/0609262", "doi": null }, { "citation": "Bedratyuk and Xin (2011): Poincaré series of invariants of ternary and quaternary forms by MacMahon partition analysis.", "url": "https://doi.org/10.1080/03081087.2010.536763", "doi": "10.1080/03081087.2010.536763" }, { "citation": "Herbig and Schwarz (2013): the Koszul complex of a moment map; irreducible non-coregular modules are 2-large.", "url": "https://arxiv.org/abs/1205.4608", "doi": "10.48550/arXiv.1205.4608" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:hilbert-series-matrix-septic-invariants", "attemptIds": [ "ep-attempt:hilbert-series-matrix-septic-invariants-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 one-variable Hilbert series of C̄(4,k), 3 ≤ k ≤ 10, and C̄(5,3), with least denominators equal to Berele's Conjecture 1; the plane-septic series agrees with the candidate R(t) through degree 760.", "scope": "Unrefined total-degree grading; septic identity conditional beyond degree 760." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Lemma 3 (exact grid evaluation) supplies residues; Proposition 5 (Van den Bergh, Hochster–Roberts, Formanek) turns agreement of finitely many coefficients into equality; certify_mat.py implements it and checks Berele's denominator. Theorem 2 rests on exact weight-counting integers and the functional-equation chain of §3.1.", "remainingRisks": [ "The C grid engine and the checkers are trusted; no formal verification.", "Van den Bergh, Formanek and Knop are used through published restatements.", "The septic identity is unconditional only through degree 760." ] }, "humanJudgmentGates": [ "Assess Proposition 5 and the grid lemma.", "Preserve candidate status, the conditional septic wording 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 Proposition 5 and an unaffiliated rerun of the matrix grid engine; the finite computation to degree 10902 for the septics.", "claimCeiling": "Unrefereed candidate. Computer-assisted theorems for nine matrix cases; the septic series proved through degree 760 and conditionally beyond; no formal verification or external review." }, "impactClaims": [ { "id": "hilbert-series-matrix-septic-invariants-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-matrix-septic-invariants-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Prospective publication-scope measurement from this registration: 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 below was frozen in the local publication state at 2026-09-26T08:23:52Z and is copied unchanged. Problem selection, the computations, certificates, the GPT-5.6 adversarial review, the publisher-supplied external model review and its repairs (applied before registration) are left-censored and are not reconstructed here; the research-phase forecasts and outcomes are in the package's PRE-REGISTERED.md. No acceleration comparison.", "forecast": { "frozenAt": "2026-09-26T08:45:03Z", "procedureClass": "mathematical-candidate-publication-small-archive", "targetOutcome": "Internal editorial acceptance, immutable public research assets (GitHub release and published Zenodo DOI 10.5281/zenodo.22974308 with byte-identical assets) and canonical Evidence Press availability.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 100, "plausibleHighMinutes": 240, "expectedUnattendedWaitMinutes": 45, "referenceClass": { "label": "Reviewed mathematical candidate publication with a sub-megabyte archive", "sampleSize": 2, "basis": "Seymour 2δ+3 (190 active minutes, 3 GB archive) and Davies–Levitin (60 active + 105 blocked minutes, 2026-09-25). Procedural prior only; no acceleration comparison. Totals first frozen in the local publication state at 2026-09-26T08:23:52Z and recorded here unchanged." }, "fermiComponents": [ { "component": "Deterministic readiness baseline and research-package assembly", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 40, "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": 30, "centralMinutesPerUnit": 40, "highMinutesPerUnit": 60, "basis": "Procedural prior; no comparative acceleration claim." }, { "component": "Two-commit seals, CI, guarded deployment and two readback cycles", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 40, "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": "2026-09-26T09:49:57Z", "calendarElapsedMinutes": 65, "activeAgentMinutes": 50, "activeHumanMinutes": null, "computeMinutes": null, "unattendedWaitMinutes": 15, "blockedMinutes": 0, "reworkMinutes": 10, "agentRuns": 9, "maxParallelAgents": 5, "modelTurns": 12, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 19, "candidateArchitecturesTested": 1, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 1, "p0Findings": 0, "p1Findings": 5, "prepublicationClaimCorrections": 3, "resultState": "target-closed", "resultSummary": "Five-role internal gate: all Minor Revision, no mathematical error. Major items: a negative control had overwritten a shipped evidence record (found independently by two roles, fixed with a manifest re-check barrier); septic result reframed as a candidate series with an explicitly unknown hsop hypothesis; Berele's conjecture stated explicitly; an a priori septic denominator added (pole set equals R's; finite route to an unconditional proof). A blind second matrix engine by a separate agent agrees over every certificate range. Two-reviewer confirmation round: Accept with minor edits, applied. Deterministic batches passed on the confirmation target and the final archive; GitHub/Zenodo byte parity verified by server checksum and download; canonical page read back after guarded deployment.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -100, "forecastRatio": 0.3333, "withinForecastInterval": false, "varianceReason": "Active minutes are an estimate from session timestamps between registration (08:45:03Z) and first canonical readback (09:49:57Z), less about 15 minutes of reviewer and CI waiting during which no other mandatory work remained. Much reviewer time overlapped with authoring (site page, art, audio, ledger), and the external-review repairs, literature pass, exact regeneration and character check preceded registration and are left-censored. modelTurns is a lower bound. No speed or impact comparison.", "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 } ] } }