{ "schemaVersion": "1.6", "slug": "degree-independent-rhombus-criterion", "title": "A Degree-Independent Rhombus Criterion in Three Variables, and the Higher-Dimensional Boundary", "shortTitle": "Degree-independent ternary rhombus criterion", "url": "https://evidencepress.org/releases/degree-independent-rhombus-criterion/", "oneLine": "A factor-3 rhombus condition is proposed as a degree-independent ternary stability criterion, while three higher-dimensional readings meet explicit obstructions and the normalized cases in four through six variables remain open.", "abstract": "This anonymous, unrefereed candidate addresses AIM-LINEAR_ALGEBRA-0011. For positive homogeneous ternary polynomials, it argues that Branden-oriented elementary rhombus quotients at least 3 imply real stability in every degree. The proof replaces a degree-dependent multiplicity estimate by two convergent theta-type sums. The higher-dimensional request is separated into three formulations. Coordinate-face inequalities fail for every prescribed finite factor from four variables onward; one precisely quantified strict multiplicative exchange margin is inconsistent on positive full-support quadratics; and normalized M-concavity fails in unrestricted dimension from seven variables onward through finite positive families converging to the prior-art Fano obstruction. The normalized cases with four, five and six variables are not classified. Deterministic producer replay, semantic mutations, internal role-separated review and a fully actioned supplied review reached PASS_WITH_NOTES. No historical-priority, unaffiliated-reproduction, formal-verification, specialist-review or peer-review claim is made.", "datePublished": "2026-08-31", "dateModified": "2026-08-31", "version": "0.2.0-candidate", "doi": "10.5281/zenodo.22211016", "doiUrl": "https://doi.org/10.5281/zenodo.22211016", "conceptDoi": "10.5281/zenodo.22211015", "pdfUrl": "https://github.com/ipitchford/degree-independent-rhombus-criterion/releases/download/v0.2.0-candidate/degree-independent-rhombus-criterion-v0.2.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22211016/files/degree-independent-rhombus-criterion-v0.2.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22211016", "repoUrl": "https://github.com/ipitchford/degree-independent-rhombus-criterion", "releaseUrl": "https://github.com/ipitchford/degree-independent-rhombus-criterion/releases/tag/v0.2.0-candidate", "markdownUrl": "https://evidencepress.org/releases/degree-independent-rhombus-criterion/index.md", "bibtexUrl": "https://evidencepress.org/releases/degree-independent-rhombus-criterion/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/degree-independent-rhombus-criterion.mp3?v=3943981857", "imageUrl": "https://evidencepress.org/assets/og/degree-independent-rhombus-criterion.png?v=add0e0f6e8", "coverArtUrl": "https://evidencepress.org/assets/art/degree-independent-rhombus-criterion.svg?v=b63234b9e4", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/degree-independent-rhombus-criterion.mp3", "name": "Audio briefing — a degree-independent ternary rhombus criterion", "description": "Plain-English synthetic-voice account of the factor-3 ternary theorem candidate, the three higher-dimensional boundaries, the unresolved normalized dimensions and the assurance limits; a communication aid, not additional evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/degree-independent-rhombus-criterion.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, AI-assisted and unrefereed partial mathematics candidate at internal PASS_WITH_NOTES. The degree-independent ternary theorem and higher-dimensional families are written arguments; the finite scripts audit identities, indexing and declared bounded searches but do not prove the universal statements. The Fano non-stability obstruction is prior work. The normalized dimensions four through six remain open, the constant 3 is not claimed optimal, and the bounded novelty search does not establish priority." }, "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/22211016", "note": "The tagged GitHub prerelease and published Zenodo record expose the canonical PDF, complete archive, checksum sidecar, claims, source bridge, reviews, exact verifiers and replay files; independently downloaded public assets match local bytes." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Digest-pinned read-only containers reproduce exact fresh Python and PDF bytes, architecture-specific C++ receipts, source-format concordance and inventory checks while rejecting semantic mutations and altered artifacts.", "evidenceUrl": "https://github.com/ipitchford/degree-independent-rhombus-criterion/actions/runs/33411540631" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No authenticated unaffiliated party has rerun the complete package and preserved a separate environment and result receipt." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The Python and C++ routes divide verification duties but were produced within one coordinated workflow and do not constitute an unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof assistant formalizes the hive normalization, parity pairing, theta bounds, stability bridge, counterexample families or Fano limit argument." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The supplied review, five-role review and confirmation were not authenticated as unaffiliated specialist review in stable-polynomial or discrete-convexity theory." }, { "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 comparable external venue has conducted editorial peer review to a decision." }, { "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/degree-independent-rhombus-criterion/blob/v0.2.0-candidate/ENVIRONMENT.txt", "note": "Digest-pinned TeX/Python and architecture-specific GCC images, commands, receipts and public Linux CI are supplied; an authenticated unaffiliated clean-environment reconstruction is not." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Python 3 standard library", "C++17 standard library", "Node.js standard library" ], "humanRole": "Research direction, candidate selection, review provision, claim-boundary decisions, publication authorization and institutional responsibility by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.", "disclosure": "AI systems contributed problem triage, proof search, exact computation, drafting, source and novelty auditing, action of the supplied review, internal role-separated review, replay and release engineering within one producer-coordinated workflow. Cross-language and cross-architecture agreement plus internal review are not unaffiliated reconstruction, authenticated specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "AIM Problem 1.1: degree-independent rhombus criteria and higher-dimensional extensions", "url": "https://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "reader-first-reference", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively against the DOI-bearing tagged v0.2.0-candidate, source bridge, supplied-review response, five-role internal editorial record, bounded confirmation, successful public digest-pinned replay and exact GitHub-Zenodo-local three-asset readback on 31 August 2026. Canonical Evidence Press deployment and terminal publication metrics are recorded separately and do not promote the mathematical assurance ceiling.", "grandfatheredAtSchemaVersion": null, "keywords": [ "real-stable polynomial", "half-plane property", "hive", "rhombus log-concavity", "M-concavity", "Fano matroid", "unrefereed candidate" ], "keyResults": [ "For every degree d at least 1, factor-3 Branden-oriented elementary rhombus inequalities are argued to imply real stability for positive ternary homogeneous polynomials.", "For every m at least 4 and every prescribed finite factor, an explicit positive quadratic family satisfies all coordinate-face rhombus bounds but is not real-stable.", "The precisely defined uniform one-exchange multiplicative margin q greater than 1 is inconsistent for positive full-support quadratics once four distinct indices occur.", "The normalized M-concavity formulation fails in unrestricted dimension from m=7 onward via finite positive families converging to the prior-art Fano obstruction.", "The normalized cases m=4,5,6 and the optimal ternary constant remain open in this release." ], "reviews": [], "evidencePackage": "A 12-page canonical candidate manuscript and accessible derivatives; a five-claim machine registry; a source bridge identifying every external theorem used; written proofs of the universal ternary statement and symbolic obstruction families; exact Python identity and mutation checks in normal and optimized modes; exact C++ enumeration of 880 M-convex supports plus a bounded long-double scan; architecture-specific receipts; deterministic PDF reconstruction; a complete 59-file manifest; a fully actioned supplied review; a five-role producer-coordinated internal review; and a bounded confirmation. Digest-pinned local replay and public GitHub Actions reproduce the declared receipts and reject altered artifacts. GitHub and Zenodo release bytes match exactly. These establish public availability, integrity, producer replay and the declared finite computations—not an independent proof of the universal theorem, historical priority, unaffiliated reproduction, formal verification, external specialist review or peer review.", "openProblems": [ "Classify the normalized M-concavity formulation for m=4,5,6, beginning with the executable support and coefficient targets in OPEN_PROBLEMS.md.", "Determine the optimal absolute ternary rhombus factor, or prove that 3 is sharp.", "Find a non-vacuous higher-dimensional quantitative condition that is stronger than coordinate-face control but does not impose the inconsistent strict exchange axiom.", "Independently reconstruct the parity-pairing proof and all higher-dimensional obstruction families in a materially separate stack.", "Formalize the ternary stability bridge, theta estimates and symbolic obstruction arguments in a proof assistant.", "Obtain an unaffiliated specialist assessment of correctness, closest prior art and historical priority." ], "mathObjects": [], "relatedWorks": [ { "citation": "American Institute of Mathematics. Theory and applications of total positivity, Problem 1.1.", "url": "https://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/", "doi": null }, { "citation": "Branden, P. (2010). Discrete concavity and the half-plane property. SIAM Journal on Discrete Mathematics 24(3), 921-933.", "url": "https://doi.org/10.1137/090758738", "doi": "10.1137/090758738" }, { "citation": "Branden, P. (2007). Polynomials with the half-plane property and matroid theory. Advances in Mathematics 216(1), 302-320.", "url": "https://doi.org/10.1016/j.aim.2007.05.011", "doi": "10.1016/j.aim.2007.05.011" }, { "citation": "Hutchinson, J. I. (1923). On a remarkable class of entire functions. Transactions of the American Mathematical Society 25(3), 325-332.", "url": "https://doi.org/10.1090/S0002-9947-1923-1501248-1", "doi": "10.1090/S0002-9947-1923-1501248-1" }, { "citation": "Borcea, J., & Branden, P. (2009). The Lee-Yang and Polya-Schur Programs I. Inventiones Mathematicae 177(3), 541-569.", "url": "https://doi.org/10.1007/s00222-009-0189-3", "doi": "10.1007/s00222-009-0189-3" }, { "citation": "Branden, P., & Huh, J. (2020). Lorentzian polynomials. Annals of Mathematics 192(3), 821-891.", "url": "https://doi.org/10.4007/annals.2020.192.3.4", "doi": "10.4007/annals.2020.192.3.4" }, { "citation": "Kummer, M., & Sawall, D. (2025). Three results related to the half-plane property of matroids. Algebraic Combinatorics 8(1), 1-15.", "url": "https://doi.org/10.5802/alco.399", "doi": "10.5802/alco.399" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:degree-independent-rhombus-criterion", "attemptIds": [ "ep-attempt:degree-independent-rhombus-criterion-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "A degree-independent ternary sufficiency theorem candidate paired with explicit higher-dimensional counterexample, impossibility and open-boundary objects.", "scope": "The factor-3 positive ternary criterion in every degree, failure of finite coordinate-face control for m at least 4, inconsistency of one strict full-support quadratic exchange rule, and a normalized Fano-derived obstruction from m=7; not the normalized m=4,5,6 cases, the optimal ternary constant, every higher-dimensional quantitative criterion, novelty, priority or external validation." }, "bottleneckTargeted": [ "assurance", "publication", "translation" ], "semanticBridge": { "state": "explicit", "description": "The package fixes Branden's rhombus orientation, normalizes by the quadratic baseline, records the parity-pairing bijection and theta bounds, states the stability bridge, and defines each higher-dimensional formulation before proving its distinct boundary. CLAIMS.json and OPEN_PROBLEMS.md keep written universal claims, finite replay and unresolved dimensions separate.", "remainingRisks": [ "The universal ternary proof and Fano limit bridge have only producer-coordinated review, not unaffiliated reconstruction.", "The coordinate-face, strict-exchange and normalized formulations do not exhaust all meaningful higher-dimensional quantitative criteria.", "The normalized m=4,5,6 cases remain open and a bounded m=4 cubic scan is non-probative.", "The targeted novelty search may miss differently phrased, unindexed or informal prior work." ] }, "humanJudgmentGates": [ "Check the parity pairing, theta-series domination and imported one-variable stability bridge as mathematics rather than inferring the theorem from finite replay.", "Keep the three higher-dimensional formulations separate and do not describe their obstructions as one complete impossibility theorem.", "Keep the prior-art Fano obstruction separate from the finite-positive limiting construction used here.", "Keep the unresolved normalized dimensions four through six prominent in summaries and media.", "Treat internal review, deterministic replay and public availability as distinct from unaffiliated validation, peer review, novelty and priority." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/", "inheritedClaim": "AIM Problem 1.1 asks whether the degree dependence in Branden's ternary sufficient rhombus condition can be removed and asks what happens for more than three variables.", "inheritedAssuranceCeiling": "The archived problem fixes the target and attribution but does not establish this candidate's theorem, definitions, obstruction families, correctness, novelty or priority." }, { "relation": "extends-result", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.1137/090758738", "inheritedClaim": "Branden's Theorem 14 supplies a degree-dependent sufficient ternary factor and the hive-to-univariate stability architecture used by the candidate.", "inheritedAssuranceCeiling": "The published theorem supplies the predecessor and bridge, not the factor-3 improvement or any higher-dimensional conclusion in this release." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain an authenticated unaffiliated reconstruction of the parity-pairing proof and symbolic families, then have a stable-polynomial specialist assess the source bridge and normalized m=4,5,6 programme.", "claimCeiling": "An anonymous, AI-assisted, unrefereed factor-3 ternary theorem candidate with explicit higher-dimensional coordinate-face, strict-exchange and Fano-derived boundaries, written proofs, public immutable assets, producer replay and internal PASS_WITH_NOTES; not a resolution of normalized dimensions four through six, a complete all-dimensional criterion, historical-priority finding, unaffiliated validation, formal verification, authenticated external specialist review or editorial peer review." }, "impactClaims": [ { "id": "degree-independent-rhombus-criterion-science-no-impact", "aim": "science", "outcome": "Faster or more reliable resolution and assurance of open questions in stable-polynomial and discrete-convexity theory", "setting": "AI-assisted proof construction, symbolic obstruction search, adversarial replay, internal review and guarded candidate publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional research, theorem-review or publication workflow was registered.", "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was 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:degree-independent-rhombus-criterion-publication", "status": "published", "resultClass": "partial", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "From prospective registration before Evidence Press page authoring through terminal canonical site readback or a preserved stop. Earlier theorem construction, source recovery, supplied-review repair, internal editorial confirmation, package freezing, GitHub identity and Zenodo publication are excluded and not reconstructed.", "forecast": { "frozenAt": "2026-08-31T16:10:16Z", "procedureClass": "reviewed-partial-mathematics-candidate-canonical-site-publication", "targetOutcome": "Publish a reader-first Evidence Press release with provenance-bound media, append-only operating records, two composite seals, hosted CI, guarded zero-cost deployment and exact canonical readback without broadening the candidate's dimensional or assurance boundaries.", "expectedActiveMinutes": 95, "plausibleLowMinutes": 60, "plausibleHighMinutes": 145, "expectedUnattendedWaitMinutes": 35, "referenceClass": { "label": "Recent reviewed exact-mathematics canonical-site publication closures", "sampleSize": 6, "basis": "Recent Evidence Press candidates used the same reader-first authoring, provenance-bound media, append-only operating records, composite sealing, hosted CI, guarded deployment and canonical-readback architecture." }, "fermiComponents": [ { "component": "Reader-first page and machine record", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 35, "basis": "One 12-page candidate with four distinct dimensional formulations and a load-bearing open m=4,5,6 boundary." }, { "component": "Deterministic art, byte-bound audio, Open Graph image and thumbnail", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 35, "basis": "One release using established generators but requiring a claim-specific triangular-rhombus visual that does not imply all-dimensional closure." }, { "component": "Operating records and composite A/B and C/D seals", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 30, "basis": "One new prospective work link and method-cluster assignment with append-only ledger reconciliation." }, { "component": "Hosted CI, guarded deployment and exact readback", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 45, "basis": "One canonical URL requiring merged-main verification, zero-cost deployment and bounded custom-domain convergence checks." } ], "probabilityPositiveSignal": 0.97, "probabilityTargetClosure": 0.86, "probabilityHorizonMinutes": 145, "assumptions": [ "The published Zenodo bytes, OpenAI fable TTS, hosted CI and zero-cost Cloudflare credentials remain available without weakening a gate or exposing a secret.", "Origin main does not acquire an irreconcilable overlapping release or operating-ledger change.", "The site generators accept the new partial-result candidate without a release-contract migration.", "Canonical custom-domain readback converges within the guarded deployment's bounded retry policy." ], "reforecastTriggers": [ "TTS, CI, Cloudflare or canonical readback remains unavailable after bounded recovery.", "A site, citation, licence, accessibility or preservation gate finds a claim-boundary conflict.", "Origin main advances with overlapping method-registry or work-ledger edits requiring reconciliation.", "Media inspection reveals a persistent representation or provenance defect." ], "stopRule": "Stop on a fatal claim, source, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 95 active minutes prioritize load-bearing release gates; at 145 active minutes preserve the exact blocker rather than weaken a gate." }, "outcome": { "measuredAt": "2026-08-31T16:47:21Z", "calendarElapsedMinutes": 37, "activeAgentMinutes": 30, "activeHumanMinutes": null, "computeMinutes": 0, "unattendedWaitMinutes": 7, "blockedMinutes": 0, "reworkMinutes": 0, "agentRuns": 1, "maxParallelAgents": 1, "modelTurns": 1, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 0, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 0, "p0Findings": 0, "p1Findings": 0, "prepublicationClaimCorrections": 0, "resultState": "target-closed", "resultSummary": "The review-repaired partial candidate reached a reader-first canonical Evidence Press release with byte-bound fable audio and transcript, deterministic triangular-rhombus art, inspected Open Graph and thumbnail media, append-only operating records, protocol revision 198, green Node 18, 20 and 22 plus accessibility CI, guarded zero-cost deployment, exact 46-release and 31-institutional-artifact canonical readback, full publication preservation and accepted IndexNow submission. The immutable machine resource fields retain the first live release-candidate snapshot; an appended correction records later observed supplemental totals of three compute minutes and five rework minutes. The first site CI run exposed a stale corpus-wide audio-provenance index, which was refreshed and resealed without changing the research package, claims or media; the first deploy attempt stopped before upload on eight byte-identical untracked duplicate files, which were verified and removed without changing tracked source. The result remains an anonymous AI-assisted unrefereed candidate proving the degree-independent ternary factor-3 criterion and three scoped higher-dimensional obstructions, while normalized dimensions four through six, a complete all-dimensional theorem, exhaustive novelty, historical priority, independent reconstruction, formal verification, external specialist review, editorial peer review and impact remain open or unestablished.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -65, "forecastRatio": 0.32, "withinForecastInterval": false, "varianceReason": "Established page, media, composite-seal, hosted-CI and guarded-deployment tooling closed the canonical-site route below the 60-minute lower active-time bound; immutable GitHub and Zenodo identity were already complete at registration, and both repairs were bounded provenance or filesystem-state corrections that did not alter the candidate's mathematics or media.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No instrument captured human direction or review time at the prospective publication-only boundary." }, { "field": "deduplicatedModelTokens", "reason": "No active fork-aware goal counter exposed exact task-local model-token usage at or after the 16:10:16Z publication-ledger boundary, so tokens are not reconstructed from conversation context." }, { "field": "uncachedInputTokens", "reason": "The runtime does not expose an uncached-input token counter." } ] } }, "missingnessReason": null } ] } }