{ "schemaVersion": "1.6", "slug": "equivariant-inverse-z-unimodality", "title": "Equivariant unimodality of inverse Z-polynomials of matroids", "shortTitle": "Unimodality within every symmetry type", "url": "https://evidencepress.org/releases/equivariant-inverse-z-unimodality/", "oneLine": "A representation-valued truncation identity shows that every irreducible symmetry type follows its own symmetric, unimodal multiplicity sequence.", "abstract": "An ordinary polynomial records totals. An equivariant polynomial also remembers how those totals split into symmetry types. This candidate proves equivariant unimodality of inverse Z-polynomials for finite loopless matroids with finite group actions: the multiplicity of each irreducible representation increases to the middle and then decreases symmetrically. A representation-ring lift of an already-public scalar truncation identity expresses each first difference as a sum of two inverse Kazhdan–Lusztig coefficients. Established equivariant positivity then supplies the inequality. The paper includes an explicit two-element example, equality criteria and a noncanonical commuting sl₂ action. It addresses Gao, Li and Xie's unimodality conjecture, with an explicit ranked-flat-lattice convention for loops; it does not prove strong equivariant log-concavity or construct a canonical geometric action.", "datePublished": "2026-11-06", "dateModified": "2026-11-06", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23995561", "doiUrl": "https://doi.org/10.5281/zenodo.23995561", "conceptDoi": "10.5281/zenodo.23995560", "pdfUrl": "https://github.com/ipitchford/equivariant-inverse-z-unimodality/releases/download/v0.1.0-candidate/equivariant-inverse-z-unimodality-v0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23995561", "repoUrl": "https://github.com/ipitchford/equivariant-inverse-z-unimodality", "releaseUrl": "https://github.com/ipitchford/equivariant-inverse-z-unimodality/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/equivariant-inverse-z-unimodality/index.md", "bibtexUrl": "https://evidencepress.org/releases/equivariant-inverse-z-unimodality/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/equivariant-inverse-z-unimodality.mp3", "imageUrl": "https://evidencepress.org/assets/og/equivariant-inverse-z-unimodality.png?v=ef5f098879", "coverArtUrl": "https://evidencepress.org/assets/art/equivariant-inverse-z-unimodality.svg?v=a27ad9ace8", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/equivariant-inverse-z-unimodality.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/equivariant-inverse-z-unimodality.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Internal simulated review is not unaffiliated specialist review or editorial peer review. No proof-assistant verification or independent reproduction is claimed." }, "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/23995561", "note": "Public GitHub and published Zenodo record verified with exact three-asset inventory and full-download SHA-256 agreement. Availability does not establish mathematical correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Frozen-package clean-extraction replay passed for eleven specified C2 examples and targeted controls in normal and optimized Python. This bounded computation does not verify the universal theorem, which rests on the written proof." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated full-package rerun established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No unaffiliated independent reimplementation established." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted reviews do not establish external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external journal editorial review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Finite exact checker source and recorded examples supplied; environment and frozen-package replay reported separately." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical development, source comparison and revision", "Internal AI-assisted review and exact finite checking", "Codex publication preparation" ], "humanRole": "Publication authority only; no human mathematical contribution inferred.", "reviewStatus": "Internal simulated review, not unaffiliated peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the revised manuscript, bibliography and scoped producer-side records on 6 November 2026. Public GitHub and Zenodo assets were downloaded and hash-matched to the frozen release files.", "grandfatheredAtSchemaVersion": null, "keywords": [ "matroid", "inverse Z-polynomial", "equivariant unimodality", "representation ring", "Kazhdan–Lusztig polynomial", "truncation" ], "keyResults": [ "A representation-ring truncation identity gives yᵢ−yᵢ₋₁=qᵢ(M)+qᵢ₋₁(Tr M) below the middle.", "Established equivariant inverse Kazhdan–Lusztig positivity yields unimodality of each irreducible multiplicity sequence, not merely of dimensions.", "Loop deletion preserves the result under the explicitly stated ranked-flat-lattice convention.", "Symmetric strings give equality criteria and a noncanonical commuting sl₂ action; no canonical geometry or strong log-concavity follows." ], "reviews": [], "evidencePackage": "Written representation-ring proof using a cited established positivity theorem; exact finite C₂-character examples and targeted checker controls. Finite examples do not prove the universal statement or establish historical priority.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Gao, Li and Xie (2026). Equivariant inverse Z-polynomials of matroids. arXiv:2601.17314v1. Conjecture 1.2 is the target; Conjecture 1.4 remains open here.", "url": "https://arxiv.org/html/2601.17314v1", "doi": null }, { "citation": "Proudfoot (2021). Equivariant incidence algebras and equivariant Kazhdan–Lusztig–Stanley theory. Algebraic Combinatorics 4(4), 675–681.", "url": "https://doi.org/10.5802/alco.174", "doi": "10.5802/alco.174" }, { "citation": "Braden, Huh, Matherne, Proudfoot and Wang (2026). Singular Hodge theory for combinatorial geometries. Theorem 8.28 in arXiv version 5 supplies the positivity input.", "url": "https://doi.org/10.1090/jams/1083", "doi": "10.1090/jams/1083" }, { "citation": "Rethlas (2026). Unimodality of inverse Z-polynomials of matroids. Public scalar predecessor, repository revision 895e3d12ba02, 11 September 2026.", "url": "https://github.com/frenzymath/Rethlas_results/blob/895e3d12ba02224c9d8ba8d52882184c77afc8af/AlgComb/MatroidKazhdanLusztig/inverse_z_unimodal.pdf", "doi": null }, { "citation": "Caiolo, Ferroni and Hoster (2026). Dual Chow polynomials of matroids and posets. arXiv:2605.28474v1. Section 7.3 provides an alias comparison, not an identification with the dual Chow polynomial.", "url": "https://arxiv.org/html/2605.28474v1", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:equivariant-inverse-z-unimodality", "attemptIds": [ "ep-attempt:equivariant-inverse-z-unimodality-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "Representation-ring truncation identity exposing effective coefficient differences.", "scope": "Finite matroids with finite group actions under the stated ranked-flat-lattice convention; no strong log-concavity or canonical geometry." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The proof tracks stabilisers, induction and virtual Möbius representations before taking dimensions; finite checks retain original matroid ranks.", "remainingRisks": [ "Bounded prior-art search is not exhaustive priority clearance.", "Finite character checks do not establish the universal theorem or all full-automorphism-group representations." ] }, "humanJudgmentGates": [ "Publication authority is not scientific endorsement.", "External specialist validation remains open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "priorityAssessment" ], "nextAction": "Obtain unaffiliated assessment of the equivariant lift and independent checking of the precise conventions and finite examples.", "claimCeiling": "Written proof and scoped producer checks; no external validation or absolute originality guarantee." }, "impactClaims": [ { "id": "equivariant-inverse-z-unimodality-no-impact", "aim": "science", "outcome": "An inspectable equivariant lift of an inherited scalar mechanism.", "setting": "Algebraic combinatorics", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched research-workflow comparison.", "estimand": "No acceleration or downstream impact effect estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. 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