{ "schemaVersion": "1.6", "slug": "knese-symmetric-multiaffine-agler", "title": "From scalar stability to an operator bound", "shortTitle": "Stable symmetric polynomials are Agler denominators", "url": "https://evidencepress.org/releases/knese-symmetric-multiaffine-agler/", "oneLine": "A positive weighted construction gives the exact Schur–Agler quotient for every stable symmetric multiaffine polynomial, in every finite dimension.", "abstract": "The candidate answers Knese’s class question affirmatively: a nonzero symmetric multiaffine polynomial with no zeros in the open unit polydisk is an Agler denominator using its actual multidegree, without an extra monomial. Inverse-binomial weights cancel unwanted permutation components. Equal-diagonal polarisation establishes every coefficient equation, positive root projectors give the boundary construction, and finite Clark interpolation plus radial approximation complete the argument. Repeated roots, common factors and boundary zeros are included. Exact finite checks supplement the written proof.", "datePublished": "2026-10-10", "dateModified": "2026-10-10", "version": "1.1.0-candidate", "doi": "10.5281/zenodo.23283382", "doiUrl": "https://doi.org/10.5281/zenodo.23283382", "conceptDoi": "10.5281/zenodo.23283381", "pdfUrl": "https://github.com/ipitchford/knese-symmetric-multiaffine-agler/releases/download/v1.1.0-candidate/knese-symmetric-multiaffine-agler-v1.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23283382/files/knese-symmetric-multiaffine-agler-v1.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23283382", "repoUrl": "https://github.com/ipitchford/knese-symmetric-multiaffine-agler", "releaseUrl": "https://github.com/ipitchford/knese-symmetric-multiaffine-agler/releases/tag/v1.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/knese-symmetric-multiaffine-agler/index.md", "bibtexUrl": "https://evidencepress.org/releases/knese-symmetric-multiaffine-agler/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/knese-symmetric-multiaffine-agler.mp3", "imageUrl": "https://evidencepress.org/assets/og/knese-symmetric-multiaffine-agler.png?v=7c97cd11aa", "coverArtUrl": "https://evidencepress.org/assets/art/knese-symmetric-multiaffine-agler.svg?v=07aa91d78c", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/knese-symmetric-multiaffine-agler.mp3", "name": "Audio briefing", "description": "Synthetic speech; communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/knese-symmetric-multiaffine-agler.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed written-proof candidate with internal finite replay. No external peer review, unaffiliated reproduction, formal verification or absolute priority is established." }, "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/23283382", "note": "Public GitHub and Zenodo assets verified by full downloads against the frozen archive; availability is not proof or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Three exact finite checkers and corrupted-weight control pass normally and under optimized Python. The all-dimensional result remains a written proof.", "evidenceUrl": "https://github.com/ipitchford/knese-symmetric-multiaffine-agler/blob/v1.1.0-candidate/VERIFICATION.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "No qualifying evidence establishing this dimension." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No qualifying evidence establishing this 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/knese-symmetric-multiaffine-agler/blob/v1.1.0-candidate/ENVIRONMENT.txt", "note": "Python/SymPy pin and LaTeX source supplied; external reproduction not established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted supplied research; originating model not established", "OpenAI Codex assistant: review actions and publication" ], "humanRole": "Publication authority only; no human research contribution asserted.", "reviewStatus": "Unrefereed written-proof candidate with internal finite replay. No external peer review, unaffiliated reproduction, formal verification or absolute priority is established." }, "attributionBoundary": null, "problem": { "name": "Knese’s symmetric multiaffine Agler-denominator question", "url": "https://arxiv.org/abs/1008.4560" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Supplied candidate and model-assisted review; publication revisions dated 10 October 2026. No historical research clock or reviewer independence reconstructed.", "grandfatheredAtSchemaVersion": null, "keywords": [ "stable polynomials", "Schur-Agler class", "multiaffine polynomials", "Schur-Weyl duality" ], "keyResults": [ "Every finite dimension: actual-multidegree reflected quotient is Schur–Agler, with no extra monomial.", "Positive inverse-binomial weighting and complete coefficient transfer, including the full-union equations.", "Explicit root-matching blocks, with orthonormal spectral convention stated." ], "reviews": [], "evidencePackage": "Written unrestricted proof; exact finite rational and symbolic checks; supplied model-assisted review and itemised response; primary-source comparison and substantive AI index.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Knese, Stable symmetric polynomials and the Schur–Agler class: Question 1.4 and coefficient criterion.", "url": "https://arxiv.org/abs/1008.4560", "doi": "10.48550/arXiv.1008.4560" }, { "citation": "Knese, Rational inner functions on the polydisk—a survey, Problem 3.", "url": "https://arxiv.org/abs/2409.14604v3", "doi": "10.48550/arXiv.2409.14604" }, { "citation": "Grinshpan, Kaliuzhnyi-Verbovetskyi and Woerdeman: the earlier all-dimensional linear case.", "url": "https://doi.org/10.1007/s11785-012-0262-6", "doi": "10.1007/s11785-012-0262-6" }, { "citation": "Grinshpan et al.: eventual Agler denominators with enlarged reflection degree.", "url": "https://doi.org/10.1007/s00209-015-1587-4", "doi": "10.1007/s00209-015-1587-4" }, { "citation": "Leake: representation-theoretic polarisation and scalar stability.", "url": "https://doi.org/10.1007/s00209-021-02825-4", "doi": "10.1007/s00209-021-02825-4" }, { "citation": "Cima, Ross and Wogen: finite Clark kernel bases.", "url": "https://files.ele-math.com/articles/oam-02-21.pdf", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:knese-symmetric-multiaffine-agler", "attemptIds": [ "ep-attempt:knese-symmetric-multiaffine-agler-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "parentLinks": [], "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "Automatic Agler denominator membership in every finite dimension.", "scope": "Nonzero symmetric multiaffine polynomials stable in the open unit polydisk." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Positive coefficient construction yields every polynomial-kernel coefficient; hereditary substitution handles commuting strict contractions without double commutativity.", "remainingRisks": [ "The unrestricted analytic argument is not formally verified.", "Priority is bounded by inspected literature." ] }, "humanJudgmentGates": [ "External specialist assessment remains open; no human research judgement claimed." ], "assuranceTarget": { "dimensions": [ "specialistReview", "formalVerification" ], "nextAction": "Audit the branching identity, complete coefficient transfer and actual-degree boundary.", "claimCeiling": "Unrefereed written-proof candidate with internal finite replay. 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