{ "schemaVersion": "1.5", "slug": "amplitude-modules-fused-support-bautin-bound", "title": "Amplitude modules, fused support, and a coefficient-Bautin bound for polynomial phases", "shortTitle": "Amplitude modules and a Bautin cutoff", "url": "https://evidencepress.org/releases/amplitude-modules-fused-support-bautin-bound/", "oneLine": "Two all-amplitude family classifications and a general ordinary coefficient-Bautin bound turn inverse support into explicit modules, attainable ranks and a finite d−1 cutoff.", "abstract": "This anonymous, unrefereed theorem candidate develops three results for polynomial phases. For every nondegenerate Dickson phase D_d(x,a), a≠0, it gives a free amplitude-module decomposition and the exact induced-rank formula q=2N+ε; odd degree realizes exactly the even ranks, while even degree realizes every rank. For the exceptional Ritt collision P=(x^4+x)^3, it exhibits three fused amplitude modules whose nonzero elements each activate exactly three channels, together with two single composition channels; the resulting formula realizes every rank from zero through eleven. For every degree-d polynomial phase, every zero-cycle and polynomial perturbations of arbitrary fixed degree, it proves that the ordinary coefficient Bautin ideal is generated by Melnikov orders one through d−1, giving b_coeff(m)≤m−1. Identification with the terse b(m) in the motivating source remains conditional on convention identity. A support-to-order construction is retained as infrastructure, not a fourth headline claim. The exact package passes deterministic producer replay, mutation controls, ordinary/optimized JUnit parity, internal model-mediated editorial repair and public asset readback. These controls do not establish independent specialist validation, formal verification, journal peer review, settled priority or that three historically posed open problems have been solved.", "datePublished": "2026-08-28", "dateModified": "2026-08-28", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22143919", "doiUrl": "https://doi.org/10.5281/zenodo.22143919", "conceptDoi": "10.5281/zenodo.22143918", "pdfUrl": "https://github.com/ipitchford/amplitude-modules-fused-support-bautin-bound/releases/download/v0.1.0-candidate/CANDIDATE_PREPRINT.pdf", "altPdfUrl": "https://zenodo.org/records/22143919/files/CANDIDATE_PREPRINT.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22143919", "repoUrl": "https://github.com/ipitchford/amplitude-modules-fused-support-bautin-bound", "releaseUrl": "https://github.com/ipitchford/amplitude-modules-fused-support-bautin-bound/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/amplitude-modules-fused-support-bautin-bound/index.md", "bibtexUrl": "https://evidencepress.org/releases/amplitude-modules-fused-support-bautin-bound/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/amplitude-modules-fused-support-bautin-bound.mp3?v=037cc71794", "imageUrl": "https://evidencepress.org/assets/og/amplitude-modules-fused-support-bautin-bound.png?v=766726bbc6", "coverArtUrl": "https://evidencepress.org/assets/art/amplitude-modules-fused-support-bautin-bound.svg?v=3fa6c290a6", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/amplitude-modules-fused-support-bautin-bound.mp3", "name": "Audio briefing — amplitude modules, fused support and the Bautin cutoff", "description": "Plain-English OpenAI API synthetic-voice summary of the three candidate results and their assurance boundary; a communication aid, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/amplitude-modules-fused-support-bautin-bound.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed theorem candidate. The written support, module and Bautin arguments are presented for specialist checking. Exact rational replay, mutation controls and public CI pass, but the programs test encoded consequences and package behaviour rather than certifying the universal proofs. The five-role substantive review and bounded confirmation were model-mediated inside the producer programme. No unaffiliated rerun, independent implementation, proof-assistant formalization, external specialist review, journal peer review or absolute priority determination has occurred. The release does not establish that three historically posed open problems are solved, does not give a real planar limit-cycle bound, and identifies the source's b(m) with b_coeff(m) only conditionally." }, "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/22143919", "note": "The tagged GitHub prerelease and published Zenodo record expose the exact candidate ZIP, paper, accessible HTML, source and release evidence; public asset checksums match their recorded identities." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "GitHub ran exact-package semantics under ordinary and optimized Python and verified the retained 9-test JUnit parity. The frozen producer receipt records 13 command executions and 14 byte-identical comparisons.", "evidenceUrl": "https://github.com/ipitchford/amplitude-modules-fused-support-bautin-bound/actions/runs/33169814484" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated party has reported rerunning the immutable candidate package in a separately provisioned environment." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The distinct checks and model-mediated reviews remain producer-coordinated; no separately authored implementation has reconstructed the mathematical predicates." }, { "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 support-rank theorem, Dickson decomposition, exceptional trace cancellation or coefficient-Bautin argument." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The five-role reports and bounded confirmation are internal model-mediated editorial controls, not 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 journal or comparable 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/amplitude-modules-fused-support-bautin-bound/blob/v0.1.0-candidate/scientific-package/ENVIRONMENT.txt", "note": "Commands, tested-host versions, manifests and receipts are public, but the environment is explicitly non-hermetic and no independent container, Nix or Guix recreation is reported." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Python 3.14.6", "SymPy 1.14.0" ], "humanRole": "Research direction, mediation, claim-boundary decisions and publication authorization by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.", "disclosure": "AI systems contributed proof exploration, code, internal review, editing, replay and release engineering in one producer-coordinated workflow. Model agreement, exact replay and matching public hashes are not independent reproduction, external specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "Polynomial-amplitude support classification and the zero-cycle coefficient Bautin index", "url": "https://doi.org/10.1016/j.bulsci.2025.103634" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively for reader-first schema 2.0 against immutable GitHub tag v0.1.0-candidate, Zenodo record 22143919, the final paired interpreter controls and the frozen internal review record.", "grandfatheredAtSchemaVersion": null, "keywords": [ "polynomial phases", "polynomial amplitudes", "cyclic rank", "Dickson polynomials", "Ritt decomposition", "monodromy", "zero-cycles", "Bautin ideals", "Melnikov functions", "computer-assisted mathematics", "unrefereed theorem candidate" ], "keyResults": [ "For every nondegenerate Dickson phase D_d(x,a), a≠0, every polynomial amplitude has a free C[D_d]-module decomposition and induced rank q=2N+ε; odd d realizes exactly even ranks and even d realizes every rank.", "For P=(x^4+x)^3, two single composition channels and three explicit fused modules partition the eleven non-trace residues; every nonzero fused component activates exactly three channels and every rank from zero through eleven is attainable.", "For every degree-d polynomial phase, every zero-cycle and arbitrary fixed perturbation degree, the ordinary coefficient Bautin ideal is generated by Melnikov orders one through d−1, so the manuscript-defined b_coeff(m) satisfies b_coeff(m)≤m−1.", "The transfer from b_coeff(m) to the motivating source's terse b(m) is conditional on the source using the identical coefficient-ideal convention.", "Inverse-series support also gives the complete-system minimal differential-operator order, while the candidate retains only corrected block bounds for a specified ray-cycle." ], "reviews": [], "evidencePackage": "A 23-page anonymous manuscript in PDF, generated TeX, canonical Markdown and accessible standalone MathML HTML; an exact 72-entry scientific manifest and 74-object package root; 13 replay command executions with 14 byte-identical artifact comparisons; exact Dickson, exceptional-module, support-rank, coefficient-Bautin and operator receipts; five semantic package-mutation controls; ordinary and optimized 9-test JUnit reports with identical testcase inventories; a bounded primary-source novelty audit; a five-role internal model-mediated substantive review, response matrix and one bounded confirmation; a deterministic closeout receipt; successful GitHub CI at immutable tag c860333d3a1ab513834a613854bd9da28d2b15b4; and nine byte-matched Zenodo assets. These establish availability, package identity, producer-side replay and internal editorial closure, not theorem truth or independent mathematical validation.", "openProblems": [ "Obtain unaffiliated specialist reconstructions of the support-rank theorem, the Dickson module decomposition, the exceptional fibre-trace cancellation and the all-order Bautin recurrence.", "Reimplement the support, fused-module and coefficient-Bautin checks in a separately authored computer-algebra stack without importing producer reducers or proof objects.", "Formalize the generic-specialization, standard-ray Fourier--Gamma, monodromy-projector and Cayley--Hamilton bridges in a proof assistant with explicit analytic hypotheses.", "Determine from an authoritative specification whether the motivating source's b(m) uses exactly the manuscript's ordinary coefficient-ideal convention.", "Derive sharper cycle-specific operator orders from the block connection and extend the standard-ray analysis to controlled contour systems without reviving the refuted raw channel-count formula." ], "mathObjects": [], "relatedWorks": [ { "citation": "Anonymous. (2026). Inverse-Root Support, Polynomial Amplitudes, and Laurent Moment Channels (Version 0.7.0-candidate). Evidence Press / Zenodo.", "url": "https://doi.org/10.5281/zenodo.22049564", "doi": "10.5281/zenodo.22049564" }, { "citation": "Bravo, J. L., Mardešić, P., Novikov, D., & Pontigo-Herrera, J. (2025). Infinitesimal and tangential 16-th Hilbert problem on zero-cycles. Bulletin des Sciences Mathématiques, 202, 103634.", "url": "https://doi.org/10.1016/j.bulsci.2025.103634", "doi": "10.1016/j.bulsci.2025.103634" }, { "citation": "Pakovich, F., & Muzychuk, M. (2009). Solution of the polynomial moment problem. Proceedings of the London Mathematical Society, 99(3), 633–657.", "url": "https://doi.org/10.1112/plms/pdp010", "doi": "10.1112/plms/pdp010" }, { "citation": "Álvarez Sánchez, A., Bravo Trinidad, J. L., & Mardešić, P. (2013). Vanishing Abelian integrals on zero-dimensional cycles. Proceedings of the London Mathematical Society, 107(6), 1302–1330.", "url": "https://doi.org/10.1112/plms/pdt012", "doi": "10.1112/plms/pdt012" }, { "citation": "Behajaina, A., Konig, J., & Neftin, D. (2026). Monodromy groups of polynomials of composition length 2. arXiv:2603.27609v2.", "url": "https://arxiv.org/abs/2603.27609", "doi": "10.48550/arXiv.2603.27609" }, { "citation": "Batenkov, D., & Yomdin, Y. (2014). Taylor Domination, Difference Equations, and Bautin Ideals. arXiv:1411.7629v1.", "url": "https://arxiv.org/abs/1411.7629", "doi": "10.48550/arXiv.1411.7629" }, { "citation": "Briskin, M., & Yomdin, Y. (1997). Algebraic Families of Analytic Functions, I. Journal of Differential Equations, 136, 248–267.", "url": "https://doi.org/10.1006/jdeq.1996.3250", "doi": "10.1006/jdeq.1996.3250" }, { "citation": "Chen, S., Kauers, M., Koutschan, C., Li, X., Wang, R.-H., & Wang, Y. (2025). Non-minimality of minimal telescopers explained by residues. arXiv:2502.03757v1.", "url": "https://arxiv.org/abs/2502.03757", "doi": "10.48550/arXiv.2502.03757" } ], "operatingModel": { "version": "1.0", "workId": "ep-work:amplitude-modules-fused-support-bautin-bound", "attemptIds": [ "ep-attempt:amplitude-modules-fused-support-bautin-bound" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": "cyclicity-programme", "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "research-lineage-reuse", "productive-failure", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "A finite-fibre module calculus connecting inverse support, monodromy blocks, polynomial amplitudes, period order and ordinary coefficient Bautin ideals.", "scope": "All polynomial amplitudes for nondegenerate Dickson phases and the named exceptional Ritt phase, plus the ordinary coefficient Bautin ideal for arbitrary degree-d polynomial phases and zero-cycles; not arbitrary contour systems, alternative Bautin conventions or real planar limit-cycle bounds." }, "bottleneckTargeted": [ "discovery", "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Formal moments and Fourier inversion identify active inverse residues with reduced generic-fibre spans; monodromy projectors organize those channels into free amplitude modules; Lagrange--Bürmann, differentiation injectivity and Cayley--Hamilton turn all Melnikov orders into a finite ordinary coefficient-Bautin generating window.", "remainingRisks": [ "No unaffiliated specialist has reconstructed the generic-specialization, standard-ray, monodromy-projector or Bautin-ideal bridges.", "The support theorem follows the Atlas proof architecture and therefore is not independent confirmation of its parent.", "Exact replay and model-mediated reviews can share a translation or specification defect.", "The targeted antecedent search and conditional b(m) source audit cannot establish absolute novelty, priority or convention identity." ] }, "humanJudgmentGates": [ "Check the written generic-specialization and Fourier--Gamma arguments as mathematics rather than inferring them from successful replay.", "Verify the Dickson invariant-field and exceptional fibre-trace module arguments, including the hidden cancellation that excludes residues four and eight.", "Audit the all-order Lagrange--Bürmann, differentiation-injectivity and coefficient-content steps before relying on the d−1 Bautin cutoff.", "Keep b_coeff(m) separate from the motivating source's b(m) until their ambient ideal conventions are authoritatively identified.", "Require external specialist assessment before any solved-open-problem, first, priority, peer-reviewed or field-acceptance language.", "Confirm Anonymous scholarly attribution, component licences, immutable identifiers and release authority before publication." ], "parentLinks": [ { "relation": "extends-result", "workId": "ep-work:cyclicity-support-fusion-atlas", "legacyReleaseSlug": null, "externalUrl": null, "inheritedClaim": "The parent supplies the inverse-support and polynomial-amplitude bridge whose family-level consequences are classified here.", "inheritedAssuranceCeiling": "The child follows the parent proof architecture and shares producer lineage, so it does not independently validate the Atlas or its support theorem." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain unaffiliated proof reconstruction and a separately authored implementation of the support, fused-module and Bautin mechanisms, paired with an authoritative source-convention audit for b(m).", "claimCeiling": "An anonymous public unrefereed theorem candidate containing two Atlas-family amplitude classifications and one ordinary coefficient-Bautin theorem candidate with producer replay and internal editorial closure; not three historically posed open problems solved, independent reproduction, formal verification, external specialist or editorial peer review, absolute priority or demonstrated research impact." }, "impactClaims": [ { "id": "amplitude-modules-science-no-impact", "aim": "science", "outcome": "Reusable amplitude-module, support-fusion and coefficient-Bautin tools for polynomial phases", "setting": "Research in polynomial exponential periods, zero-cycle perturbations, monodromy, symbolic integration and computer-assisted theorem development", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional mathematical-research or publication workflow was registered.", "estimand": "No effect on discovery time, proof quality, error rate, review effort, independent-assurance time, reuse, uptake or citation was estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] } }