{
  "schemaVersion": "1.3",
  "slug": "cyclicity-support-fusion-atlas",
  "title": "Inverse-Root Support and Fourier Fusion for Polynomial Exponential Periods",
  "shortTitle": "Inverse-root support and Fourier fusion",
  "url": "https://evidencepress.org/releases/cyclicity-support-fusion-atlas/",
  "oneLine": "A formal-moment theorem removes the stationary-phase condition from fixed-seed cyclic rank, a Fourier formula classifies fusion for every polynomial decomposition and Ritt move, and an exact coefficient atlas covers degrees through twenty.",
  "abstract": "This anonymous, unrefereed theorem candidate gives an unconditional algebraic formula for the cyclic rank generated by [dx] in the twisted de Rham module of any complex polynomial phase: it is both the number of nonzero congruence channels in the normalized inverse-at-infinity series and the dimension of the reduced constant span of the inverse roots. A fixed-sector Fourier-Gamma determinant separately supplies the period realization without a moving-cycle gap. For every actual decomposition P=R composed with Q, a finite Fourier-submatrix rank formula determines every block-sum dimension, including equality, partial fusion, full fusion and pairwise intersections. Equality classes expose a canonical polynomial right factor, and complete decomposition chains are compatible with Ritt moves; at global rank at most three, exactly eleven unordered Ritt rank diamonds occur. A fail-closed exact-rational Groebner sweep of 1,152 support cases, together with direct degrees two through four, yields a finite computer-assisted affine coefficient atlas for degrees at most twenty. The all-degree atlas is not proved: its first uncovered quartic lift has degree thirty-two and the supplied descent contains an identified circular step. Producer-side replay and internal model-assisted review pass within the declared trust boundary. No unaffiliated rerun or reimplementation, proof-assistant formalization, external specialist review, editorial peer review, or novelty or priority determination has occurred.",
  "datePublished": "2026-08-20",
  "dateModified": "2026-08-20",
  "version": "0.5.0-candidate",
  "doi": "10.5281/zenodo.22036030",
  "doiUrl": "https://doi.org/10.5281/zenodo.22036030",
  "conceptDoi": "10.5281/zenodo.22036029",
  "pdfUrl": "https://github.com/ipitchford/cyclicity-support-fusion-atlas/releases/download/v0.5.0-candidate/Inverse-Root-Support-and-Fourier-Fusion.pdf",
  "altPdfUrl": "https://zenodo.org/records/22036030/files/Inverse-Root-Support-and-Fourier-Fusion.pdf?download=1",
  "zenodoUrl": "https://zenodo.org/records/22036030",
  "repoUrl": "https://github.com/ipitchford/cyclicity-support-fusion-atlas",
  "releaseUrl": "https://github.com/ipitchford/cyclicity-support-fusion-atlas/releases/tag/v0.5.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/cyclicity-support-fusion-atlas/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/cyclicity-support-fusion-atlas/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/cyclicity-support-fusion-atlas.mp3?v=47b452afb6",
  "imageUrl": "https://evidencepress.org/assets/og/cyclicity-support-fusion-atlas.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/cyclicity-support-fusion-atlas.svg",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/cyclicity-support-fusion-atlas.mp3",
      "name": "Audio briefing — inverse-root support and Fourier fusion",
      "description": "Plain-English OpenAI API synthetic-voice summary of the unconditional support theorem, complete decomposition fusion calculus, finite degree-twenty atlas and open all-degree descent; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/cyclicity-support-fusion-atlas.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 universal inverse-support, cyclic-rank, reduced-root-span, fixed-sector period and Fourier block-fusion claims are presented as written theorems. The affine coefficient atlas is a finite computer-assisted theorem candidate only for degrees two through twenty. The all-degree atlas, zero-cycle programme and Laurent-polynomial programme remain respectively conjectural or exploratory. Producer-side ordinary and optimized replay passes with exact receipts and semantic failure controls. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, identified external specialist review, editorial peer review, settled novelty or priority assessment, or demonstrated field impact 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://doi.org/10.5281/zenodo.22036030",
      "note": "The immutable GitHub candidate and Zenodo version record expose the same release assets under persistent identities; availability and byte identity do not establish correctness."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "The one-command gate verifies the complete manifest, runs ordinary and optimized structural and semantic controls, regenerates all 1,152 cases, compares frozen receipts and requires injected failures to be detected.",
      "evidenceUrl": "https://github.com/ipitchford/cyclicity-support-fusion-atlas/actions"
    },
    {
      "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 public candidate."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The exact Fraction audit and separate checks were created within the same AI-assisted programme and are fault-diversity controls, not unaffiliated reimplementation."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No formal-moment, rapid-decay, Fourier-fusion, Ritt or coefficient-atlas theorem has been formalized in a proof assistant."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "One adversarial review and one confirmation re-review are internal model-assisted records; no identified external specialist has signed off."
    },
    {
      "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/cyclicity-support-fusion-atlas/blob/v0.5.0-candidate/REPRODUCIBILITY.md",
      "note": "Exact Python dependencies, authoritative environment, deterministic archives and cross-patch portable semantic receipts are supplied, but no container, Nix or Guix lock is included."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Anthropic Claude in a producer-coordinated review path"
    ],
    "humanRole": "Ian Pitchford supplied research direction, mediated review and correction, authorized publication, and acts as repository maintainer and publisher. The scholarly creator is cited as Anonymous.",
    "disclosure": "AI systems contributed mathematical development, exact computation, literature work, drafting, review, revision and release engineering within one producer-coordinated programme. Cross-model agreement, clean-room controls and hosted replay are not independent reproduction, external specialist review or editorial peer review."
  },
  "problem": {
    "name": "Fixed-seed cyclicity for polynomial exponential periods",
    "url": "https://doi.org/10.5281/zenodo.21853682"
  },
  "corrections": [],
  "keywords": [
    "polynomial exponential periods",
    "twisted de Rham module",
    "inverse roots",
    "polynomial decomposition",
    "Ritt moves",
    "discrete Fourier transform",
    "coefficient atlas",
    "computer-assisted mathematics",
    "reproducible research",
    "unrefereed theorem candidate"
  ],
  "keyResults": [
    "For every complex polynomial P of degree at least two, the cyclic rank generated by [dx] equals both the number of nonzero inverse-at-infinity congruence channels and the dimension of the reduced constant inverse-root span.",
    "Fixed rapid-decay contours on sufficiently narrow sectors give an invertible Fourier-Gamma period matrix, supplying an unconditional analytic realization without a moving-cycle assumption.",
    "For every actual polynomial decomposition and every subset of root blocks, a finite Fourier-submatrix formula determines the block-sum dimension, pairwise intersections, partial fusion, full fusion and equality.",
    "Equality fusion exposes a canonical coarser polynomial right factor; complete decomposition chains are Ritt-invariant, and global rank at most three permits exactly eleven unordered Ritt rank diamonds.",
    "The degree-twelve collision (x^4+x)^3 has support {1,7,10}; the quartic-block profile [1,1,1] and cubic-block profile [3] explain both sides of its Ritt move in one three-dimensional span.",
    "Direct low degrees plus 1,152 exact-rational support cases classify every affine family of fixed-seed rank at most three through degree twenty, with zero anomalies and zero missing certificates.",
    "The same coefficient list in all degrees remains Conjecture 13.2; the first uncovered quartic lift has degree thirty-two and requires a non-circular descent or new monodromy argument."
  ],
  "reviews": [],
  "evidencePackage": "A 32-page tagged anonymous candidate paper with canonical Markdown, DOCX and HTML; machine-readable claim, evidence, assurance and provenance records; every supplied historical file inventoried and preserved; exact support, Fourier, fusion, heredity, Ritt and coefficient-atlas code; a frozen per-case inventory for 1,152 degree-five-to-twenty support cases; 45 full-family, 884 power-only and 223 union certificates with zero anomaly or missing certificate; ordinary and optimized fail-closed replay; exact Fraction clean-room controls, 278 support/rank comparisons, 200 heredity pairs and structured fusion cases; deliberate anomaly, missing-certificate, support, Fourier, receipt and manifest corruptions; internal adversarial review and confirmation re-review; citation and rights audits; deterministic manifest and archive tooling; and byte-identity checks across the public release surfaces. This establishes bounded producer-side evidence and availability, not independent theorem verification.",
  "openProblems": [
    "Prove or refute the all-degree coefficient atlas by replacing the circular quartic and cubic lift step with a non-circular twisted-factor descent or a new monodromy argument, beginning with the degree-thirty-two lift.",
    "Obtain focused external specialist review of the formal-moment isomorphism, fixed-sector Fourier-Gamma determinant, equality-factor argument and Ritt fibre-product classification.",
    "Reimplement the complete degree-twenty atlas independently in another open-source computer-algebra stack and compare normalized per-case proof objects.",
    "Formalize the support/root-span identity and finite Fourier block-rank formula in a proof assistant with a declared axiom and computer-algebra trust footprint.",
    "Classify higher-rank Ritt diamonds and identify which abstract Fourier fusion profiles are realizable by polynomial decompositions.",
    "Develop the generic ODE corollary into explicit minimal operators, and pursue the zero-cycle and Laurent-polynomial channel programmes without importing them into the current theorem boundary."
  ],
  "relatedWorks": [
    {
      "citation": "Anonymous. (2026). Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods (Version 0.2.1-candidate). Zenodo.",
      "url": "https://doi.org/10.5281/zenodo.21853682"
    },
    {
      "citation": "Anonymous. (2026). Fixed-seed cyclic rank as reduced inverse-root span (Version 0.1.0-candidate). Zenodo.",
      "url": "https://doi.org/10.5281/zenodo.21864884"
    },
    {
      "citation": "Ritt, J. F. (1922). Prime and composite polynomials. Transactions of the American Mathematical Society, 23(1), 51-66.",
      "url": "https://doi.org/10.1090/S0002-9947-1922-1501189-9"
    },
    {
      "citation": "Zieve, M. E., and Mueller, P. (2008). On Ritt's polynomial decomposition theorems. arXiv:0807.3578.",
      "url": "https://arxiv.org/abs/0807.3578"
    },
    {
      "citation": "Girstmair, K. (1999). Linear relations between roots of polynomials. Acta Arithmetica, 89(1), 53-96.",
      "url": "https://doi.org/10.4064/aa-89-1-53-96"
    },
    {
      "citation": "Sabbah, C. (1999). On a twisted de Rham complex. Tohoku Mathematical Journal, 51(1), 125-140.",
      "url": "https://doi.org/10.2748/tmj/1178224856"
    },
    {
      "citation": "Hien, M. (2009). Periods for flat algebraic connections. Inventiones Mathematicae, 178, 1-22.",
      "url": "https://doi.org/10.1007/s00222-009-0185-7"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:cyclicity-support-fusion-atlas",
    "attemptIds": [
      "ep-attempt:cyclicity-support-fusion-atlas"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "method-demonstration",
      "communication"
    ],
    "lineageId": "cyclicity-programme",
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "adversarial-controls",
      "research-lineage-reuse",
      "productive-failure",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "reusable-method",
      "description": "A support-to-rank and Fourier block-fusion calculus for polynomial exponential periods, paired with a finite exact coefficient-atlas protocol.",
      "scope": "Universal support, root-span and fusion theorems for actual polynomial decompositions; finite affine coefficient classification only through degree twenty; all-degree atlas remains conjectural."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The normalized inverse-at-infinity coefficients are mapped to congruence support, active Kummer channels, reduced inverse-root spans, decomposition-block Fourier ranks, affine coefficient families and per-case Groebner witnesses.",
      "remainingRisks": [
        "No unaffiliated specialist has validated the formal-moment, analytic or Ritt bridges.",
        "The coefficient-atlas source-to-encoding bridge and exact arithmetic trust Python and SymPy.",
        "The quartic and cubic descent needed for the all-degree atlas is open."
      ]
    },
    "humanJudgmentGates": [
      "Assess the formal-moment and active-channel proofs as mathematics rather than inferring truth from successful replay.",
      "Assess whether every claimed fusion profile is tied to an actual polynomial decomposition and whether equality classes really define the stated factor.",
      "Keep the degree-twenty finite atlas distinct from Conjecture 13.2 and the first open degree-thirty-two lift.",
      "Require a new release decision before claiming independent verification, external specialist endorsement, priority or an all-degree theorem.",
      "Confirm anonymous scholarly attribution, component licences and public-release authority."
    ],
    "parentLinks": [
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": "cyclicity-loci-exponential-periods",
        "externalUrl": null,
        "inheritedClaim": "The predecessor introduced exact fixed-seed cyclicity loci and the stationary-phase bridge that originally connected algebraic rank to periods.",
        "inheritedAssuranceCeiling": "The predecessor is an anonymous unrefereed candidate with producer-side replay; it is not independent verification of this successor."
      },
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": "cyclicity-root-span-low-rank",
        "externalUrl": null,
        "inheritedClaim": "The predecessor identified reduced inverse-root span and block fusion as the route to a decomposable low-rank classification.",
        "inheritedAssuranceCeiling": "The predecessor's fixed-seed identity was conditional and its decomposable classification remained open; its internal reviews and replay are not independent confirmation."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain focused unaffiliated specialist review of the formal-moment, fixed-sector and Fourier-fusion proofs, paired with an independently authored coefficient-atlas implementation and a targeted attack on the degree-thirty-two lift.",
      "claimCeiling": "Universal written theorem candidates for inverse support, cyclic rank, root span and Fourier fusion; a producer-side finite computer-assisted atlas through degree twenty; not an all-degree atlas, independent reconstruction, formal verification, external specialist or editorial peer review, absolute priority, or demonstrated research impact."
    },
    "impactClaims": [
      {
        "id": "cyclicity-science-no-impact",
        "aim": "science",
        "outcome": "Reusable support and fusion tools for polynomial decomposition and related period problems",
        "setting": "Research in polynomial exponential periods, inverse-root relations, Ritt theory and computer-assisted classification",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional mathematical-research or publication workflow was registered.",
        "estimand": "No effect on discovery time, error rate, proof quality, review effort, theorem production, reuse, uptake or citation was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}