{
  "schemaVersion": "1.2",
  "slug": "cyclicity-loci-exponential-periods",
  "title": "Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods",
  "shortTitle": "Fixed-seed cyclicity loci for polynomial exponential periods",
  "url": "https://evidencepress.org/releases/cyclicity-loci-exponential-periods/",
  "oneLine": "Exact Krylov determinants locate where a chosen exponential-period amplitude loses differential rank, with complete quartic strata and a finite exact odd-quintic certificate.",
  "abstract": "This anonymous, unrefereed candidate studies a fixed-seed question for univariate polynomial exponential periods: where does one specified amplitude cease to generate the ambient differential module under repeated Gauss-Manin differentiation? Over an explicitly localized parameter ring, exact polynomial reduction produces finite Krylov matrices whose minors define scheme-theoretic fibre-rank loci. The candidate gives complete depressed-quartic ranks and distinguishes a persistent even component from a generally pointwise scale defect. For an odd quintic it supplies the full four-by-four matrix, four exact reduction witnesses, its determinant, all sixteen size-three minors and a complete rank-one/two/three/four stratification. The known Chebyshev-Bessel family is used only as a prior-art benchmark. Producer-side replay and targeted failure controls pass, but no independent reproduction, proof-assistant formalization, specialist review, analytic contour validation, or novelty or priority determination has occurred.",
  "datePublished": "2026-08-08",
  "dateModified": "2026-08-08",
  "version": "0.2.1-candidate",
  "doi": "10.5281/zenodo.21853682",
  "doiUrl": "https://doi.org/10.5281/zenodo.21853682",
  "conceptDoi": "10.5281/zenodo.21853681",
  "pdfUrl": "https://github.com/ipitchford/cyclicity-loci-exponential-periods/releases/download/v0.2.1-candidate/main.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/cyclicity-loci-exponential-periods/v0.2.1-candidate/main.pdf",
  "zenodoUrl": "https://zenodo.org/records/21853682",
  "repoUrl": "https://github.com/ipitchford/cyclicity-loci-exponential-periods",
  "releaseUrl": "https://github.com/ipitchford/cyclicity-loci-exponential-periods/releases/tag/v0.2.1-candidate",
  "markdownUrl": "https://evidencepress.org/releases/cyclicity-loci-exponential-periods/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/cyclicity-loci-exponential-periods/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/cyclicity-loci-exponential-periods.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/cyclicity-loci-exponential-periods.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/cyclicity-loci-exponential-periods.svg",
  "media": [],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed candidate theoretical and computer-assisted result. The immutable archive and paper are public under version DOI 10.5281/zenodo.21853682 and CC0 for original content. Producer-side ordinary, optimized, coefficient-matching, certificate and targeted failure-control replays pass from fresh extracted copies. The alternative reducer and Wolfram transcript were produced inside the same workflow and are not independent reproduction. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, specialist review, conventional peer review, analytic rapid-decay-cycle or Stokes validation, or novelty or priority determination is claimed. The scalar period equations retain their stated local integral-curve and contour hypotheses."
  },
  "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/21853682",
      "note": "The exact candidate ZIP, paper and checksum sidecar are public under a version DOI; Zenodo file checksums match the GitHub release assets."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Producer-side ordinary, optimized, coefficient-matching, certificate, mutation and negative-control layers pass, including fresh-extraction replay under Python 3.12 and 3.13.",
      "evidenceUrl": "https://raw.githubusercontent.com/ipitchford/cyclicity-loci-exponential-periods/v0.2.1-candidate/REPLAY_RECEIPT.json"
    },
    {
      "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 running the exact archived candidate."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The coefficient-matching path and Wolfram transcript are algorithmically or computationally distinct but remain producer-workflow evidence."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No structural theorem or end-to-end claim has been formalized in a proof assistant."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "A supplied major-revision review had unspecified reviewer or model provenance; it is not an external specialist endorsement."
    },
    {
      "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 venue peer review has occurred."
    },
    {
      "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-loci-exponential-periods/blob/v0.2.1-candidate/requirements.txt",
      "note": "Python compatibility and SymPy 1.14.0 are recorded and multiple interpreter versions replay, but no full runtime container, Nix or Guix lock is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI GPT-5.6 Pro",
      "Wolfram Language 15.0.1"
    ],
    "humanRole": "Research direction, supplied review, correction and publication authorization; Ian Pitchford is repository maintainer and publisher, while the scholarly creator is cited as Anonymous.",
    "disclosure": "The package reports GPT-5.6 Pro contributions to mathematical development, implementation, literature work, drafting, revision and packaging, without an immutable platform run identifier. Wolfram Language supplied an author-run cross-CAS spot check. Neither system agreement nor producer-side replay is independent verification."
  },
  "problem": {
    "name": "Fixed-seed cyclicity in univariate polynomial exponential Gauss-Manin systems",
    "url": "https://doi.org/10.5281/zenodo.21745937"
  },
  "corrections": [],
  "keywords": [
    "computer-assisted mathematics",
    "polynomial exponential periods",
    "Gauss-Manin connection",
    "cyclic vectors",
    "determinantal loci",
    "Picard-Fuchs equations",
    "twisted de Rham reduction",
    "odd quintic",
    "reproducible research",
    "AI-generated mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "On the recorded localization, the selected polynomial twisted quotient has a monomial normal form of rank deg(Phi)-1, and fixed-seed Krylov minors define scheme-theoretic fibre-rank loci.",
    "For phi=x^4+a*x^2+b*x, the candidate obtains Delta_4=b[s(8a^3+27b^2)-6a]/(64s) with complete ranks one, two and three; the component b=0 is persistent under the scale connection.",
    "For phi=x^5+a*x^3+b*x, a finite exact reduction certificate gives Delta_5=-64(a^2-5b)^2 P/(9765625 s^2), freezes every size-three minor and supports the stated rank-one/two/three/four stratification.",
    "An explicit tangent-family counterexample shows why substituting a constraint before differentiation can produce the wrong Krylov matrix and scalar equation.",
    "Affine Chebyshev phases provide a known prior-art benchmark with exact fixed-seed rank two inside ambient rank n-1; the integral and modified-Bessel equation are not claimed as discoveries."
  ],
  "reviews": [],
  "evidencePackage": "A 24-page anonymous candidate manuscript in PDF, generated TeX and Markdown; an exact SymPy implementation and CLI; 129 ordinary and 129 optimized checks; a 17-check coefficient-matching path packaged in the same workflow; a 47-check finite odd-quintic reduction certificate; deterministic certificate regeneration; nine detected mutation or invalid-input controls; an author-run Wolfram Language transcript; a 72-file immutable tagged manifest and fresh-extraction replay under Python 3.12.11 and 3.13.5; machine-readable claim, status, assurance, provenance, licence and archival records; and a 14-reference citation/context receipt. These are producer-side and packaging checks, not independent theorem verification.",
  "openProblems": [
    "Reconstruct the quartic and odd-quintic arguments independently and obtain a specialist audit of the localized base-ring, rank-stratification and persistence proofs.",
    "Reimplement the load-bearing reductions in an unaffiliated open-source CAS path, such as SageMath with Singular, without consulting the production reducers beyond the public statements.",
    "Formalize the localized normal-form, determinantal-locus, tangent-persistence and component-multiplicity theorems in a proof assistant with a reported axiom footprint.",
    "Supply analytic rapid-decay cycles, justify contour differentiation uniformly where needed, and determine the associated Stokes data.",
    "Extend the fixed-seed construction to multivariate twisted complexes, relative or logarithmic amplitudes, and explicit scientific applications.",
    "Conduct a broader recognition and priority search for the quartic and odd-quintic factorizations across Brieskorn lattices, moment determinants, special functions, theses and non-English literature."
  ],
  "relatedWorks": [
    {
      "citation": "Evidence Press. (2026). Irreducible Pushforwards and Constrained Quartic Transitions, version 1.0.0-preprint, Theorem 6.2 and Proposition 6.3.",
      "url": "https://doi.org/10.5281/zenodo.21745937"
    },
    {
      "citation": "Charbonnier, S., Chidambaram, N. K., Garcia-Failde, E., & Giacchetto, A. (2024). Shifted Witten classes and topological recursion. Transactions of the AMS, 377(2), 1069-1110.",
      "url": "https://doi.org/10.1090/tran/9046"
    },
    {
      "citation": "Fantini, V., & Fenyes, A. (2025). The regularity of ODEs and thimble integrals with respect to Borel summation. SIGMA, 21, 101.",
      "url": "https://doi.org/10.3842/SIGMA.2025.101"
    },
    {
      "citation": "Johnson-Freyd, T. (2015). Homological perturbation theory for nonperturbative integrals. Letters in Mathematical Physics, 105, 1605-1632.",
      "url": "https://doi.org/10.1007/s11005-015-0791-9"
    },
    {
      "citation": "Katz, N. M. (1968). On the differential equations satisfied by period matrices. Publications Mathematiques de l'IHES, 35, 71-106.",
      "url": "https://doi.org/10.1007/BF02698924"
    },
    {
      "citation": "Lairez, P., & Vanhove, P. (2023). Algorithms for minimal Picard-Fuchs operators of Feynman integrals. Letters in Mathematical Physics, 113, 37.",
      "url": "https://doi.org/10.1007/s11005-023-01661-3"
    },
    {
      "citation": "Hien, M. (2007). Periods for irregular singular connections on surfaces. Mathematische Annalen, 337, 631-669.",
      "url": "https://doi.org/10.1007/s00208-006-0050-6"
    }
  ]
}