{
  "schemaVersion": "1.2",
  "slug": "cyclicity-root-span-low-rank",
  "title": "Fixed-seed cyclic rank as reduced inverse-root span",
  "shortTitle": "Fixed-seed cyclic rank from inverse roots",
  "url": "https://evidencepress.org/releases/cyclicity-root-span-low-rank/",
  "oneLine": "A conditional theorem turns fixed-seed cyclic rank into a count of independent inverse-root functions, classifies every indecomposable rank-one to rank-three phase, and exposes block fusion as the remaining compositional obstruction.",
  "abstract": "This anonymous, unrefereed candidate studies the differential system generated by the fixed class [dx] for a complex polynomial exponential phase. Conditional on the fixed-seed stationary-phase bridge established in the parent release, it proves that the cyclic rank equals the dimension of the reduced constant span of the inverse roots. The result gives an arbitrary-degree classification for indecomposable phases of rank at most three: powers have rank one, Chebyshev phases rank two, and indecomposable quartics rank three. A finite exact reducer computes cyclic rank for trusted rational polynomial inputs. Right-factor heredity is proved, but a degree-twelve rank-three example shows that symmetry can fuse separate root blocks under composition, so the full decomposable classification remains open. Producer-side replay and internal review tracks pass. No independent reproduction, formal verification, external specialist review, editorial peer review, or novelty or priority determination has occurred.",
  "datePublished": "2026-08-09",
  "dateModified": "2026-08-09",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.21864884",
  "doiUrl": "https://doi.org/10.5281/zenodo.21864884",
  "conceptDoi": "10.5281/zenodo.21864883",
  "pdfUrl": "https://github.com/ipitchford/cyclicity-root-span-low-rank/releases/download/v0.1.0-candidate/paper.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/cyclicity-root-span-low-rank/v0.1.0-candidate/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21864884",
  "repoUrl": "https://github.com/ipitchford/cyclicity-root-span-low-rank",
  "releaseUrl": "https://github.com/ipitchford/cyclicity-root-span-low-rank/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/cyclicity-root-span-low-rank/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/cyclicity-root-span-low-rank/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/cyclicity-root-span-low-rank.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/cyclicity-root-span-low-rank.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/cyclicity-root-span-low-rank.svg",
  "media": [],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed candidate, explicitly conditional on the fixed-seed stationary-phase bridge inherited from the parent release. The versioned paper and evidence package are public under DOI 10.5281/zenodo.21864884. Producer-side ordinary and optimized replay receipts are byte-identical, and the archived bytes match the GitHub release. Internal AI reviews found no material contradiction in the stated scope. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, signed external specialist review, conventional peer review, or novelty or priority determination is claimed. The indecomposable q<=3 classification is complete under the bridge; the decomposable classification remains open."
  },
  "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/21864884",
      "note": "The exact candidate ZIP, paper, manifest and ordinary/optimized receipts are public under a version DOI; Zenodo bytes 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": "The 86-entry package boundary, fixed targets, exact Krylov tests, composition controls and clean-room controls pass under ordinary and optimized Python with byte-identical receipts.",
      "evidenceUrl": "https://github.com/ipitchford/cyclicity-root-span-low-rank/releases/download/v0.1.0-candidate/VERIFY_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 clean-room and held-out paths were produced inside the same AI-assisted research programme and are 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 stationary-phase bridge, conductor-span theorem, low-rank classification or implementation has been formalized in a proof assistant."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Internal specialist-style reviews are included, but no signed external specialist report has been obtained."
    },
    {
      "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-root-span-low-rank/blob/v0.1.0-candidate/requirements.txt",
      "note": "Python, SymPy and pypdf versions are recorded and the verifier is archive-relative, but no complete container, Nix or Guix environment is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "OpenAI GPT-5.6",
      "OpenAI GPT-5.6 Pro"
    ],
    "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 to mathematical development, computation, literature work, drafting, adversarial review, revision and packaging. Internal model-to-model review and producer replay are not independent verification."
  },
  "problem": {
    "name": "Fixed-seed cyclicity for polynomial exponential periods",
    "url": "https://doi.org/10.5281/zenodo.21853682"
  },
  "corrections": [],
  "keywords": [
    "computer-assisted mathematics",
    "polynomial exponential periods",
    "cyclic vectors",
    "inverse roots",
    "polynomial monodromy",
    "twisted de Rham reduction",
    "Ritt decomposition",
    "reproducible research",
    "AI-generated mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "Conditional on the parent fixed-seed stationary-phase bridge, q(P,dx) equals the dimension of the reduced constant inverse-root span for every complex polynomial phase of degree at least two.",
    "For indecomposable polynomial phases in arbitrary degree, q at most three occurs exactly for powers with q=1, Chebyshev phases with q=2, and indecomposable quartics with q=3, up to affine changes.",
    "A finite exact twisted-de-Rham Krylov reducer computes the fixed-seed cyclic rank for trusted rational polynomial and amplitude inputs and returns exact independence or dependence witnesses.",
    "Every right factor inherits the cyclic-rank upper bound, but P=(x^4+x)^3 has q=3 because symmetry fuses three root-block spans.",
    "The complete decomposable low-rank classification is not claimed; equivariant block fusion under Ritt moves is the stated frontier."
  ],
  "reviews": [],
  "evidencePackage": "A 10-page anonymous candidate paper and canonical Markdown; a finite exact SymPy reducer; an 86-entry static manifest that includes nested audit manifests; byte-identical ordinary and optimized producer receipts; fixed-target hash checks; exact Krylov, composition-mutation and independently written clean-room control tracks; held-out implementation checks; internal fixed-text stationary-phase and polynomial-monodromy reviews; a conductor-span follow-up audit; scoped primary-source prior-art ledgers; a rights map; and byte-identical GitHub and Zenodo release assets. This evidence establishes availability, package integrity and producer-side replay, not independent theorem verification.",
  "openProblems": [
    "Obtain signed external specialist reviews of the fixed-seed stationary-phase bridge and the root-relation and polynomial-monodromy arguments.",
    "Reproduce the exact package and reimplement the rank calculator independently in another open-source computer-algebra stack.",
    "Classify equivariant root-block fusion under complete polynomial decompositions and Ritt moves, including monomial collisions such as (x^4+x)^3.",
    "Formalize the stationary-phase dependency, conductor-span equality and indecomposable low-rank classification in a proof assistant with a reported axiom footprint.",
    "Extend the fixed-seed criterion beyond dx to nonconstant amplitudes, relative or logarithmic families, and multivariate twisted complexes."
  ],
  "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": "Sabbah, C. (1999). On a twisted de Rham complex. Tohoku Mathematical Journal, 51, 125-140.",
      "url": "https://arxiv.org/abs/math/9805087"
    },
    {
      "citation": "Muller, P. (2002). Finiteness results for Hilbert's irreducibility theorem. Annales de l'Institut Fourier, 52, 983-1015.",
      "url": "https://doi.org/10.5802/aif.1907"
    },
    {
      "citation": "Fried, M. D. (1999). Variables separated polynomials, the genus 0 problem and moduli spaces. In Number Theory in Progress, 169-228.",
      "url": "https://doi.org/10.1515/9783110285581.169"
    },
    {
      "citation": "Alvarez, M., Bravo, J. L., and Mardesic, P. (2013). Vanishing Abelian integrals on zero-dimensional cycles. Proceedings of the London Mathematical Society, 107, 1302-1330.",
      "url": "https://doi.org/10.1112/plms/pdt012"
    }
  ]
}