{
  "schemaVersion": "1.2",
  "slug": "irreducible-pushforwards-quartic-transitions",
  "title": "Irreducible Pushforwards and Constrained Quartic Transitions: Two Reusable Methods from a Study of Furter's R(3) Conjecture",
  "shortTitle": "Irreducible pushforwards and quartic transitions",
  "url": "https://evidence-press.pages.dev/releases/irreducible-pushforwards-quartic-transitions/",
  "oneLine": "A stalled attack yields two standalone tools: one turns irreducibility into geometric separation, and the other detects exact rank collapse in a quartic transition; Furter's R(3) remains open.",
  "abstract": "This anonymous methods preprint extracts two proved pieces of mathematics from an unfinished study of Furter's R(3) conjecture. The first shows that, under an exact divisor-pushforward hypothesis, irreducibility of one polynomial forces adjacent rational functions on a curve to have disjoint zero sets; on an elliptic double cover, a nontorsion condition also prevents collision with the deck-transformed divisor. The second reduces a constrained quartic phase in twisted polynomial de Rham cohomology and obtains the exact determinant (9b/64)(-3K^3+K+3b^2), together with its rank strata and a first-order cancellation. For Furter's family the geometric conclusion is conditional at each fixed index on irreducibility of an explicit deck norm. Exact checks for 3 <= n <= 40 are finite evidence only. Uniform irreducibility, the global contour and period estimates, and R(3) itself remain open.",
  "datePublished": "2026-08-01",
  "dateModified": "2026-08-02",
  "version": "1.0.0",
  "doi": "10.5281/zenodo.21745937",
  "doiUrl": "https://doi.org/10.5281/zenodo.21745937",
  "conceptDoi": "10.5281/zenodo.21745936",
  "pdfUrl": "https://github.com/ipitchford/irreducible-pushforwards-quartic-transitions/releases/download/v1.0.0-preprint/irreducible-pushforwards-quartic-transitions-v1.0.0.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/irreducible-pushforwards-quartic-transitions/main/build/main.pdf",
  "zenodoUrl": "https://zenodo.org/records/21745937",
  "repoUrl": "https://github.com/ipitchford/irreducible-pushforwards-quartic-transitions",
  "releaseUrl": "https://github.com/ipitchford/irreducible-pushforwards-quartic-transitions/releases/tag/v1.0.0-preprint",
  "markdownUrl": "https://evidence-press.pages.dev/releases/irreducible-pushforwards-quartic-transitions/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/irreducible-pushforwards-quartic-transitions/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/irreducible-pushforwards-quartic-transitions.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/irreducible-pushforwards-quartic-transitions.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/irreducible-pushforwards-quartic-transitions.svg",
  "media": [],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-preprint",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed methods preprint. The standalone irreducible-pushforward, elliptic deck-collision, and constrained-quartic identities are proved in the manuscript and replayed exactly. The Furter application is conditional at each fixed index on deck-norm irreducibility; checks through n = 40 remain finite evidence and do not imply uniform irreducibility. The work has not been independently reproduced, formally verified in a proof assistant, accepted by experts, or peer reviewed; internal AI-assisted audits are not independent assessment. The remaining global contour, cycle, Stokes, and period estimates and Furter's R(3) conjecture are open."
  },
  "provenance": {
    "aiGenerated": false,
    "aiAssisted": true,
    "generatedBy": [
      "Automated research and coding tools"
    ],
    "humanRole": "Problem selection, mediation, review, pivot approval, and publication management; the scholarly creator is cited as Anonymous.",
    "disclosure": "Automated tools assisted symbolic exploration, implementation, replay, critique, drafting, and publication checks; they are not authors. The methods preprint is cited as Anonymous."
  },
  "problem": {
    "name": "Furter's R(3) conjecture for polynomial-composition rigidity",
    "url": "https://doi.org/10.1112/jlms/jdu064"
  },
  "keywords": [
    "algebraic geometry",
    "elliptic curves",
    "irreducible pushforward",
    "deck involution",
    "divisor disjointness",
    "quartic transition",
    "twisted de Rham cohomology",
    "polynomial automorphisms",
    "computer-assisted mathematics",
    "Furter R(3)",
    "unrefereed preprint",
    "reproducible research"
  ],
  "keyResults": [
    "If an effective zero divisor of degree h pushes forward with coefficient one to the prime divisor of an irreducible degree-h polynomial, then it is one reduced closed point; with pole orders h and h+1, this forces adjacent base-point freeness.",
    "On an elliptic curve with degree-two deck involution Q -> S-Q, the same prime-pushforward hypothesis and hS != O prevent the zero divisor from meeting its deck image; nontorsion of S gives the conclusion for every h.",
    "For the constrained quartic phase Psi(W)=W^4-(3/2)KW^2+bW, the twisted-de-Rham coordinate determinant of 1, Psi, Psi^2 is (9b/64)(-3K^3+K+3b^2). It vanishes on the aligned transition curve, with rank two away from the origin, rank one at the origin, and cancellation of the first moving-alignment correction.",
    "For each fixed Furter index, irreducibility of the exact deck norm implies the required adjacent and self-deck divisor disjointness. The package verifies irreducibility only for 3 <= n <= 40; no uniform all-index theorem or proof of R(3) is claimed."
  ],
  "evidencePackage": "A 14-page anonymous methods preprint; exact symbolic replay under normal and optimized Python with SymPy 1.14.0; a self-contained exact Furter deck-norm audit for 3 <= n <= 40 plus auxiliary deck diagnostics through n = 20; 22-file SHA-256 manifest; fresh git-archive extraction and complete replay; negative assurance boundaries and a seven-mode publication-integrity report; byte-identical GitHub and Zenodo release assets.",
  "openProblems": [
    "Prove or disprove uniform irreducibility of the exact Furter deck norms for every index, without treating a longer finite computation as the principal result.",
    "Complete the global analytic step: construct a contour deformation and Stokes-compatible cycle control with uniform period estimates through the constrained quartic transition.",
    "Determine whether the exact determinant and first-correction cancellation admit a broader invariant interpretation for coalescing saddles, Gauss-Manin systems, or Stokes-filtered twisted de Rham families.",
    "Find further settings in which coefficient-one irreducible pushforward plus a one-degree pole gap gives useful base-point freeness or deck-collision exclusion.",
    "Independently reproduce the theorems and exact calculations without reusing the repository's symbolic code, and formalise the algebraic arguments in a proof assistant."
  ],
  "relatedWorks": [
    {
      "citation": "Furter, J.-P. (2015). Polynomial Composition Rigidity and Plane Polynomial Automorphisms. Journal of the London Mathematical Society, 91(1), 180-202.",
      "url": "https://doi.org/10.1112/jlms/jdu064"
    },
    {
      "citation": "Lewis, D., Perry, K., & Straub, A. (2019). An Algorithmic Approach to the Polydegree Conjecture for Plane Polynomial Automorphisms. Journal of Pure and Applied Algebra, 223(12), 5346-5359.",
      "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
    },
    {
      "citation": "Silverman, J. H. (2009). The Arithmetic of Elliptic Curves (2nd ed.). Springer.",
      "url": "https://doi.org/10.1007/978-0-387-09494-6"
    },
    {
      "citation": "Ursell, F. (1972). Integrals with a Large Parameter: Several Nearly Coincident Saddle Points. Mathematical Proceedings of the Cambridge Philosophical Society, 72(1), 49-65.",
      "url": "https://doi.org/10.1017/S0305004100050945"
    },
    {
      "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), 1-22.",
      "url": "https://doi.org/10.1007/s00222-009-0185-7"
    }
  ]
}