{ "schemaVersion": "1.2", "slug": "exact-smith-invariants-affine-determinant-lines", "title": "Exact Smith Invariants and Affine Determinant Lines of Binary-Form Factorisation", "shortTitle": "Exact Smith invariants of binary-form factorisation", "url": "https://evidencepress.org/releases/exact-smith-invariants-affine-determinant-lines/", "oneLine": "A candidate all-factor determinant-line theorem and exact Smith calculation replace the rank-one degree difference by a complete character-lattice invariant.", "abstract": "This anonymous, unrefereed candidate develops an integral determinant-line and character-lattice theory for multiplication maps of several binary forms. It gives an explicit affine all-factor maximal-minor identity, a multi-border contraction formula, graph formulae for resultant-character determinants, and an exact local and global Smith calculation for the complete lattice of pairwise-resultant characters. For primitive degree data the residual cokernel is cyclic, with exact prime valuations and bad-characteristic support; arbitrary degrees follow by scaling. The geometric application identifies the finite diagonalizable residual group scheme and the generic finite-locally-free degree of normalized factorisation charts. The public package contains a 35-page manuscript, a 96-artifact manifest, exact SymPy and FLINT checks, exhaustive character-lattice tests, 129,352 positive checks, 18 deliberate negative controls, review records, and a fresh-extraction replay. The universal results rest on the written proofs. Independent reproduction, proof-assistant formalization, external specialist review, editorial peer review, and absolute novelty or priority are not claimed.", "datePublished": "2026-08-09", "dateModified": "2026-08-09", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.21861347", "doiUrl": "https://doi.org/10.5281/zenodo.21861347", "conceptDoi": "10.5281/zenodo.21861346", "pdfUrl": "https://github.com/ipitchford/exact-smith-invariants-affine-determinant-lines/releases/download/v0.1.0-candidate/exact-smith-invariants-affine-determinant-lines-v0.1.0-candidate.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/exact-smith-invariants-affine-determinant-lines/v0.1.0-candidate/release/exact-smith-invariants-affine-determinant-lines-v0.1.0-candidate.pdf", "zenodoUrl": "https://zenodo.org/records/21861347", "repoUrl": "https://github.com/ipitchford/exact-smith-invariants-affine-determinant-lines", "releaseUrl": "https://github.com/ipitchford/exact-smith-invariants-affine-determinant-lines/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/exact-smith-invariants-affine-determinant-lines/index.md", "bibtexUrl": "https://evidencepress.org/releases/exact-smith-invariants-affine-determinant-lines/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/exact-smith-invariants-affine-determinant-lines.mp3", "imageUrl": "https://evidencepress.org/assets/og/exact-smith-invariants-affine-determinant-lines.png", "coverArtUrl": "https://evidencepress.org/assets/art/exact-smith-invariants-affine-determinant-lines.svg", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/exact-smith-invariants-affine-determinant-lines.mp3", "name": "Audio briefing — exact Smith invariants and determinant lines", "description": "Plain-English AI-generated voice summary of the candidate results, lineage, evidence, and assurance boundary.", "transcriptUrl": "https://evidencepress.org/assets/audio/exact-smith-invariants-affine-determinant-lines.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed candidate mathematical release. Availability, deterministic packaging, public checksum readback, and producer-side exact replay pass. The packaged evidence reports 129,352 positive checks and 18 detected negative controls in the recorded Python 3.13.5 environment. The main all-factor, Smith, and geometric assertions are universal mathematical theorems supported by prose proofs, not established by finite computation alone. The antecedent audit found classical monic and square Jacobian-resultant identities, multi-factor coprimality criteria, signed-incidence methods, and arithmetic-matroid multiplicity frameworks; it did not locate the complete affine Pluecker tensor or the exact cross-weighted local Smith presentation in its bounded search. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, external specialist review, editorial peer review, or absolute priority determination 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://zenodo.org/records/21861347", "note": "The version DOI, concept DOI, manuscript, source, archive, receipts, bibliography, audio, graphics, and checksum ledger are public. Independent downloads from Zenodo match the release SHA-256 inventory." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "The deterministic archive passed manifest and claims validation, normal and optimized exact replay, deliberate negative controls, receipt comparison, and manuscript reconstruction from a fresh extraction.", "evidenceUrl": "https://github.com/ipitchford/exact-smith-invariants-affine-determinant-lines/releases/tag/v0.1.0-candidate" }, { "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 replaying the immutable public archive." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The SymPy, FLINT, graph, and local-Smith implementations were produced inside one coordinated workflow and do not constitute independent reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "evidenceUrl": "https://github.com/ipitchford/exact-smith-invariants-affine-determinant-lines/blob/v0.1.0-candidate/package/formal/STATUS.md", "note": "The package includes a formalisation scaffold and machine-readable claim boundary, but no theorem in this release has been completed in a proof assistant." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Proof, geometry, arithmetic, citation, originality, and reproducibility reviews were conducted within the producer workflow; no external specialist review has occurred." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No journal submission or venue peer-review decision 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/exact-smith-invariants-affine-determinant-lines/blob/v0.1.0-candidate/package/verification/README.md", "note": "The replay records Python 3.13.5, SymPy 1.14.0, python-flint 0.9.0, jsonschema 4.26.0, commands, hashes, and normal/optimized receipts, but no independently recreated container, Nix, or Guix environment is supplied." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "GPT-5.6 Sol review workflow", "SymPy 1.14.0", "python-flint 0.9.0" ], "humanRole": "Research direction, continuation authorization, mediation, and publication authorization; Ian Pitchford is repository maintainer and publisher, while scholarly attribution remains Anonymous.", "disclosure": "The work was developed and reviewed within one producer-directed AI-assisted workflow. Multiple simulated reviews, exact computational backends, archive replay, and public readback reduce specified risks but are not independent reproduction, external specialist review, formal verification, or editorial peer review." }, "problem": { "name": "Bordered Jacobian foundations for binary-form multiplication", "url": "https://doi.org/10.5281/zenodo.21855302" }, "corrections": [], "keywords": [ "algebraic geometry", "binary forms", "factorisation maps", "resultants", "determinant lines", "Smith normal form", "arithmetic matroids", "torus characters", "finite diagonalizable group schemes", "computer-assisted mathematics", "AI-generated mathematics", "unrefereed candidate" ], "keyResults": [ "For any number of positive-degree binary-form factors, the candidate identifies every maximal minor of the affine multiplication Jacobian as the product of all pairwise resultants times the complementary Pluecker coordinate of the relative-scaling kernel, with an explicit integral orientation.", "A multi-border Cauchy-Binet contraction turns the complementary-minor tensor into a square determinant formula; semi-invariant normalisers contribute the determinant of their integer character matrix.", "For the complete lattice of pairwise-resultant characters, the primitive cokernel is cyclic. A local presentation supplies exact prime valuations, an efficient symmetric formula, the scaled global Smith form, and the precise bad-characteristic support.", "The resultant-unit map has a canonical finite diagonalizable residual group scheme on the pairwise-coprime locus; a square Laurent basis has degree equal to the lattice index times the root-partition degree.", "On the squarefree generic locus, normalized factorisation charts are finite locally free of degree equal to the torus-isogeny degree times the multinomial number of labelled root partitions, with separate separable and inseparable factors." ], "reviews": [], "evidencePackage": "A 35-page manuscript and generated TeX; 28-source bibliography and citation audit; a 96-artifact deterministic manifest; seven machine-readable claims; three inherited determinant/character evidence sets plus a new exact local-Smith tier; 129,352 exact positive checks and 18 deliberate negative controls under ordinary and optimized Python; exact SymPy, FLINT, graph, Smith and local-presentation calculations; source and receipt hashes; proof, geometry, arithmetic-priority, originality, citation and reproducibility review records; a deterministic 118-entry release ZIP; and a full fresh-extraction replay that rebuilt the manuscript. The immutable GitHub and Zenodo downloads match the deposited SHA-256 ledger. These establish availability, package integrity, finite producer-side checks, and same-environment replay. The universal theorems rest on the written proofs, and none of these mechanics is independent mathematical verification.", "openProblems": [ "Obtain an unaffiliated reconstruction of the all-factor affine maximal-minor theorem, including its integral orientation and multiplicity-one statement.", "Reimplement the complete-edge local Smith presentation and valuation formula in a fresh exact system and compare the resulting group schemes prime by prime.", "Formalize the determinant-line induction, graph-minor classification, local module presentation, and finite-locally-free descent in Lean or another proof assistant.", "Extend the bounded antecedent audit across determinant complexes, Hensel lifting, signed and gain graphs, arithmetic matroids, toric arrangements, non-English literature, and theses.", "Classify all regular-unit and polynomial normalisers on the pairwise-coprime locus, beyond semi-invariant and Laurent resultant coordinates.", "Determine which normalized target slices have affine-space inverse images and whether the known three-dimensional Keller mechanism is isolated inside this architecture.", "Seek either a construction or a no-go theorem connecting factorisation-derived maps with the unresolved four-dimensional Hessian frontier, without treating the determinant identity alone as evidence for such a connection." ], "relatedWorks": [ { "citation": "Anonymous. (2026). The Bordered Jacobian of Binary-Form Multiplication (Version 0.3-candidate). Parent candidate release.", "url": "https://doi.org/10.5281/zenodo.21855302" }, { "citation": "Artin, M. (2022). Algebraic Geometry: Notes on a Course. Graduate Studies in Mathematics 222. See Lemma 1.8.5 for a square polynomial-multiplication Jacobian and the Sylvester matrix.", "url": "https://doi.org/10.1090/gsm/222" }, { "citation": "Bhargava, M., Cremona, J. E., Fisher, T., & Gajovic, S. (2022). The density of polynomials of degree n over Z_p having exactly r roots in Q_p.", "url": "https://doi.org/10.1112/plms.12429" }, { "citation": "Chaperon, M., & Lopez de Medrano, S. (2009). Some regularities and singularities appearing in the study of polynomials and operators. Asterisque 323, 123-160.", "url": "https://www.numdam.org/item/AST_2009__323__123_0/" }, { "citation": "Moci, L. (2012). A Tutte polynomial for toric arrangements. Transactions of the American Mathematical Society, 364, 1067-1088.", "url": "https://doi.org/10.1090/S0002-9947-2011-05465-3" }, { "citation": "Zaslavsky, T. (1982). Signed graphs. Discrete Applied Mathematics, 4, 47-74, together with the 1983 erratum.", "url": "https://doi.org/10.1016/0166-218X(82)90033-6" } ] }