{ "schemaVersion": "1.1", "site": "Evidence Press", "baseUrl": "https://evidence-press.pages.dev", "description": "An independent press office for new research released with complete, replayable evidence — code, data, certificates, and archived deposits. Each release explains one paper twice: once for curious readers, once for specialists. Every page states plainly what has and has not been verified.", "schema": "https://evidence-press.pages.dev/api/schema.json", "count": 7, "papers": [ { "schemaVersion": "1.1", "slug": "exact-low-length-recht-re-inequalities", "title": "Exact low-length Recht–Ré inequalities: complete status through five factors and six-factor balanced families", "shortTitle": "Exact low-length Recht–Ré inequalities", "url": "https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/", "oneLine": "The first exact account of where a leading mathematical justification for shuffling data works, where it fails, and where it recovers — with a surprise: the usual theoretical measure can call reshuffling worse even when it reduces error.", "abstract": "This paper asks a basic question behind many machine-learning algorithms: is it better to shuffle a dataset and use each example once, or keep sampling examples at random? It gives the first exact account of where a leading mathematical justification for reshuffling works, where it fails, and where it recovers. It also finds something more surprising: the usual theoretical measure can say reshuffling is worse even when it actually reduces error. The proof is computer-assisted, exact, and fully replayable.", "datePublished": "2026-07-30", "dateModified": "2026-07-30", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.21709239", "doiUrl": "https://doi.org/10.5281/zenodo.21709239", "conceptDoi": "10.5281/zenodo.21709238", "pdfUrl": "https://github.com/ipitchford/exact-low-length-recht-re-inequalities/releases/download/v1.0.0-candidate/exact-low-length-recht-re-inequalities-v1.0.0-candidate.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/exact-low-length-recht-re-inequalities/main/paper/exact_low_length_recht_re.pdf", "zenodoUrl": "https://zenodo.org/records/21709239", "repoUrl": "https://github.com/ipitchford/exact-low-length-recht-re-inequalities", "releaseUrl": "https://github.com/ipitchford/exact-low-length-recht-re-inequalities/releases", "markdownUrl": "https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/exact-low-length-recht-re-inequalities.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/exact-low-length-recht-re-inequalities.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/exact-low-length-recht-re-inequalities.svg", "media": [], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "This is an unrefereed candidate computer-assisted proof. Its internal exact replay and adversarial checks pass, but it has not received independent external reproduction, proof-assistant formalisation, expert acceptance, or peer review. Deterministic replay validates the certificates as shipped; it does not constitute formal verification." }, "provenance": { "aiGenerated": true, "generatedBy": [ "Anonymous" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Recht–Ré noncommutative arithmetic–geometric mean conjecture (low-length cases)", "url": "https://doi.org/10.48550/arXiv.1202.4184" }, "keywords": [ "Recht–Ré conjecture", "matrix inequalities", "random reshuffling", "stochastic gradient descent", "noncommutative sums of squares", "positive semidefinite matrices", "computer-assisted proof", "exact arithmetic", "sum-of-squares certificates", "semidefinite programming", "randomised numerical linear algebra", "reproducible research" ], "keyResults": [ "Four factors: the two-sided inequality holds for every n ≥ 4.", "Five factors: the upper spectral bound holds for every n ≥ 5; the lower bound fails at n = 5 (exact rational counterexample with eigenvalue −285/2 against a permitted −120) and is restored for every n ≥ 6, so the threshold is sharp.", "Six factors: the upper bound holds when 7 | n, the lower bound when 8 | n, hence the complete norm inequality whenever 56 | n; remaining six-factor cases are open.", "Metric reversal: on an exact quadratic example, reshuffling worsens the norm of the expected iterate while improving expected squared error and objective value." ], "evidencePackage": "Exact rational sum-of-squares certificates; FLINT characteristic-polynomial positive-semidefiniteness certificates; 2,312 six-factor coefficient identity checks; deterministic replay scripts with pinned dependencies; SHA-256 manifests; mutation and negative controls; hash-bound release certificate.", "openProblems": [ "Characterise the metric reversal: find conditions under which the norm of the expected iterate, the expected squared error, and the expected objective value must agree in ranking sampling schemes, and conditions under which they may diverge (candidates: commutativity, simultaneous diagonalisation, near-identity updates, bounded condition number, small step size).", "Finish the six-factor classification: determine whether the two-sided result extends to every sufficiently large n, whether upper and lower bounds have different thresholds, and whether further exceptional congruence classes contain counterexamples.", "Obtain an independent proof: reimplement the computation without reusing the repository's orbit encoder, certificate schema, or extraction code; formalise the balanced-seed continuation theorem in Lean, Isabelle, or Coq; check the rational positivity certificates independently.", "Run recently proposed data-ordering schemes (block reshuffling, paired reversal) on the paper's exact counterexample and ask which orderings improve all three metrics simultaneously.", "Translate the low-length inequalities into short-horizon or block-iteration guarantees for randomised numerical linear algebra, and test whether structured matrices from real linear systems avoid the pathological behaviour." ], "relatedWorks": [ { "citation": "Recht, B., & Ré, C. (2012). Beneath the valley of the noncommutative arithmetic–geometric mean inequality: Conjectures, case-studies, and consequences (COLT 2012 version: Toward a noncommutative arithmetic-geometric mean inequality, JMLR W&CP 23).", "url": "https://arxiv.org/abs/1202.4184" }, { "citation": "Lai, Z., & Lim, L.-H. (2020). Recht–Ré noncommutative arithmetic–geometric mean conjecture is false. ICML 2020, PMLR 119.", "url": "https://arxiv.org/abs/2006.01510" }, { "citation": "Liu, Z. (2026). Random reshuffling dominates stochastic gradient descent. COLT 2026, PMLR 336.", "url": "https://arxiv.org/abs/2606.32005" }, { "citation": "Nguyen, L. M., Phan, D. T., & Kalagnanam, J. (2026). Learning to shuffle: Block reshuffling and reversal schemes for stochastic optimization.", "url": "https://arxiv.org/abs/2604.00260" } ] }, { "schemaVersion": "1.1", "slug": "z20-equals-6", "title": "z(20) = 6: resolving the first open case of Erdős problem 758", "shortTitle": "z(20) = 6 (Erdős problem 758)", "url": "https://evidence-press.pages.dev/releases/z20-equals-6/", "oneLine": "Every graph on 20 vertices can be split into six parts, each a clique or an independent set — and 6 is best possible. A computer-assisted proof, released with its full certificate chain, settles the first open case of an Erdős–Gimbel problem.", "abstract": "How few colours do you need to partition any 20-vertex graph so that every colour class is either a clique (all pairs connected) or an independent set (no pairs connected)? The answer for up to 19 vertices was known; at 20 it was either 6 or 7. This computer-assisted proof establishes that the answer is 6. The lower bound is a short, hand-checkable argument about the Paley graph on 17 vertices; the upper bound reduces all possible counterexamples to two SAT problems whose impossibility is certified with proof objects checked by four independent checkers, including a formally verified one.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "candidate-2026-07-26", "doi": "10.5281/zenodo.21647645", "doiUrl": "https://doi.org/10.5281/zenodo.21647645", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/z20-cochromatic/releases/download/candidate-2026-07-26/z20_equals_6_proof.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/z20-cochromatic/master/paper/z20_equals_6_proof.pdf", "zenodoUrl": "https://zenodo.org/records/21647645", "repoUrl": "https://github.com/ipitchford/z20-cochromatic", "releaseUrl": "https://github.com/ipitchford/z20-cochromatic/releases/tag/candidate-2026-07-26", "markdownUrl": "https://evidence-press.pages.dev/releases/z20-equals-6/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/z20-equals-6/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/z20-equals-6.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/z20-equals-6.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/z20-equals-6.svg", "media": [], "authors": [ "GPT 5.6 Sol", "Claude Opus 4.8 / 5" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate proof. No human mathematician has verified the complete argument, and there is no independent external reproduction or end-to-end proof-assistant formalisation. A dated replay on one machine (macOS/arm64) covered catalogue regeneration, cocolourability checks, CNF regeneration, and multiple proof-checker validations. Encoding soundness rests on semantic spot-testing, not proof, so implementation errors remain logically possible. The mathematics was generated by AI systems with human problem selection and mediation." }, "provenance": { "aiGenerated": true, "generatedBy": [ "GPT 5.6 Sol", "Claude Opus 4.8 / 5" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Erdős problem 758 (Erdős–Gimbel, cochromatic numbers of small graphs)", "url": "https://www.erdosproblems.com/758" }, "keywords": [ "cochromatic number", "Erdős problem 758", "Erdős–Gimbel", "Ramsey theory", "R(4,4)", "Paley graph", "SAT certificates", "DRUP", "LRAT", "cake_lpr", "computer-assisted proof", "AI-generated mathematics" ], "keyResults": [ "z(20) = 6: the maximum cochromatic number over all 20-vertex graphs is exactly 6 (previously known to be 6 or 7).", "Lower bound, hand-checkable: the Paley graph on 17 vertices has no clique or independent set of size 4, so its cochromatic number is at least ⌈17/3⌉ = 6.", "Upper bound: any 20-vertex counterexample reduces, via R(4,4) = 18, to cases built around the two 16-vertex (4,4)-Ramsey graphs; the resulting two CNF formulas (64 variables, 104,524 clauses each) are unsatisfiable, with DRUP and LRAT certificates.", "Supporting recomputations: R(4,4) = 18, regeneration of the (4,4)-Ramsey graph catalogue, and the known values z(8) ≤ 3 and z(12) ≤ 4." ], "evidencePackage": "Two SAT instances with DRUP and LRAT unsatisfiability certificates, checked by a custom Python RUP checker, a C RUP checker, drat-trim, and the formally verified cake_lpr; a hand-verifiable Paley(17) lower-bound argument; a 13-page proof PDF; a machine-readable AI_INDEX; SHA-256 manifests; portable replay script.", "openProblems": [ "Independently reproduce the reduction: regenerate the (4,4)-Ramsey catalogue and the two core CNFs from a fresh implementation, and confirm unsatisfiability with independent tooling.", "Formally verify the encoding: the LRAT certificates are already checked by a formally verified checker (cake_lpr), but the reduction from 'z(20) ≤ 6' to the two CNFs is not itself machine-checked — formalising it in Lean or Isabelle would close the main gap.", "Extend the method to z(n) for n = 21 and beyond, where the Erdős–Gimbel table is open.", "Study whether the Paley(17)-based lower-bound construction generalises to give improved lower bounds for larger n.", "Use the release as a benchmark case for AI-generated mathematics workflows: claim–evidence indexing, certificate exchange, and independent agent reproduction." ], "relatedWorks": [ { "citation": "Erdős problem 758 (Erdős & Gimbel): determine z(n) for small n — records the known values through n = 19 and Mehta's computational confirmation that z(12) = 4.", "url": "https://www.erdosproblems.com/758" }, { "citation": "Companion release: VR2(K4) = 20, which reuses this paper's upper-bound proof objects.", "url": "https://doi.org/10.5281/zenodo.21647654" }, { "citation": "Greenwood, R. E., & Gleason, A. M. (1955). Combinatorial relations and chromatic graphs. Canadian Journal of Mathematics, 7, 1–7 — establishes R(4,4) = 18.", "url": "https://doi.org/10.4153/CJM-1955-001-4" }, { "citation": "McKay, B. Ramsey graphs data page — lists the two 16-vertex (4,4)-Ramsey graphs used in the reduction.", "url": "https://users.cecs.anu.edu.au/~bdm/data/ramsey.html" }, { "citation": "Tan, Y. K., Heule, M. J. H., & Myreen, M. O. cake_lpr: a verified LRAT proof checker (CakeML) — the formally verified checker used on the certificates.", "url": "https://github.com/tanyongkiam/cake_lpr" } ] }, { "schemaVersion": "1.1", "slug": "vr2-k4-equals-20", "title": "VR2(K4) = 20: twenty vertices force two vertex-disjoint monochromatic K4s", "shortTitle": "VR2(K4) = 20", "url": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/", "oneLine": "Colour every edge among 20 points red or blue and you cannot avoid two completely separate single-colour foursomes — while a carefully built 19-point colouring can. A companion result to z(20) = 6.", "abstract": "The classical Ramsey number R(4,4) = 18 says that among 18 points with red/blue connections there is always a single-colour foursome. This release asks for more: how many points guarantee two such foursomes sharing no points at all? The claimed answer is exactly 20. A hand-checkable 19-vertex colouring (a 'Paley twin') has no two disjoint monochromatic K4s, while the impossibility of avoiding them at 20 vertices reuses the SAT certificates from the companion z(20) = 6 release.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "0.1-candidate", "doi": "10.5281/zenodo.21647654", "doiUrl": "https://doi.org/10.5281/zenodo.21647654", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/z20-cochromatic/releases/download/vr2-k4-v0.1-candidate/vr2_k4_equals_20_candidate.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/z20-cochromatic/master/applications/vr2-k4/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21647654", "repoUrl": "https://github.com/ipitchford/z20-cochromatic", "releaseUrl": "https://github.com/ipitchford/z20-cochromatic/releases/tag/vr2-k4-v0.1-candidate", "markdownUrl": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/vr2-k4-equals-20/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/vr2-k4-equals-20.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/vr2-k4-equals-20.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/vr2-k4-equals-20.svg", "media": [], "authors": [ "GPT 5.6 Sol", "Claude Opus 4.8 / 5" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate result, marked as a prerelease. No independent external reproduction, peer review, or full formal verification is claimed. The lower-bound witness is directly inspectable and replayable; the upper bound inherits the assurance boundary of the z(20) = 6 release, including the fact that encoding soundness is spot-tested rather than proved. The definition and notation VR2(K4) are as given by the project; the release is authored by AI systems with human project mediation." }, "provenance": { "aiGenerated": true, "generatedBy": [ "GPT 5.6 Sol", "Claude Opus 4.8 / 5" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Vertex-disjoint Ramsey-type quantities (related context: Erdős problem 758)", "url": "https://www.erdosproblems.com/758" }, "keywords": [ "Ramsey theory", "vertex-disjoint monochromatic subgraphs", "VR2(K4)", "K4", "R(4,4)", "Paley graph", "SAT certificates", "LRAT", "cake_lpr", "computer-assisted proof", "AI-generated mathematics" ], "keyResults": [ "VR2(K4) = 20: every red–blue colouring of the edges of K20 contains two vertex-disjoint monochromatic copies of K4.", "Sharpness: an explicit 19-vertex 'Paley-twin' colouring admits no two vertex-disjoint monochromatic K4s, so VR2(K4) > 19.", "The upper bound reuses the z(20) = 6 reduction and SAT unsatisfiability certificates (DRUP/LRAT, checked including by the formally verified cake_lpr)." ], "evidencePackage": "A human-readable candidate paper with TeX source; a hand-checkable 19-vertex lower-bound construction; a dependency-free direct verifier with its output included; reused DRUP/LRAT proof objects from the z20 release (pinned to a specific commit); an AI-readable claim/evidence index; SHA-256 manifest.", "openProblems": [ "Verify the 19-vertex Paley-twin construction independently — it is small enough for a fresh implementation in an afternoon.", "Determine VR2(K5), or more generally VRk(Km) for small k and m, for which the same reduction-plus-SAT architecture is a plausible route.", "Establish whether VR2(K4) = 20 admits a certificate-free human proof, given that R(4,4) = 18 does.", "Search the literature for prior appearances of this quantity under other notation, and connect it to known results on vertex-disjoint monochromatic structures.", "Formalise the reduction linking the z(20) proof objects to the VR2 claim." ], "relatedWorks": [ { "citation": "Companion release: z(20) = 6 (Erdős problem 758), whose upper-bound proof objects this result reuses.", "url": "https://doi.org/10.5281/zenodo.21647645" }, { "citation": "Greenwood, R. E., & Gleason, A. M. (1955). Combinatorial relations and chromatic graphs. Canadian Journal of Mathematics, 7, 1–7 — establishes R(4,4) = 18.", "url": "https://doi.org/10.4153/CJM-1955-001-4" }, { "citation": "McKay, B. Ramsey graphs data page — the two 16-vertex (4,4)-Ramsey graphs.", "url": "https://users.cecs.anu.edu.au/~bdm/data/ramsey.html" }, { "citation": "Erdős problem 758 (related context).", "url": "https://www.erdosproblems.com/758" } ] }, { "schemaVersion": "1.1", "slug": "reducible-incidence-divisors", "title": "Reducible incidence divisors and the isolation of affine slices in binary-form factorisation spaces", "shortTitle": "Reducible incidence divisors", "url": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/", "oneLine": "A classification of exactly when a natural family of divisors breaks into pieces — and a screening theorem narrowing where Jacobian-conjecture-style behaviour could possibly live.", "abstract": "This paper classifies the reducible members of a marked-common-root incidence linear system for binary forms: for adjacent degrees, a divisor becomes reducible precisely when its defining functional lies on the tangent developable of the rational normal curve of evaluation functionals. It proves that the corresponding affine slices are never isomorphic to affine space, computes their precise motivic and Hodge-theoretic defects, obstructs non-adjacent-degree slices via finite cyclic actions, and — conditionally on an upstream classification — isolates a unique slice that could in principle carry a nonzero-constant-Jacobian polynomial map.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "1.0-candidate (v1.0.1 archived as 10.5281/zenodo.21653119)", "doi": "10.5281/zenodo.21647616", "doiUrl": "https://doi.org/10.5281/zenodo.21647616", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices/releases/download/v1.0.1/paper.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/reducible-incidence-divisors-affine-slices/main/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21647616", "repoUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices", "releaseUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices/releases", "markdownUrl": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/reducible-incidence-divisors.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/reducible-incidence-divisors.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/reducible-incidence-divisors.svg", "media": [], "authors": [ "OpenAI Codex 5.6 Sol", "Anthropic Fable 5" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate manuscript. The release states it is not independent external reproduction, expert human review, peer review, proof-assistant formalisation, or a literature-wide novelty determination; its internal review disposition was 'minor revision for candidate publication'. The isolation theorem is conditional on an unverified upstream cubic classification from a companion release. Authored by AI systems with human project mediation." }, "provenance": { "aiGenerated": true, "generatedBy": [ "OpenAI Codex 5.6 Sol", "Anthropic Fable 5" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Structure of incidence divisors in factorisation spaces (screening for the Jacobian conjecture)", "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture" }, "keywords": [ "algebraic geometry", "binary forms", "incidence divisor", "rational normal curve", "tangent developable", "catalecticant", "Hodge–Deligne polynomial", "Jacobian conjecture", "affine slices", "computer-assisted mathematics" ], "keyResults": [ "Classification: for consecutive degrees (m, m+1) with m ≥ 2, the divisor D_ℓ = {ℓ(P²A′B′) = 0} is reducible precisely when [ℓ] lies on the tangent developable of the rational normal curve of evaluation functionals.", "Component counts: rank-one functionals yield three reduced components; first-jet functionals give two; genuine two-point secants and higher catalecticant ranks give irreducible divisors.", "For all m ≥ 2 and nonzero ℓ, the slice X_ℓ^{m,m+1} is not isomorphic to affine space A^{2m+1}.", "Precise defects: rank-one has Grothendieck class L^{2m+1} − L^{2m}; rank-two secants and first-jets show diagnostic Hodge coefficients −2 and −1.", "Non-adjacent degrees (|r − s| ≥ 2) give non-contractible slices via finite cyclic group actions.", "Conditional on an upstream cubic classification: the tangent nonosculating linear–quadratic slice is the unique positive-bidegree affine source permitting a nonzero constant Jacobian determinant." ], "evidencePackage": "Manuscript (PDF and TeX); deterministic SymPy audit scripts (verify_paper.py, verify_rank_two.py); claim-to-evidence mapping (AI_INDEX.md/.json); assurance-boundary documents (STATUS.md, ASSURANCE.md); SHA-256 manifest; immutable source snapshots and citation gates; archived on Zenodo as v1.0-candidate with a v1.0.1 revision.", "openProblems": [ "Verify the tangent-developable classification by hand — the statement is precise, classical in flavour, and a natural target for expert review or refutation.", "Discharge or refute the conditional hypothesis: the isolation theorem depends on an upstream cubic classification that has itself had no external verification.", "Check novelty against the classical literature on catalecticants, secant varieties, and tangent developables, where fragments of the classification may already be known.", "Extend the motivic-defect calculations to non-adjacent degree pairs beyond the cyclic-action obstruction.", "Determine whether the unique surviving slice actually carries a nonzero-constant-Jacobian map — either constructing one or excluding it, which would close this branch of the screening programme." ], "relatedWorks": [ { "citation": "Foundational companion release: The degree-difference principle and affine slices of binary-form factorisation spaces.", "url": "https://doi.org/10.5281/zenodo.21647593" }, { "citation": "Continues: Exotic affine three-spheres and the quadratic–cubic obstruction (upstream cubic classification).", "url": "https://doi.org/10.5281/zenodo.21647607" }, { "citation": "Keller, O.-H. (1939) — the Jacobian conjecture, which remains open; this release proves screening results only, not the conjecture.", "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture" } ] }, { "schemaVersion": "1.1", "slug": "exotic-affine-three-spheres", "title": "Exotic affine three-spheres and the quadratic–cubic obstruction", "shortTitle": "Exotic affine three-spheres", "url": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/", "oneLine": "A natural slice of polynomial factorisation space is claimed to be an exotic affine three-sphere — a shape that mimics a familiar one without being it — while every quadratic–cubic slice is obstructed from being ordinary affine space.", "abstract": "Some algebraic shapes impersonate familiar ones: they are indistinguishable by the tools of smooth topology yet are provably different as algebraic varieties. This paper argues that a natural slice of the space of polynomial factorisations is exactly such an impostor — a known exotic affine three-sphere (the Dubouloz–Finston torsor) rather than the standard SL2 quadric it resembles. It further claims that every normalised quadratic–cubic slice fails to be affine five-space, by an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction, and that in characteristic 3 the tangent slice remains affine three-space with an Artin–Schreier collision in the induced Keller map.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "0.1-candidate (v0.1.1 'citation robustness' archived as 10.5281/zenodo.21653108)", "doi": "10.5281/zenodo.21647607", "doiUrl": "https://doi.org/10.5281/zenodo.21647607", "conceptDoi": "10.5281/zenodo.21647606", "pdfUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic/releases/download/v0.1.1/paper.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/exotic-affine-spheres-quadratic-cubic/main/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21647607", "repoUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic", "releaseUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic/releases", "markdownUrl": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/exotic-affine-three-spheres.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/exotic-affine-three-spheres.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/exotic-affine-three-spheres.svg", "media": [], "authors": [ "OpenAI Codex / Anthropic models" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate manuscript. No peer review, no independent external reproduction, no complete external human review, and no exhaustive novelty check are documented. The symbolic and finite-field checks cover selected formulae only; the Grothendieck-class and Hodge–Deligne arguments themselves are not machine-verified. The repository's latest revision (v0.1.1) fixes citation routing after a cache incident, with manuscript content unchanged. AI-attributed with human direction." }, "provenance": { "aiGenerated": true, "generatedBy": [ "OpenAI Codex / Anthropic models" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Recognition of exotic affine spheres (Dubouloz–Finston)", "url": "https://arxiv.org/abs/1106.2900" }, "keywords": [ "algebraic geometry", "exotic affine sphere", "Dubouloz–Finston torsor", "binary forms", "Grothendieck ring of varieties", "Hodge–Deligne polynomial", "Keller map", "Artin–Schreier", "Jacobian conjecture", "finite-field verification", "computer-assisted mathematics" ], "keyResults": [ "The transverse linear–quadratic slice is claimed to be a specific exotic affine three-sphere (the Dubouloz–Finston torsor) rather than SL2.", "Every normalised quadratic–cubic slice fails to be affine five-space, via an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction.", "In characteristic 3, the tangent slice remains affine three-space, with an Artin–Schreier collision in the induced Keller map." ], "evidencePackage": "Manuscript (PDF and TeX); verify_paper.py running exact symbolic checks plus finite-field censuses over F2 through F11 covering 23,941 projective functionals, with byte-for-byte validation of generated tables against checked-in versions and a deliberate semantic control; claim-to-evidence mapping (AI_INDEX.md); integrity documentation (ASSURANCE.md); SHA-256-manifested Zenodo deposit.", "openProblems": [ "Have the torsor identification checked by specialists in affine algebraic geometry — in particular, experts on Dubouloz–Finston exotic spheres — since this is the manuscript's most consequential claim.", "Verify the Grothendieck-class formula and the rank-by-rank Hodge–Deligne obstruction by hand; the shipped checks validate consequences, not the arguments.", "Rerun the finite-field censuses (F2 through F11, 23,941 projective functionals) from an independent implementation.", "Search the literature for prior art on slices of binary-form factorisation spaces to establish novelty.", "Investigate whether the characteristic-3 Artin–Schreier phenomenon has characteristic-p analogues for other small primes." ], "relatedWorks": [ { "citation": "Dubouloz, A., & Finston, D. On exotic affine 3-spheres.", "url": "https://arxiv.org/abs/1106.2900" }, { "citation": "Dubouloz, A., & Finston, D. Families of exotic affine 3-spheres.", "url": "https://hal.science/hal-01374364" }, { "citation": "Foundational companion release: The degree-difference principle and affine slices of binary-form factorisation spaces.", "url": "https://doi.org/10.5281/zenodo.21647593" }, { "citation": "Continuation: Reducible incidence divisors and the isolation of affine slices.", "url": "https://doi.org/10.5281/zenodo.21647616" } ] }, { "schemaVersion": "1.1", "slug": "erdos-848-all-n", "title": "Erdős problem 848: an exact answer for every N", "shortTitle": "Erdős 848: exact answer for all N", "url": "https://evidence-press.pages.dev/releases/erdos-848-all-n/", "oneLine": "A certificate-backed determination that the answer to an Erdős–Sárközy extremal problem is exactly ⌊(N+18)/25⌋ for every N — closing the gap between known asymptotic results and small cases.", "abstract": "Erdős and Sárközy asked: how large can a set A of integers from 1 to N be if the product of any two members (including a member with itself), plus one, is never squarefree? Recent work resolved the question for all sufficiently large N. This release establishes the exact answer f(N) = ⌊(N+18)/25⌋ for every positive integer N, stitching together exact colouring certificates for small N, structural decompositions and exact-rational envelope arguments for intermediate ranges, and a pinned explicit-threshold analytic theorem for N beyond 2.64 × 10^17.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "0.1-candidate", "doi": "10.5281/zenodo.21647629", "doiUrl": "https://doi.org/10.5281/zenodo.21647629", "conceptDoi": "10.5281/zenodo.21647628", "pdfUrl": "https://github.com/ipitchford/erdos-848-all-n/releases/download/v0.1-candidate/paper.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/erdos-848-all-n/main/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21647629", "repoUrl": "https://github.com/ipitchford/erdos-848-all-n", "releaseUrl": "https://github.com/ipitchford/erdos-848-all-n/releases/tag/v0.1-candidate", "markdownUrl": "https://evidence-press.pages.dev/releases/erdos-848-all-n/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/erdos-848-all-n/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/erdos-848-all-n.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/erdos-848-all-n.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/erdos-848-all-n.svg", "media": [], "authors": [ "OpenAI Codex" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate result. The release itself states it is not independent external reproduction, external human peer review, or end-to-end formal verification; correctness rests on local certificate replay under documented compiler, runtime, and hardware assumptions, and on an external third-party explicit-threshold theorem (a pinned PDF source). The catalogue at erdosproblems.com records the resolution for sufficiently large N but does not (as of this release) acknowledge an all-N determination." }, "provenance": { "aiGenerated": true, "generatedBy": [ "OpenAI Codex" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Erdős problem 848 (Erdős–Sárközy, nonsquarefree ab+1)", "url": "https://www.erdosproblems.com/848" }, "keywords": [ "Erdős problem 848", "squarefree numbers", "extremal number theory", "combinatorial number theory", "computer-assisted proof", "certificate replay", "exact-rational certificates", "AI-generated mathematics", "reproducible research" ], "keyResults": [ "Claim: f(N) = ⌊(N+18)/25⌋ for every positive integer N, where f(N) is the largest size of A ⊆ {1,…,N} with ab+1 nonsquarefree for all a, b ∈ A (including a = b).", "1 ≤ N ≤ 100,000,006: exact finite colouring certificates with compact endpoint induction.", "10^8 ≤ N ≤ 10^9: exhaustive structural decomposition with exact lower-range validation.", "10^9 ≤ N ≤ 10^12: exact-rational short-shift envelopes.", "10^12 ≤ N ≤ 2.64 × 10^17: exact-rational rank envelopes.", "N ≥ 2.64 × 10^17: an explicit analytic threshold theorem (external, pinned source)." ], "evidencePackage": "A 75.6 MB archived deposit containing LaTeX/PDF documentation, a Python replay framework with semantic mutations and sanitizer controls, a C++ verifier for the range up to 100,000,006 with a compressed colouring-delta binary, SHA-256 manifests for all principal components, and machine-readable claim indexes (AI_INDEX.json/md, STATUS, ASSURANCE, PROVENANCE).", "openProblems": [ "Independently replay the certificates on different hardware and toolchains — the C++ verifier for N ≤ 10^8 with its compressed colouring-delta binary is the most accessible entry point.", "Independently verify the exact-rational envelope arguments covering 10^9 ≤ N ≤ 2.64 × 10^17, which are the least conventional part of the architecture.", "Confirm the hand-off: check that the pinned explicit-threshold theorem genuinely covers all N ≥ 2.64 × 10^17 under the same normalisation of f(N).", "Formalise the endpoint-induction scheme in a proof assistant.", "Ask whether the same five-regime architecture (certificates, structural decomposition, envelopes, analytic threshold) transfers to neighbouring extremal problems such as Erdős problem 844." ], "relatedWorks": [ { "citation": "Erdős problem 848 (Erdős & Sárközy): maximum size of A ⊆ {1,…,N} with ab+1 never squarefree — the problem page also presents van Doorn's upper bound |A| ≤ (0.108…+o(1))N.", "url": "https://www.erdosproblems.com/848" }, { "citation": "Sawhney, M. Resolution for all sufficiently large N, with the structural statement that near-extremal sets lie in {n ≡ 7 (mod 25)} or {n ≡ 18 (mod 25)}.", "url": "https://www.math.columbia.edu/~msawhney/Problem_848.pdf" }, { "citation": "Sothanaphan, N. An explicit threshold in Erdős Problem #848 — the pinned analytic input for N ≥ 2.64 × 10^17.", "url": "https://drive.google.com/file/d/1ujhm4_WYpgRV_rd1rJXIfHyvx16COEKe/view" } ] }, { "schemaVersion": "1.1", "slug": "degree-difference-affine-slices", "title": "The degree-difference principle and affine slices of binary-form factorisation spaces", "shortTitle": "Degree-difference principle and affine slices", "url": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/", "oneLine": "A clean determinant identity for the map that multiplies two polynomials and records their resultant — and a study of the geometric slices where Jacobian-conjecture-style questions live.", "abstract": "Multiply two polynomials together and, alongside the product, record their resultant — a single number measuring whether they share a root. This paper proves an exact formula for the Jacobian determinant of that combined operation: up to sign, it is the degree difference of the two polynomials times the square of their resultant. From this 'degree-difference principle' the paper develops the geometry of natural affine slices of binary-form factorisation spaces — the setting in which the project's related Jacobian-conjecture investigations take place — including classifications of low-degree slices, explicit coordinates for Keller maps, and Euler-characteristic obstructions in higher degrees.", "datePublished": "2026-07-28", "dateModified": "2026-07-28", "version": "0.1-candidate", "doi": "10.5281/zenodo.21647593", "doiUrl": "https://doi.org/10.5281/zenodo.21647593", "conceptDoi": "10.5281/zenodo.21647592", "pdfUrl": "https://github.com/ipitchford/degree-difference-affine-slices/releases/download/v0.1-candidate/paper.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/degree-difference-affine-slices/main/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21647593", "repoUrl": "https://github.com/ipitchford/degree-difference-affine-slices", "releaseUrl": "https://github.com/ipitchford/degree-difference-affine-slices/releases", "markdownUrl": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/index.md", "bibtexUrl": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/cite.bib", "audioUrl": "https://evidence-press.pages.dev/assets/audio/degree-difference-affine-slices.mp3", "imageUrl": "https://evidence-press.pages.dev/assets/og/degree-difference-affine-slices.png", "coverArtUrl": "https://evidence-press.pages.dev/assets/art/degree-difference-affine-slices.svg", "media": [], "authors": [ "OpenAI Codex / Anthropic models" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate manuscript. No peer review, no independent reproduction, and no documented external human review. Computational verification covers only the explicit low-degree polynomial identities (bidegrees r+s ≤ 4; sign checks to degree eight); the general all-degree determinant argument and the theoretical proofs (torsor, divisor-class, cubic-orbit classification) are asserted but not machine-checked. The manuscript is AI-attributed with human direction." }, "provenance": { "aiGenerated": true, "generatedBy": [ "OpenAI Codex / Anthropic models" ], "humanRole": "problem selection, mediation, and publication management", "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution." }, "problem": { "name": "Structure of binary-form factorisation spaces (adjacent to the Jacobian conjecture)", "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture" }, "keywords": [ "algebraic geometry", "binary forms", "resultant", "Sylvester matrix", "Jacobian conjecture", "Keller maps", "affine slices", "factorisation spaces", "computer-assisted mathematics", "symbolic verification", "SymPy" ], "keyResults": [ "Determinant identity: det DΦ_{r,s} = (−1)^{s(r+1)} (r−s) · Res(A,B)² for the multiplication–resultant map Φ_{r,s}(A,B) = (AB, Res(A,B)).", "Associated torsor and divisor-class statements for the factorisation spaces.", "Classification of normalised linear–quadratic slices.", "Explicit coordinates for tangent nonosculating slices and the induced Keller maps.", "Fibre and image characterisation of the map.", "Euler-characteristic obstructions for higher-degree cases." ], "evidencePackage": "Manuscript (PDF and TeX); a SymPy verification script with intentional negative-control tests, confirming the polynomial identities through bidegree r+s ≤ 4 and base-point signs through degree eight; a claim-level evidence map (AI_INDEX.md); assurance-boundary documentation (STATUS.md, ASSURANCE.md); provenance records; SHA-256 manifests; Zenodo and Software Heritage archives.", "openProblems": [ "Verify the all-degree determinant identity by an independent conceptual proof — or locate it in prior literature on resultants and multiplication maps, where such identities may already exist in another form.", "Subject the torsor and divisor-class arguments, which the symbolic checker deliberately does not cover, to expert human review.", "Extend the symbolic verification beyond bidegree 4 and automate the identity checks at general degree.", "Determine whether the Euler-characteristic obstructions can be upgraded to full non-isomorphism proofs in the higher-degree cases.", "Connect the classified slices explicitly to the companion releases on exotic affine spheres and reducible incidence divisors, which build on this framework." ], "relatedWorks": [ { "citation": "Keller, O.-H. (1939). Ganze Cremona-Transformationen — origin of the Jacobian conjecture, which remains open even for two variables.", "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture" }, { "citation": "Companion release: Exotic affine three-spheres and the quadratic–cubic obstruction.", "url": "https://doi.org/10.5281/zenodo.21647607" }, { "citation": "Companion release: Reducible incidence divisors and the isolation of affine slices.", "url": "https://doi.org/10.5281/zenodo.21647616" } ] } ] }