{ "schemaVersion": "1.2", "site": "Evidence Press", "baseUrl": "https://evidencepress.org", "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://evidencepress.org/api/schema.json", "count": 10, "papers": [ { "schemaVersion": "1.2", "slug": "random-reshuffling-baseline-policy", "title": "Random Reshuffling as a Baseline Policy: Exact Value Certificates and Common-Hessian Ordering Hardness", "shortTitle": "Random reshuffling as a baseline policy", "url": "https://evidencepress.org/releases/random-reshuffling-baseline-policy/", "oneLine": "In an exact quadratic model of learning, some fair choices inside ordinary data shuffling admit a same-baseline value certificate; popular ordering proxies can fail without bound, while global threshold ordering is NP-complete when the stable step is instance-encoded and approaches 2.", "abstract": "This anonymous, unrefereed candidate studies one-epoch data order in common-Hessian convex quadratics. It derives exact order-resolved terminal loss with a dense positive-semidefinite time kernel of rank at most the model dimension. A four-item rational family shows that global minimizers of either an ideal maximum-prefix criterion or a dynamics-weighted diagonal proxy can have an unbounded multiplicative terminal-loss ratio; the encoded theorem core is formalized in pinned Lean 4/mathlib. Decomposing a uniform random permutation into an unoriented skeleton and independent fair pair-orientation bits yields a restricted policy: exact conditional value resolves exposed bits with a pathwise decrement certificate and expected loss no greater than the same random-reshuffling law. For an instance-encoded stable step approaching 2, the candidate proves the scalar threshold problem NP-complete via PARTITION and, unless P=NP, excludes deterministic polynomial-time approximation within any fixed finite factor. It does not prove strong, fixed-step, randomized, or average-case hardness. A fixed stability margin instead admits normalized additive guarantees. A preregistered 32-instance matrix-free pilot was negative/mixed: every safety control passed and no false certified sign occurred on the frozen grid, but no eligible degree passed every feasibility stratum. No neural-network, model-scale, wall-clock, or generalization improvement is established.", "datePublished": "2026-08-02", "dateModified": "2026-08-02", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.21757895", "doiUrl": "https://doi.org/10.5281/zenodo.21757895", "conceptDoi": "10.5281/zenodo.21757894", "pdfUrl": "https://github.com/ipitchford/random-reshuffling-baseline-policy/releases/download/v1.0.0-candidate/random-reshuffling-baseline-policy-v1.0.0-candidate.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/random-reshuffling-baseline-policy/main/paper/paper.pdf", "zenodoUrl": "https://zenodo.org/records/21757895", "repoUrl": "https://github.com/ipitchford/random-reshuffling-baseline-policy", "releaseUrl": "https://github.com/ipitchford/random-reshuffling-baseline-policy/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/random-reshuffling-baseline-policy/index.md", "bibtexUrl": "https://evidencepress.org/releases/random-reshuffling-baseline-policy/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/random-reshuffling-baseline-policy.mp3", "imageUrl": "https://evidencepress.org/assets/og/random-reshuffling-baseline-policy.png", "coverArtUrl": "https://evidencepress.org/assets/art/random-reshuffling-baseline-policy.svg", "media": [ { "type": "video", "url": "https://youtu.be/HKknZ3QlGkk", "name": "Video briefing — Random reshuffling as a baseline policy", "description": "Video overview of this release on the Evidence Press YouTube channel." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed public candidate. The internal Stage 3-prime editorial outcome is Major Revision. Only the declared encoded core of Theorem 2 is formalized in Lean; the rest of the paper is conventionally proved and locally checked. Three provenance-separated external work packages remain open, covering the RR/value identity, policy and Bellman/full-conditional-expectation reconstruction, and approximation complexity. Independent reproduction, external specialist review, human peer review, whole-paper formal verification, novelty or priority certification, and evidence of neural-network or model-scale gain are absent. A DOI, release certificate, passing replay, or agreement among producer-workflow models does not establish mathematical truth or acceptance." }, "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/21757895", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Anonymous, unrefereed public candidate. The internal Stage 3-prime editorial outcome is Major Revision. Only the declared encoded core of Theorem 2 is formalized in Lean; the rest of the paper is conventionally proved and locally checked. Three provenance-separated external work packages remain open, covering the RR/value identity, policy and Bellman/full-conditional-expectation reconstruction, and approximation complexity. Independent reproduction, external specialist review, human peer review, whole-paper formal verification, novelty or priority certification, and evidence of neural-network or model-scale gain are absent. A DOI, release certificate, passing replay, or agreement among producer-workflow models does not establish mathematical truth or acceptance." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "OpenAI and Anthropic research systems" ], "humanRole": "Research direction and publication authorization; the scholarly creator is cited as Anonymous.", "disclosure": "Generative systems assisted hypothesis generation, mathematical drafting, code generation, adversarial checking, source triage, editing, and release preparation. They are not authors and their agreement is not independent verification." }, "problem": { "name": "Certified data-order control relative to random reshuffling", "url": "https://proceedings.mlr.press/v23/recht12.html" }, "corrections": [], "keywords": [ "random reshuffling", "stochastic gradient descent", "data ordering", "conditional expectation", "common-Hessian quadratics", "permutation optimization", "value-aware ordering", "computational complexity", "NP-completeness", "Lean 4", "formal verification", "reproducible research", "machine learning theory", "unrefereed candidate" ], "keyResults": [ "G1: Exact common-Hessian terminal loss factors through a dense positive-semidefinite time kernel whose rank is at most the model dimension.", "S1: In an exact four-item rational family, every global minimizer of either an ideal maximum-prefix criterion or a dynamics-weighted diagonal proxy has an unbounded multiplicative terminal-loss ratio; the encoded theorem core is formalized in pinned Lean 4/mathlib.", "P1-P2: A uniform random permutation decomposes into an unoriented skeleton and independent fair orientation bits. Resolving exposed bits by exact conditional value gives realized decrement terms and a conditional identity whose expected one-epoch loss is no greater than the same RR law under the stated predictable-exposure coupling.", "C1: With an instance-encoded stable step approaching 2, Scalar Common-Hessian Final-Loss Ordering is NP-complete via PARTITION and has no deterministic polynomial-time fixed-factor approximation unless P=NP. Strong, fixed-step, randomized, and average-case hardness are not established.", "C2: At a fixed margin from the stability endpoint, ordered-suffix enumeration gives normalized additive endpoint and loss guarantees plus an optimum interval; this is not a multiplicative PTAS or FPTAS.", "E2: The preregistered 32-instance exact-arithmetic matrix-free pilot passed all safety controls and produced no false certified sign on its frozen grid, but failed its overall feasibility gate and selected no approximation degree." ], "reviews": [], "evidencePackage": "A 55-page anonymous candidate paper; canonical Markdown, TeX, and bibliography; exact rational Python verifiers and mutation controls; 91 tests under ordinary and optimized Python; a pinned Lean 4.32.1/mathlib formalization of the encoded Theorem 2 core with 15 manuscript anchors and no project-local proof placeholders; a preregistered 32-instance matrix-free pilot; manifests, receipts, deterministic tagged-tree archive tooling, release replay, and a detached publication certificate. These are producer-side checks, not independent reproduction.", "openProblems": [ "Prove a perturbation theorem for heterogeneous or drifting component Hessians with an explicit margin that certifies when a common-Hessian orientation decision remains valid.", "Replace the conservative binomial-action bounds with low-rank, diagonal, Krylov, sketching, or probabilistic error controls that maximize certified sign coverage while charging preprocessing, memory, communication, and Hessian-vector products.", "Run a preregistered transfer ladder from expanded synthetic generators to frozen-backbone or LoRA workloads and then small nonlinear networks, comparing RR, established balancing baselines, exact LAPO, and certified approximations on net time or energy to a fixed target.", "Close the exact complexity gap between near-endpoint NP-completeness and fixed-margin normalized additive tractability: determine fixed-step complexity, strong hardness, the nonoscillatory range, and structured or pseudo-polynomial cases.", "Characterize joint matching and decision-order selection: test approximate submodularity, exchange properties, greedy guarantees, and hardness on small exact instances.", "Extend the finite and discounted value theorems to decaying steps, variance reduction, interpolation, or an average-cost objective with the required uniform-integrability or transversality conditions.", "Independently reconstruct the paper's central proofs and executable results without using the production verifiers, and obtain specialist audits of the probability/value identities and complexity reduction." ], "relatedWorks": [ { "citation": "Recht, B., & Ré, C. (2012). Toward a Noncommutative Arithmetic-Geometric Mean Inequality: Conjectures, Case-studies, and Consequences. COLT 2012.", "url": "https://proceedings.mlr.press/v23/recht12.html" }, { "citation": "Lai, Z., & Lim, L.-H. (2020). Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False. ICML 2020.", "url": "https://proceedings.mlr.press/v119/lai20a.html" }, { "citation": "De Sa, C. M. (2020). Random Reshuffling is Not Always Better. NeurIPS 2020.", "url": "https://proceedings.neurips.cc/paper/2020/hash/42299f06ee419aa5d9d07798b56779e2-Abstract.html" }, { "citation": "Lu, Y., Guo, W., & De Sa, C. M. (2022). GraB: Finding Provably Better Data Permutations than Random Reshuffling. NeurIPS 2022.", "url": "https://proceedings.neurips.cc/paper_files/paper/2022/hash/3acb49252187efa352a1ae0e4b066ced-Abstract-Conference.html" }, { "citation": "Cooper, A. F., et al. (2023). Coordinating Distributed Example Orders for Provably Accelerated Training. NeurIPS 2023.", "url": "https://proceedings.neurips.cc/paper_files/paper/2023/hash/af9ac087ed9123957bb3a45dca56b9d4-Abstract-Conference.html" }, { "citation": "Kubo, S., Makino, K., & Sakamoto, S. (2024). Composition Orderings for Linear Functions and Matrix Multiplication Orderings. ISAAC 2024.", "url": "https://doi.org/10.4230/LIPIcs.ISAAC.2024.44" }, { "citation": "Pearlmutter, B. A. (1994). Fast Exact Multiplication by the Hessian. Neural Computation, 6(1), 147-160.", "url": "https://doi.org/10.1162/neco.1994.6.1.147" } ] }, { "schemaVersion": "1.2", "slug": "stocks-are-not-flows", "title": "Stocks Are Not Flows: An Identification Audit of Investor Pressure and Tenure Change in England", "shortTitle": "Stocks are not flows", "url": "https://evidencepress.org/releases/stocks-are-not-flows/", "oneLine": "England's tenure shift is real; the aggregate housing data do not reveal how much, if any, was caused by rising resources at the top.", "abstract": "Among English households whose household reference person was aged 25 to 44, owner occupation fell by 11.95 percentage points and private renting rose by 12.33 points between 2008-09 and 2018-19. This reproducible audit shows why those near-mirror movements, together with public price, rent, mortgage-rate, supply, and dwelling-stock series, cannot uniquely attribute any share of the shift to wealthy investors. The registered design admits observationally equivalent credit, supply, demographic, tax, and resource mechanisms, so the planned structural estimation was stopped. The result is neither a positive nor a zero causal estimate: it is a documented identification boundary and a concrete agenda for better measurement.", "datePublished": "2026-08-01", "dateModified": "2026-08-01", "version": "1.0.0", "doi": "10.5281/zenodo.21740339", "doiUrl": "https://doi.org/10.5281/zenodo.21740339", "conceptDoi": "10.5281/zenodo.21740338", "pdfUrl": "https://github.com/ipitchford/stocks-are-not-flows/releases/download/v1.0.0-preprint/stocks-are-not-flows-v1.0.0.pdf", "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/stocks-are-not-flows/main/paper/stocks-are-not-flows-v1.0.0.pdf", "zenodoUrl": "https://zenodo.org/records/21740339", "repoUrl": "https://github.com/ipitchford/stocks-are-not-flows", "releaseUrl": "https://github.com/ipitchford/stocks-are-not-flows/releases/tag/v1.0.0-preprint", "markdownUrl": "https://evidencepress.org/releases/stocks-are-not-flows/index.md", "bibtexUrl": "https://evidencepress.org/releases/stocks-are-not-flows/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/stocks-are-not-flows.mp3", "imageUrl": "https://evidencepress.org/assets/og/stocks-are-not-flows.png", "coverArtUrl": "https://evidencepress.org/assets/art/stocks-are-not-flows.svg", "media": [ { "type": "video", "url": "https://youtu.be/L-a2vj9b8zM", "name": "Video briefing — Stocks are not flows", "description": "Narrated video overview of this release on the Evidence Press YouTube channel." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-preprint", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed public discussion preprint. The public accounting paths and registered synthetic design checks were deterministically replayed, but the work has not been independently reproduced, peer reviewed, formally verified, or accepted by experts. The response design is synthetic rather than an estimated model of England. The audit establishes a design-conditioned non-identification result, not a positive or zero causal effect of wealthy investors." }, "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/21740339", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Anonymous, unrefereed public discussion preprint. The public accounting paths and registered synthetic design checks were deterministically replayed, but the work has not been independently reproduced, peer reviewed, formally verified, or accepted by experts. The response design is synthetic rather than an estimated model of England. The audit establishes a design-conditioned non-identification result, not a positive or zero causal effect of wealthy investors." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": false, "aiAssisted": true, "generatedBy": [ "Automated research and coding tools" ], "humanRole": "Project selection, mediation, review, and publication management; the scholarly creator is cited as Anonymous.", "disclosure": "Automated tools assisted discovery, implementation, testing, critique, and drafting; they are not authors. The public discussion preprint is cited as Anonymous." }, "problem": null, "corrections": [], "keywords": [ "housing tenure", "private renting", "homeownership", "wealth inequality", "housing investors", "structural identification", "partial identification", "English Housing Survey", "reproducible research", "public discussion preprint" ], "keyResults": [ "For English households with a household reference person aged 25-44, the reconstructed owner share fell from 60.5270% in 2008-09 to 48.5783% in 2018-19, while private renting rose from 22.1325% to 34.4594%.", "The near-offsetting stock changes do not reveal gross owner-to-landlord transactions, purchaser type, purchaser resources, or the origin of funds.", "In the registered synthetic response design, the focal top-resource response lies in the span of admitted rival and nuisance responses, so the planned structural fit cannot uniquely attribute the historical shift.", "The correct decision is FAIL-IDENT-0: stop unique causal attribution, publish the boundary, and move to better measurement and source-backed partial identification.", "The audit reports no causal coefficient, no historical contribution share, and no empirical England partial-identification interval; it also does not establish that wealthy investors had no effect." ], "reviews": [], "evidencePackage": "A 30-page anonymous preprint; Python source and a pinned dependency lock; 135 tests passed under normal and optimized Python; frozen IDENT-0, public Gate A, deterministic replay, and partial-identification reports; seven redistributed official public-source objects with receipts and Open Government Licence attribution; derived accounting tables and deterministic figures; SHA-256 manifests, negative and sign-reversing controls, and a publication-integrity report.", "openProblems": [ "Construct a linked measurement that jointly records gross property-use conversion, purchaser landlord status, and the purchaser's pre-purchase non-housing resources, while documenting linkage error, anticipation, legal-form switching, reclassification, and equilibrium spillovers.", "Develop source-backed bounds for the focal and rival response classes, then compute an empirical sharp identified set for England; near-zero, signed, and sign-ambiguous results must all remain admissible.", "Reproduce the public accounting layer independently from the original official downloads and implement the observation-map audit without reusing the repository's code or intermediate files.", "Obtain the survey-design information needed to attach defensible uncertainty intervals to the combined 25-44 English Housing Survey tenure shares.", "Compare alternative linked-listing definitions of investor acquisition and owner-to-landlord conversion, and quantify their sensitivity to listing windows, platform coverage, purchaser type, and funding-source misclassification.", "Use the discriminator theorem as a pre-analysis value-of-information test for candidate administrative, tax, mortgage, ownership, and transaction linkages before committing to a full structural estimation." ], "relatedWorks": [ { "citation": "Stevenson, G. (2019). The Impact of Inequality on Asset Prices When Households Care About Wealth. MPhil thesis manuscript; Oxford INET seminar record.", "url": "https://www.inet.ox.ac.uk/events/the-impact-of-inequality-on-asset-prices-when-households-care-about-wealth" }, { "citation": "Head, A., Lloyd-Ellis, H., & Stacey, D. (2023). Heterogeneity, Frictional Assignment, and Home-Ownership. International Economic Review, 64(3), 1265-1308.", "url": "https://doi.org/10.1111/iere.12624" }, { "citation": "Bracke, P. (2021). How Much Do Investors Pay for Houses? Real Estate Economics, 49(S1), 41-73.", "url": "https://doi.org/10.1111/1540-6229.12280" }, { "citation": "Bank of England (2023). Has the Private Rental Sector Been Shrinking? Bank Overground.", "url": "https://www.bankofengland.co.uk/bank-overground/2023/has-the-private-rental-sector-been-shrinking" }, { "citation": "Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer.", "url": "https://doi.org/10.1007/b97478" }, { "citation": "Tamer, E. (2010). Partial Identification in Econometrics. Annual Review of Economics, 2, 167-195.", "url": "https://doi.org/10.1146/annurev.economics.050708.143401" } ] }, { "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://evidencepress.org/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://evidencepress.org/releases/irreducible-pushforwards-quartic-transitions/index.md", "bibtexUrl": "https://evidencepress.org/releases/irreducible-pushforwards-quartic-transitions/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/irreducible-pushforwards-quartic-transitions.mp3", "imageUrl": "https://evidencepress.org/assets/og/irreducible-pushforwards-quartic-transitions.png", "coverArtUrl": "https://evidencepress.org/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." }, "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/21745937", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "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" }, "corrections": [], "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." ], "reviews": [], "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" } ] }, { "schemaVersion": "1.2", "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://evidencepress.org/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://evidencepress.org/releases/exact-low-length-recht-re-inequalities/index.md", "bibtexUrl": "https://evidencepress.org/releases/exact-low-length-recht-re-inequalities/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/exact-low-length-recht-re-inequalities.mp3", "imageUrl": "https://evidencepress.org/assets/og/exact-low-length-recht-re-inequalities.png", "coverArtUrl": "https://evidencepress.org/assets/art/exact-low-length-recht-re-inequalities.svg", "media": [ { "type": "video", "url": "https://youtu.be/PXBlZLk753g", "name": "Video briefing — Exact low-length Recht–Ré inequalities", "description": "Narrated video overview of this release on the Evidence Press YouTube channel." } ], "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." }, "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/21709239", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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." ], "reviews": [], "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": "1.2", "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://evidencepress.org/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://evidencepress.org/releases/z20-equals-6/index.md", "bibtexUrl": "https://evidencepress.org/releases/z20-equals-6/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/z20-equals-6.mp3", "imageUrl": "https://evidencepress.org/assets/og/z20-equals-6.png", "coverArtUrl": "https://evidencepress.org/assets/art/z20-equals-6.svg", "media": [ { "type": "video", "url": "https://youtu.be/m6LBNDwhs5I", "name": "Video explainer — Solving Erdős 758", "description": "Video explainer for this release on the Evidence Press YouTube channel." } ], "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." }, "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/21647645", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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." ], "reviews": [], "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": "1.2", "slug": "vr2-k4-equals-20", "title": "VR2(K4) = 20: twenty vertices force two vertex-disjoint monochromatic K4s", "shortTitle": "VR2(K4) = 20", "url": "https://evidencepress.org/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://evidencepress.org/releases/vr2-k4-equals-20/index.md", "bibtexUrl": "https://evidencepress.org/releases/vr2-k4-equals-20/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/vr2-k4-equals-20.mp3", "imageUrl": "https://evidencepress.org/assets/og/vr2-k4-equals-20.png", "coverArtUrl": "https://evidencepress.org/assets/art/vr2-k4-equals-20.svg", "media": [ { "type": "video", "url": "https://youtu.be/lsPcDxhbUkk", "name": "Video explainer — VR2(K4) = 20", "description": "Video explainer for this release on the Evidence Press YouTube channel." } ], "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." }, "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/21647654", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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)." ], "reviews": [], "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": "1.2", "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://evidencepress.org/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://evidencepress.org/releases/reducible-incidence-divisors/index.md", "bibtexUrl": "https://evidencepress.org/releases/reducible-incidence-divisors/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/reducible-incidence-divisors.mp3", "imageUrl": "https://evidencepress.org/assets/og/reducible-incidence-divisors.png", "coverArtUrl": "https://evidencepress.org/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." }, "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/21647616", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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." ], "reviews": [], "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": "1.2", "slug": "exotic-affine-three-spheres", "title": "Exotic affine three-spheres and the quadratic–cubic obstruction", "shortTitle": "Exotic affine three-spheres", "url": "https://evidencepress.org/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://evidencepress.org/releases/exotic-affine-three-spheres/index.md", "bibtexUrl": "https://evidencepress.org/releases/exotic-affine-three-spheres/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/exotic-affine-three-spheres.mp3", "imageUrl": "https://evidencepress.org/assets/og/exotic-affine-three-spheres.png", "coverArtUrl": "https://evidencepress.org/assets/art/exotic-affine-three-spheres.svg", "media": [ { "type": "video", "url": "https://youtu.be/dQtm5zfWJRk", "name": "Video briefing — Exotic affine three-spheres", "description": "Narrated video overview of this release on the Evidence Press YouTube channel." } ], "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." }, "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/21647607", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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." ], "reviews": [], "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": "1.2", "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://evidencepress.org/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://evidencepress.org/releases/erdos-848-all-n/index.md", "bibtexUrl": "https://evidencepress.org/releases/erdos-848-all-n/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/erdos-848-all-n.mp3", "imageUrl": "https://evidencepress.org/assets/og/erdos-848-all-n.png", "coverArtUrl": "https://evidencepress.org/assets/art/erdos-848-all-n.svg", "media": [ { "type": "video", "url": "https://youtu.be/h2dUsd0F-Sk", "name": "Video explainer — Erdős 848: an exact answer for every N", "description": "Video explainer for this release on the Evidence Press YouTube channel." } ], "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." }, "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/21647629", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [ { "date": "2026-08-05", "scope": "presentation", "fixedIn": "Evidence Press 1.1.0", "summary": "From publication until 5 August 2026 this page displayed the main result as holding “for every $N$ from $\\\\lfloor (N+18)/25 \\\\rfloor$ upwards”. It should have read, and now reads, “for every $N$ from 1 upwards”.", "detail": "The cause was a defect in this site's Markdown renderer, which used plain integers as internal placeholders for mathematics and could therefore replace a literal number in prose with an unrelated formula. The error was introduced in rendering only: the archived deposit, the manuscript PDF, and this release's Markdown and JSON representations always carried the correct statement. No result, proof, certificate or item of evidence is affected, and the release's assurance state is unchanged." } ], "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)." ], "reviews": [], "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": "1.2", "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://evidencepress.org/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://evidencepress.org/releases/degree-difference-affine-slices/index.md", "bibtexUrl": "https://evidencepress.org/releases/degree-difference-affine-slices/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/degree-difference-affine-slices.mp3", "imageUrl": "https://evidencepress.org/assets/og/degree-difference-affine-slices.png", "coverArtUrl": "https://evidencepress.org/assets/art/degree-difference-affine-slices.svg", "media": [ { "type": "video", "url": "https://youtu.be/imXySLswk3M", "name": "Video explainer — the degree-difference principle", "description": "Video explainer for this release on the Evidence Press YouTube channel." } ], "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." }, "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/21647593", "note": "Archived Zenodo deposit under a DOI." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "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." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed" }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed" }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed" }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed" }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed" }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed" } ], "provenance": { "aiGenerated": true, "aiAssisted": 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" }, "corrections": [], "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." ], "reviews": [], "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" } ] } ] }