{ "name": "State of Proof Paper Watch", "url": "https://stateofproof.org/paper-watch", "description": "A public intake tracker for consequential new mathematical claims selected for possible examination.", "updated": "2026-09-20T13:03:54Z", "contentModified": "2026-09-20T13:03:54Z", "notice": "Inclusion is not validation. Every item is unexamined unless a separate proof docket records completed work.", "counts": { "total": 67, "docketReady": 13, "candidate": 48, "watch": 6 }, "papers": [ { "id": "arxiv-2609-20809", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20809", "contentModified": "2026-09-20T13:03:54Z", "arxivId": "2609.20809", "intakeDate": "2026-09-20", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "The linear instability of Kasner spacetimes", "authors": "Oliver Petersen.", "source": { "url": "https://arxiv.org/abs/2609.20809v1", "version": "submitted 2026-09-17 17:58:33 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-20T13:03:54Z" }, "attributedClaim": "The abstract says the paper proves linear stability toward the Big Bang modulo an explicit finite-dimensional non-decaying self-similar space, then gives a complete linear description of the expected instability without symmetry assumptions. Because the title foregrounds instability while the first abstract sentence foregrounds stability modulo modes, the exact theorem statement and conventions are a required first check.", "publicOverview": { "updated": "2026-09-20", "what": "A paper analyzes which small disturbances of an idealized early-universe spacetime persist or grow in the linearized equations.", "who": "Oliver Petersen", "meaning": "If the stated analysis holds, it sharpens a mathematical map of which modes matter near a singularity. It does not by itself describe our universe or settle the nonlinear Einstein equations." }, "whyItMatters": "a fresh, source-backed PDE/geometry theorem with a precise linearized scope. Intake does not resolve nonlinear Einstein evolution, establish a Big-Bang model, or verify the paper’s analysis.", "publicImpact": { "updated": "2026-09-20", "headline": "Which ripples survive near a spacetime singularity?", "plainEnglish": "Kasner spacetimes are exact solutions used to study an extreme mathematical limit of gravity. This paper claims to sort their linear disturbances into decaying behavior and a finite set of persistent self-similar modes.", "ifHolds": "Foundational: it would give a sharper linear map for a difficult PDE-and-geometry regime, including the stated Taub-transition mode.", "ifFails": "The exact exceptional modes, stability norm, or linearization may need revision, clarifying where the proposed early-time picture stops applying.", "horizon": "Foundational", "areas": [ "PDE", "Differential geometry", "Mathematical relativity" ] }, "whyTracked": "a fresh, source-backed PDE/geometry theorem with a precise linearized scope. Intake does not resolve nonlinear Einstein evolution, establish a Big-Bang model, or verify the paper’s analysis.", "availableArtifacts": "arXiv exposes PDF, experimental HTML, and TeX source. The inspected record names no formal proof, code repository, numerical certificate, or executable artifact; AI involvement is not established.", "proposedCheckRoute": "Read the main theorem and definitions of the linearized variables, quasinormal modes, stability norm, and excluded self-similar modes; then check exactly how the Taub Bianchi-II linearization and the claimed no-symmetry conclusion follow. Do not execute manuscript artifacts.", "highestRiskDependency": "The abstract’s stability-modulo-modes and instability language must not be flattened into either a nonlinear stability theorem or a general cosmological prediction. Gauge choice, norm, time direction, and the finite-dimensional exceptional space are load-bearing.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Read the main theorem and definitions of the linearized variables, quasinormal modes, stability norm, and excluded self-similar modes; then check exactly how the Taub Bianchi-II linearization and the claimed no-symmetry conclusion follow. Do not execute manuscript artifacts." }, { "id": "arxiv-2609-20811", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20811", "contentModified": "2026-09-19T13:04:00Z", "arxivId": "2609.20811", "intakeDate": "2026-09-19", "disposition": "watch", "examinationState": "not-started", "validationResult": null, "title": "A note on generating polyhedra and quadrangulations", "authors": "Luisa Andreis, Riccardo W. Maffucci, and Federico Polito.", "source": { "url": "https://arxiv.org/abs/2609.20811v1", "version": "submitted 2026-09-17 17:58:54 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-19T13:04:00Z" }, "attributedClaim": "The abstract says two graph transformations generate every polyhedron other than pyramids from the square pyramid; it separately gives a unique-transformation construction for the specified facial-four-cycle sphere quadrangulations. It is a current constructive classification claim with a bounded combinatorial check surface.", "publicOverview": { "updated": "2026-09-19", "what": "A claimed compact recipe for building broad classes of polyhedral graphs and sphere quadrangulations.", "who": "Luisa Andreis, Riccardo W. Maffucci, and Federico Polito.", "meaning": "If complete, the transformations would give a more economical constructive view of the stated graph families." }, "whyItMatters": "a fresh, source-backed constructive graph-theory result. The exact transformations, completeness argument and excluded classes remain unexamined.", "publicImpact": { "updated": "2026-09-19", "headline": "Two moves build many polyhedral graphs", "plainEnglish": "The work asks whether complex graph families can grow from one seed through a few reliable moves. Such recipes can make a large mathematical family easier to organize and explore.", "ifHolds": "The construction would provide a concise route to generate the stated non-exceptional polyhedra and related quadrangulations.", "ifFails": "An omitted graph family or failed inverse step would locate the limit of the proposed construction.", "horizon": "Foundational", "areas": [ "Graph theory", "Combinatorics", "Constructive mathematics" ] }, "whyTracked": "a fresh, source-backed constructive graph-theory result. The exact transformations, completeness argument and excluded classes remain unexamined.", "availableArtifacts": "arXiv exposes PDF, experimental HTML and TeX source. No formal proof artifact, code repository, certificate or AI role is identified in the inspected record.", "proposedCheckRoute": "Inspect the exact transformations, invariants and inverse/reduction argument; verify the scope of pyramids, antibipyramids and facial four-cycles before treating the construction as complete.", "highestRiskDependency": "A local transformation description does not establish completeness or uniqueness without the paper's reduction and exceptional-class argument.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Inspect the exact transformations, invariants and inverse/reduction argument; verify the scope of pyramids, antibipyramids and facial four-cycles before treating the construction as complete." }, { "id": "arxiv-2609-20785", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20785", "contentModified": "2026-09-19T13:04:00Z", "arxivId": "2609.20785", "intakeDate": "2026-09-19", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "Boolean Small-Ball Inequalities for Discrepancy Theory", "authors": "Emrullah Akbas and Suvrit Sra.", "source": { "url": "https://arxiv.org/abs/2609.20785v1", "version": "submitted 2026-09-17 17:52:14 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-19T13:04:00Z" }, "attributedClaim": "The abstract states a small-ball inequality for Boolean matrix series under bounded trace and variance conditions, then claims an interlacing-free proof of Kadison–Singer and further discrepancy corollaries. The explicit method and dependencies make the claimed advance an appropriate unexamined proof-method intake.", "publicOverview": { "updated": "2026-09-19", "what": "A proposed route for controlling signed matrix sums with a small-ball inequality.", "who": "Emrullah Akbas and Suvrit Sra.", "meaning": "If the argument holds, one proof technique may connect several discrepancy results without spectral interlacing." }, "whyItMatters": "a fresh, source-backed discrepancy proof method with a stated main inequality and claimed consequence chain. Intake does not independently establish the inequality, its dependencies, or the asserted Kadison–Singer route.", "publicImpact": { "updated": "2026-09-19", "headline": "A new route through matrix discrepancy", "plainEnglish": "The paper studies how random plus-or-minus choices can keep a matrix sum controlled. It offers a different proof mechanism for results about balancing many competing effects.", "ifHolds": "The method could give mathematicians another reusable way to derive matrix-balancing results and inspect which assumptions carry the work.", "ifFails": "Pinpointing a failed inequality or dependency would clarify which part of the proposed proof route cannot support its advertised consequences.", "horizon": "Methods", "areas": [ "Discrepancy theory", "Matrix analysis", "Proof methods" ] }, "whyTracked": "a fresh, source-backed discrepancy proof method with a stated main inequality and claimed consequence chain. Intake does not independently establish the inequality, its dependencies, or the asserted Kadison–Singer route.", "availableArtifacts": "arXiv exposes PDF, experimental HTML and TeX source. The inspected source lists no formalization, code repository, certificate or replay package; AI involvement is not established.", "proposedCheckRoute": "Inspect the main inequality and every boundedness hypothesis; then trace the stated reciprocal-estimate, signing-theorem and replica dependencies to the claimed Kadison–Singer consequence. Do not execute source artifacts for this intake.", "highestRiskDependency": "The abstract-level consequence chain may conceal representation, normalization, or prerequisite assumptions; a new proof route does not by itself establish each advertised corollary.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Inspect the main inequality and every boundedness hypothesis; then trace the stated reciprocal-estimate, signing-theorem and replica dependencies to the claimed Kadison–Singer consequence. Do not execute source artifacts for this intake." }, { "id": "arxiv-2609-20492", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20492", "contentModified": "2026-09-18T14:36:13Z", "arxivId": "2609.20492", "intakeDate": "2026-09-18", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "Nonexistence of a Leech Tree of Order 18: A Computer-Assisted Proof", "authors": "Maseeh Ghodsi.", "source": { "url": "https://arxiv.org/abs/2609.20492v1", "version": "submitted 2026-09-17 14:41:05 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-18T14:36:13Z" }, "attributedClaim": "The abstract claims that no Leech tree of order 18 exists. It says Lean 4 verifies structural facts reducing a putative example to eight local configurations; conventional mathematics establishes an exact-cover condition and search completeness; exhaustive computation closes the eight cases. The source explicitly calls this a computer-assisted proof rather than an end-to-end Lean formalization.", "publicOverview": { "updated": "2026-09-18", "what": "A paper claims a weighted tree whose distances are exactly 1 through 153 cannot exist when it has 18 vertices.", "who": "Maseeh Ghodsi", "meaning": "A finite graph puzzle can be impossible for reasons a computer helps expose. This work also makes the proof boundary visible: Lean checks some structure, while a separate search still needs its own scrutiny." }, "whyItMatters": "a fresh computer-assisted finite nonexistence claim with a clearly stated formal/computational boundary. Intake is not an end-to-end proof validation or an independent reproduction.", "publicImpact": { "updated": "2026-09-18", "headline": "When a proof has both a kernel and a search", "plainEnglish": "Can one weighted tree realize every whole-number distance from 1 through 153 exactly once? This paper says no—and carefully divides its evidence between Lean-checked reductions and an exhaustive search.", "ifHolds": "Methods: it would resolve this finite graph puzzle while offering a candid map of what a proof kernel certifies and what remains in the computation-and-checker trust boundary.", "ifFails": "The split record helps locate the problem: a formal reduction, a conventional argument, the search program, its run, or the checker may need correction.", "horizon": "Methods", "areas": [ "Graph theory", "Computer-assisted proof", "Formal verification" ] }, "whyTracked": "a fresh computer-assisted finite nonexistence claim with a clearly stated formal/computational boundary. Intake is not an end-to-end proof validation or an independent reproduction.", "availableArtifacts": "The arXiv record links a version-v1.0.0 Lean artifact, a version-v1.0.0 computational-evidence release, and an earlier Zenodo preprint. The source expressly says the search program, its execution, and certificate checker are not formalized in Lean. No artifact was downloaded, executed, or replayed.", "proposedCheckRoute": "Source-lock the tagged artifacts; verify hashes and provenance; inspect the Lean declarations and axiom closure; then map the structural reduction to the separate computation/checker boundary. Any future execution must be isolated and must not silently turn a component replay into a whole-proof verdict.", "highestRiskDependency": "Kernel checking of the structural layer does not certify the unformalized search, its execution, or the theorem-to-artifact correspondence. Conversely, a gap in the computational layer would not automatically negate every formal structural fact.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Source-lock the tagged artifacts; verify hashes and provenance; inspect the Lean declarations and axiom closure; then map the structural reduction to the separate computation/checker boundary. Any future execution must be isolated and must not silently turn a component replay into a whole-proof verdict." }, { "id": "arxiv-2609-19536", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-19536", "contentModified": "2026-09-18T14:36:13Z", "arxivId": "2609.19536", "intakeDate": "2026-09-18", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "The smallest square tileable by pairwise incomparable integer rectangles", "authors": "George M. Georgiou.", "source": { "url": "https://arxiv.org/abs/2609.19536v1", "version": "submitted 2026-09-17 01:00:56 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-18T14:36:13Z" }, "attributedClaim": "The abstract claims that for every integer (nleq26) and every number of tiles (kgeq2), the (n× n) square has no tiling by pairwise incomparable integer rectangles, establishing that the known 27-by-27 example is smallest. Its finite stage is reported to search 167,538 candidates using two independently written programs.", "publicOverview": { "updated": "2026-09-18", "what": "A paper claims that a 27-by-27 square is the smallest square tileable by integer rectangles no two of which contain one another in both dimensions.", "who": "George M. Georgiou", "meaning": "An apparently tiny tiling puzzle can hide a hard impossibility proof. If this holds, every smaller square is ruled out—and the public search records show exactly where finite checking enters the argument." }, "whyItMatters": "a fresh finite combinatorial theorem with public dual-implementation evidence artifacts. Intake does not mean the finite search, logs, checksums, or theorem correspondence have been independently replayed.", "publicImpact": { "updated": "2026-09-18", "headline": "Why 26-by-26 may be forever too small", "plainEnglish": "Cut a square into integer-sided rectangles, but forbid any rectangle from being at least as wide and tall as another. A 27-by-27 example is known; this paper claims every smaller square is impossible.", "ifHolds": "Foundational: it would close a clean finite geometry puzzle and show how structural reasoning can shrink an enormous search to a checkable set of cases.", "ifFails": "A missing tile family, an incomplete reduction, or a mismatched search record would show exactly where the claimed impossibility needs repair.", "horizon": "Foundational", "areas": [ "Combinatorics", "Discrete geometry", "Computer-assisted proof" ] }, "whyTracked": "a fresh finite combinatorial theorem with public dual-implementation evidence artifacts. Intake does not mean the finite search, logs, checksums, or theorem correspondence have been independently replayed.", "availableArtifacts": "arXiv lists a complete ancillary verification package with two implementations, build instructions, SHA256SUMS, C/Python source, output logs, and certificate/round-trip logs. No source, log, checksum, or program was downloaded, executed, or replayed.", "proposedCheckRoute": "First source-lock and hash-check the ancillary release, then examination the structural reduction from the theorem to the finite candidate list. Any dual-implementation replay belongs in a separate isolated environment and must compare its output to the stated 167,538 count and exact (n,k) scope.", "highestRiskDependency": "Finite enumeration supports only the candidate universe reached by the structural reductions. A successful rerun would not establish the theorem if the reduction omits a permitted rectangle family, changes incomparability conventions, or mismatches the stated quantifiers.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: First source-lock and hash-check the ancillary release, then examination the structural reduction from the theorem to the finite candidate list. Any dual-implementation replay belongs in a separate isolated environment and must compare its output to the stated 167,538 count and exact (n,k) scope." }, { "id": "arxiv-2609-19118", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-19118", "contentModified": "2026-09-18T14:36:13Z", "arxivId": "2609.19118", "intakeDate": "2026-09-18", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "On the Strong Matroid Secretary Conjecture and Beyond", "authors": "Hamed Abdi, Kiarash Banihashem, MohammadTaghi Hajiaghayi, and Danny Mittal.", "source": { "url": "https://arxiv.org/abs/2609.19118v1", "version": "submitted 2026-09-16 17:43:06 UTC", "sourceDate": "2026-09-16", "retrieved": "2026-09-18T14:36:13Z" }, "attributedClaim": "The abstract claims a 1/e-competitive ordinal secretary algorithm for every linear matroid, maintaining expected-intersection-dimension bounds while selecting online. It separately claims a 1/2 single-sample prophet algorithm for arbitrary matroids and a black-box 1/64 secretary reduction. This is a newly posted, concrete advance on structured online selection, not a blanket result for all allocation problems.", "publicOverview": { "updated": "2026-09-18", "what": "A paper claims an online algorithm can choose a strong feasible set even when options appear one by one.", "who": "Hamed Abdi, Kiarash Banihashem, MohammadTaghi Hajiaghayi, and Danny Mittal", "meaning": "Choosing now can mean missing the best offer that arrives later. If this holds, it gives a sharp guarantee for making good choices under a specific mathematical kind of constraint—not every real-world marketplace." }, "whyItMatters": "a fresh, source-backed online-selection theorem with a precise claimed guarantee. Intake is not a verification of the algorithm, its constants, or an application to any operational allocation problem.", "publicImpact": { "updated": "2026-09-18", "headline": "How do you choose before every offer arrives?", "plainEnglish": "Hiring, booking, and bidding all share a cruel timing problem: accept too early and a better option may appear; wait too long and nothing remains. This paper claims a sharp online-selection rule for a structured family of feasible choices.", "ifHolds": "Enabling: it would settle the strong secretary guarantee for linear matroids, giving algorithm designers a precise benchmark for online selection under that structure.", "ifFails": "The sharp guarantee or its scope would need revision, revealing which arrival, independence, or representation assumption carries more weight than claimed.", "horizon": "Enabling", "areas": [ "Online algorithms", "Combinatorial optimization", "Decision-making under uncertainty" ] }, "whyTracked": "a fresh, source-backed online-selection theorem with a precise claimed guarantee. Intake is not a verification of the algorithm, its constants, or an application to any operational allocation problem.", "availableArtifacts": "arXiv exposes PDF, experimental HTML, and TeX source. The inspected primary record identifies no Lean/Rocq/Coq/Isabelle development, code repository, certificate, or replay package. AI involvement is not established by this source.", "proposedCheckRoute": "Inspect the exact theorem statements and define their arrival-order, value, representation, and oracle assumptions; verify that the linear-matroid 1/e theorem is not conflated with the separate arbitrary-matroid prophet or 1/64 reduction claims. No code needs to be run for this first source review.", "highestRiskDependency": "A linear-matroid feasible set is not automatically a real capacity, pricing, authority, or lifecycle model. The guarantee's online-information and representation assumptions determine what it says; a claimed 1/e result does not make it an all-purpose offer-selection rule.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Inspect the exact theorem statements and define their arrival-order, value, representation, and oracle assumptions; verify that the linear-matroid 1/e theorem is not conflated with the separate arbitrary-matroid prophet or 1/64 reduction claims. No code needs to be run for this first source review." }, { "id": "arxiv-2609-20803", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20803", "contentModified": "2026-09-18T13:41:39Z", "arxivId": "2609.20803", "intakeDate": "2026-09-18", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "Regularity of asymptotically axisymmetric solutions to the 3D Navier–Stokes equations with analytic forcing", "authors": "Peter Constantin, Mihaela Ignatova, and Vlad Vicol.", "source": { "url": "https://arxiv.org/abs/2609.20803v1", "version": "submitted 2026-09-17 17:57:45 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-18T13:41:39Z" }, "attributedClaim": "The abstract and Theorem 1.3 claim regularity for a specified class of forced 3D Navier–Stokes solutions. As a consequence, a singular construction with the named core and anisotropic properties cannot have force that both remains (C²)-bounded to singular time and is locally uniformly real analytic in space. It is a timely mathematical boundary on the September 8 forced Navier–Stokes intake, not a refutation of that claim.", "publicOverview": { "updated": "2026-09-18", "what": "A theorem claims a proposed kind of Navier–Stokes singularity cannot occur when its forcing is spatially real analytic.", "who": "Peter Constantin, Mihaela Ignatova, and Vlad Vicol", "meaning": "How a fluid is driven matters. Under specified symmetry and scaling assumptions, analytic forces rule out blowup. OpenAI’s construction uses non-analytic forcing, so this is a boundary, not a refutation." }, "whyItMatters": "a fresh, source-backed conditional regularity theorem materially constraining an active forced Navier–Stokes claim. It does not by itself decide the OpenAI construction, whose forcing need not meet the theorem’s analytic hypothesis.", "publicImpact": { "updated": "2026-09-18", "headline": "A sharp new boundary around a fluid singularity claim", "plainEnglish": "Navier–Stokes singularities are notoriously hard to rule in or out. This paper says a particular proposed route cannot work with a force that remains real analytic in space, narrowing the terrain without resolving every case.", "ifHolds": "Foundational: it would impose a concrete regularity constraint on this class of forced singularity constructions and focus scrutiny on the exact behavior of their forcing.", "ifFails": "The proposed restriction would weaken, revealing which symmetry, scale, or analyticity step needs a more careful argument.", "horizon": "Foundational", "areas": [ "Fluid dynamics", "Partial differential equations", "Mathematical analysis" ] }, "whyTracked": "a fresh, source-backed conditional regularity theorem materially constraining an active forced Navier–Stokes claim. It does not by itself decide the OpenAI construction, whose forcing need not meet the theorem’s analytic hypothesis.", "availableArtifacts": "arXiv exposes PDF, HTML, and TeX source. The primary record did not identify a formal proof, code repository, numerical certificate, or executable artifact. No artifact was downloaded, executed, or replayed.", "proposedCheckRoute": "examination Theorem 1.3’s force regularity, anisotropic-bound, and axisymmetric-core assumptions; compare them line by line with the cited OpenAI manuscript’s Theorem 1.1 and Appendix-A properties; then review the ancient-limit and local-regularity argument without executing manuscript code.", "highestRiskDependency": "The consequence is conditional. Equating smooth compactly supported forcing with real-analytic forcing, or treating this theorem as a general no-blowup result, would overstate its scope.", "boundary": "State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: examination Theorem 1.3’s force regularity, anisotropic-bound, and axisymmetric-core assumptions; compare them line by line with the cited OpenAI manuscript’s Theorem 1.1 and Appendix-A properties; then review the ancient-limit and local-regularity argument without executing manuscript code." }, { "id": "arxiv-2609-20805", "canonicalUrl": "https://stateofproof.org/paper-watch/arxiv-2609-20805", "contentModified": "2026-09-18T13:41:37Z", "arxivId": "2609.20805", "intakeDate": "2026-09-18", "disposition": "candidate", "examinationState": "not-started", "validationResult": null, "title": "Universal completeness of exponentials", "authors": "Susanna Bertolini, Enric Florit-Simon, Lukas Liehr, and Mitchell A. Taylor.", "source": { "url": "https://arxiv.org/abs/2609.20805v1", "version": "submitted 2026-09-17 17:58:15 UTC", "sourceDate": "2026-09-17", "retrieved": "2026-09-18T13:41:37Z" }, "attributedClaim": "The abstract claims uniformly discrete density-one frequency sets complete in every (L^p(S)) for (|S|<1), density-(v) integer-frequency counterparts for (|S|0) and (X₀) such that every (X≥ X₀) has at least (cX) positive starts (n