{ "archive": { "archive": "math_research_02_of_08.zip", "sha256": "40a51f33992504fe2298832d269280bc82c12610885dc08dcac0e2c68cdc662c", "bytes": 67558242, "crc_ok": true, "files": 1776, "physical_work_folders": 112, "preprint_pdfs": 115, "groups": [ "068", "069", "087", "088", "089", "090", "091", "092", "093", "094", "095", "096", "097", "098", "099", "100", "101", "155", "156", "157", "158", "159", "160", "161", "162", "164", "165", "166", "167", "168", "169", "170", "171", "172", "173", "174", "175", "176", "177", "178", "179", "180", "181", "182", "183", "184", "185", "186", "187", "188", "189", "190", "191", "192", "333", "334", "335", "336", "337", "338", "339", "340", "341", "342", "343", "344", "345", "346", "347", "348", "349", "350" ], "current_manuscripts": 112 }, "reading_policy": "Archive catalog parsed; PDFs opened structurally; only explicit read_detail entries represent actual reading.", "works": [ { "id": "068-05", "group": "068", "title": "Metric descent and rank-preserving contractions", "pdf": "preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf", "folder": "preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026", "abstract_original": "From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 1738, "coverage": "catalog_only", "read_detail": null, "pages": 76, "sha256": "35f14f53ac82dc8a58373ec70b2a5b17625b541dc2db8226e7bb60245d6dc4d3", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "069-01", "group": "069", "title": "Global quantum geometric Langlands at irrational level", "pdf": "preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf", "folder": "preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026", "abstract_original": "We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level $`c\\in\\mathbb C\\setminus\\mathbb Q`$, including non-real levels. For every smooth projective connected complex curve X , it identifies the full twisted D -module categories on $`\\mathop{\\mathrm{Bun}}\\nolimits _G(X)`$ and $`\\mathop{\\mathrm{Bun}}\\nolimits _{G^\\vee}(X)`$ at the dual shifted levels c and $`-1/(rc)`$, where r is the lacing number. The equivalence uses the given global forms and includes all connected components.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 1772, "coverage": "catalog_only", "read_detail": null, "pages": 119, "sha256": "8d9bc135865dbc97a76efbf10fd7b9631bbe1107bd87c467b64db60eb3f229f1", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "087-01", "group": "087", "title": "The symmetric Mahler conjecture and its equality cases", "pdf": "preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf", "folder": "preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026", "abstract_original": "We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝ n has volume product at least $`4^n/n!`$, with equality exactly for invertible linear images of Hanner polytopes.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2134, "coverage": "intro_theorems", "read_detail": { "work": "087-01", "pages": "1–3", "scope": "Точная симметричная теорема и введение; компаньоны не разобраны" }, "pages": 26, "sha256": "0560a02b890de6637a1e007f2d438e8dec93575b4e12a66a6313511348ddd6a6", "bundled_pdf": "sources/papers/087-01/paper.pdf", "local_href": "../math_sources/math_research_02_of_08/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "087-02", "group": "087", "title": "The Mahler Conjecture for General Convex Bodies", "pdf": "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf", "folder": "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026", "abstract_original": "We resolve the Mahler conjecture for general convex bodies positively. For every convex body $`K\\subset\\mathbb R^n`$, n ≥ 1, with Santaló point $`s(K)`$, $`|K|\\,|(K-s(K))^\\circ|\\ge (n+1)^{n+1}/(n!)^2`$, with equality exactly for simplices.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2143, "coverage": "catalog_only", "read_detail": null, "pages": 72, "sha256": "f2d35982e2e7dfe55474092a7db4c558419fb892b412bf122b20d43b51229eff", "bundled_pdf": "sources/papers/087-02/paper.pdf", "local_href": "../math_sources/math_research_02_of_08/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "087-03", "group": "087", "title": "Symplectic Balls in Symmetric Polar Products", "pdf": "preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf", "folder": "preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026", "abstract_original": "For every integer n ≥ 2 and every origin-symmetric convex body $`K\\subset\\mathbb R^n`$, we prove that the Gromov width of $`\\mathop{\\mathrm{int}}\\nolimits K\\times\\mathop{\\mathrm{int}}\\nolimits K^\\circ`$ is 4. We construct smooth symplectic embeddings of standard balls of every capacity $`0\\lt c\\lt 4`$ into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2152, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "643930d71faeb8ba5aa5cda35465d20697f2955dce9a91ff729fb3ebd90c76ee", "bundled_pdf": "sources/papers/087-03/paper.pdf", "local_href": "../math_sources/math_research_02_of_08/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "088-01", "group": "088", "title": "Petty’s projection-volume conjecture in dimensions at least four", "pdf": "preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf", "folder": "preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026", "abstract_original": "We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2168, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "ce25657d3f42aebbc5c6623367c257e46ff79cde62069bebc3324daa282d98e1", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "088-02", "group": "088", "title": "A product counterexample to the simplex maximum for projection-body volume", "pdf": "preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf", "folder": "preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026", "abstract_original": "The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2177, "coverage": "catalog_only", "read_detail": null, "pages": 7, "sha256": "1d2211bb8d175f40fb55554586f02538fc9f4c04ecdc9793999a6f7ec514720a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "089-01", "group": "089", "title": "Planar Graph Metrics Embed into L1 with Constant Distortion", "pdf": "preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf", "folder": "preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026", "abstract_original": "We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L 1 with a universal distortion bound. This resolves the planar embedding conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2193, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "11b3a102e7333bf836a1317d774647394e580a5078027d1b9153a0b7fbc9e0e5", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "089-02", "group": "089", "title": "L1 Embeddings of Graphs of Bounded Treewidth", "pdf": "preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf", "folder": "preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026", "abstract_original": "For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L 1 with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2202, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "13276d0ab77859c506912444eab88bd09e3a47fba8a8d1b4c0511740bfe80dbb", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "090-01", "group": "090", "title": "An atomic certificate for triangular-lattice universal optimality", "pdf": "preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf", "folder": "preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026", "abstract_original": "We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2218, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "83e0d874119abae56db94fb1ecd6ebb794753d4982464b4f9788e5158708be93", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "090-02", "group": "090", "title": "Universal optimality of the triangular lattice", "pdf": "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf", "folder": "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026", "abstract_original": "We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with $`0\\lt s\\lt 2`$ in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2227, "coverage": "catalog_only", "read_detail": null, "pages": 70, "sha256": "50daa477c52b1f3e179764794506f0037d26c2ec130d1d6e817626f4e0b689f3", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "090-03", "group": "090", "title": "A sharp Fourier certificate for planar circle packing", "pdf": "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf", "folder": "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026", "abstract_original": "We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density $`\\pi/(2\\sqrt3)`$. We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2236, "coverage": "catalog_only", "read_detail": null, "pages": 49, "sha256": "3a532659c4aa989d4d9cca9d33810940b61c835597bb0b1b50c16d3487001329", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "090-04", "group": "090", "title": "Triangular minimality for planar Coulomb renormalized energy", "pdf": "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf", "folder": "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026", "abstract_original": "We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2245, "coverage": "catalog_only", "read_detail": null, "pages": 49, "sha256": "6f640e977aed5d3b876e6a275a85c8549575051fb21181fb3159fba8596fcc4a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "091-01", "group": "091", "title": "The logarithmic Brunn–Minkowski conjecture", "pdf": "preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf", "folder": "preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026", "abstract_original": "We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric L p Brunn–Minkowski inequality for every $`0\\lt p\\lt 1`$. Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation $`(B)`$-conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2261, "coverage": "catalog_only", "read_detail": null, "pages": 20, "sha256": "d5c7e764dee425e8172446d0a9dacc2d657a69e8f8b46ac5b68957200a5f09cc", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "092-01", "group": "092", "title": "A single-lattice covering bound of order n log n", "pdf": "preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf", "folder": "preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026", "abstract_original": "Every convex body in ℝ n , n ≥ 2, admits a covering by translates along one full-rank lattice with density at most $`Cn\\log n`$, for an absolute constant C . No symmetry or boundary regularity is assumed.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2277, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "c7dc74641f89bce3f456410280429a47b3e214365a9467b494e14ee6b2348cbb", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "092-02", "group": "092", "title": "Translative covering densities of order n log n", "pdf": "preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf", "folder": "preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026", "abstract_original": "For every sufficiently large dimension n , we construct a centrally symmetric convex body whose translative covering density exceeds $`c n\\log n`$, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2286, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "2944d4b3135b4d794626b13db292b869dfc58b4668a29f194e8fbcb542290a3e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "093-01", "group": "093", "title": "A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "pdf": "preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf", "folder": "preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026", "abstract_original": "We prove that every centered log-concave probability measure with a Lebesgue density on ℝ n and linear subgaussian parameter a satisfies $`\\mathop{\\mathrm{Ent}}\\nolimits _\\mu(f^2)\\le Ca^2\\int|Df|^2\\,d\\mu`$ for compactly supported smooth f , with one universal constant C . This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2302, "coverage": "catalog_only", "read_detail": null, "pages": 65, "sha256": "767a5eb7cd819af7e33c8b867c41403aac36ec015c89c1d2beb186abca9ec89e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "094-01", "group": "094", "title": "Subpolynomial dimension reduction in Lp", "pdf": "preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf", "folder": "preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026", "abstract_original": "For every fixed $`1\\lt p\\lt \\infty`$ and D > 1, every n -point subset of a real L p space embeds into $`\\ell_p^d`$ with distortion at most D and subpolynomial dimension $`d=n^{o(1)}`$. The target has the same exponent p , and the embedding need not be linear.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2318, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "001a29a154753ee273e236ccb13788a8dbba09f9ac4fd59a7707a42ef7d6dc03", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "095-01", "group": "095", "title": "Hyperbolicity Cones Without Semidefinite Lifts", "pdf": "preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf", "folder": "preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026", "abstract_original": "We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2334, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "48c3a89da2a380d18a71abf7d31ce49cc3488675607de304fcade3a8ae457b1a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "095-02", "group": "095", "title": "A nonspectrahedral hyperbolicity cone", "pdf": "preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf", "folder": "preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026", "abstract_original": "We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2343, "coverage": "catalog_only", "read_detail": null, "pages": 14, "sha256": "c0522a9b099318b2b79a90219749b18e1f917b4c0c0c5337e420640d1989dddf", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "095-03", "group": "095", "title": "An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone", "pdf": "preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf", "folder": "preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026", "abstract_original": "We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X . Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2352, "coverage": "catalog_only", "read_detail": null, "pages": 10, "sha256": "6b16fbcc207da5cceeb0aa1160ab8706a482340a7af3c46a38071c2cc76b3e75", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "096-01", "group": "096", "title": "The Gaussian propeller bound in every dimension", "pdf": "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf", "folder": "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026", "abstract_original": "We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most $`9/(8\\pi)`$. For dimension at least two and at least three cells, three planar sectors of angle $`2\\pi/3`$, extended by an orthogonal Euclidean factor, attain the bound.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2368, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "b546ae5fd2ea7011510d5d582148b642e82d39c6ddfb23c16bc9a0a399ffee09", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "097-01", "group": "097", "title": "The Euclidean Steinitz–Bergström theorem", "pdf": "preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf", "folder": "preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026", "abstract_original": "Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ d has one signing for which every signed prefix has norm at most $`C\\sqrt d`$, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $`S_2(d)=\\Theta(\\sqrt d)`$, and resolves the Euclidean Steinitz–Bergström conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2384, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "d8924ac01b6d4dc0e70bd3fcb025132dc7fb66aa239dd7cb77d2310aa8f7b326", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "098-01", "group": "098", "title": "A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding", "pdf": "preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf", "folder": "preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026", "abstract_original": "Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2400, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "9c9b7e794d3a66305eda83409e40ca56eed2c09eb687c6abd5ec367f6e980b15", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "099-01", "group": "099", "title": "Edit Distance in l1: Matching Bounds up to Constants in the Exponent", "pdf": "preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf", "folder": "preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026", "abstract_original": "We determine the exponential scale of the least ℓ 1 distortion of unit-cost edit distance on all strings of length at most d . For every sufficiently large d , uniformly over finite alphabets of size at least two, the distortion lies between $`\\exp(c\\sqrt{\\log d\\,\\log\\log d})`$ and $`\\exp(C\\sqrt{\\log d\\,\\log\\log d})`$ for absolute constants $`c,C\\gt 0`$. The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2416, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "c554cfee6ede3e18215a79cd4de7fb89a8aad26f8255c0ea72b9ed1aaef302b3", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "099-02", "group": "099", "title": "Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance", "pdf": "preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf", "folder": "preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026", "abstract_original": "We give two finite-circle constructions of binary strings whose least ℓ 1 distortion is $`\\exp(\\Omega(\\sqrt{\\log d\\,\\log\\log d}))`$, where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d , including the empty word.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2425, "coverage": "catalog_only", "read_detail": null, "pages": 33, "sha256": "8b30fa34409155e66c2d653aa31e61eacb441f91124caeee4c8d64dc528e70c8", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "099-03", "group": "099", "title": "Tree Constructions for the l1 Distortion of Binary Edit Distance", "pdf": "preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf", "folder": "preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026", "abstract_original": "We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ 1 distortion $`\\exp(\\Omega(\\sqrt{\\log d\\,\\log\\log d}))`$ for every sufficiently large d . We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2434, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "696bf7848724daa24c77dbb61626c434d129a6b74dca595585f4204657c22cae", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "100-01", "group": "100", "title": "Finite angular cylinder covers below the half-area bound", "pdf": "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf", "folder": "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026", "abstract_original": "A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2450, "coverage": "catalog_only", "read_detail": null, "pages": 20, "sha256": "edbcf76e6573c125cf3cf085ef7a2b23c3028768ff683dd76f9af5756e7bd6be", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "100-02", "group": "100", "title": "Finite cylinder approximation of ruled sets", "pdf": "preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf", "folder": "preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026", "abstract_original": "We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C 1 , and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2459, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "06959ab19716522a939868729297d06016fa80b763407815ef5eecd470c279ae", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "100-03", "group": "100", "title": "Slope-field perturbations of the two-cylinder covering", "pdf": "preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf", "folder": "preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026", "abstract_original": "The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2468, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "6e6336ef3860f900f59ea95a73637d67f24d4f6de20a67f98a8b19396473e7ef", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "100-04", "group": "100", "title": "Finite triangular approximation of radial sweeps", "pdf": "preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf", "folder": "preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026", "abstract_original": "We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2477, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "6e7f63022190f390e00a26a7f822d46b73290167bab9d9f1e94ef099f519fefd", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "101-01", "group": "101", "title": "A sharp entropy bound and the simplex inequality for isotropic constants", "pdf": "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf", "folder": "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026", "abstract_original": "We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound $`h(f)\\ge m+\\tfrac12\\log\\det\\mathop{\\mathrm{Cov}}\\nolimits (f)`$ for every log-concave probability density on ℝ m , with equality precisely for invertible affine images of products of one-sided exponential laws.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 2493, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "b6cb0115cd04d4eb0c24db7e600ba8dd9519848a92e87fa8a661f435e41e95a0", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "155-01", "group": "155", "title": "A translational tile with no fully periodic tiling in dimension three", "pdf": "preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf", "folder": "preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026", "abstract_original": "We construct a finite translational tile in ℤ 3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ 3 , even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3709, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "195ccaae8617ea5d18a532aa4913a1ba2ca29e95a791f686808f01a9e6242089", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "156-01", "group": "156", "title": "A nine-dimensional counterexample to Borsuk's covering assertion", "pdf": "preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf", "folder": "preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026", "abstract_original": "The compact set of rank-one orthogonal projectors on ℝ 4 , with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3725, "coverage": "catalog_only", "read_detail": null, "pages": 18, "sha256": "e7ffa6f812d97395aa7794f4dc0f394ea36f24b00ced0acad29157c4c3bcca87", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "157-01", "group": "157", "title": "A counterexample to Hadwiger's conjecture", "pdf": "preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026", "abstract_original": "We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3741, "coverage": "catalog_only", "read_detail": null, "pages": 104, "sha256": "a97293542586d3518b9355dcea732012b09d4ce809126da73d285ba0d3906128", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "157-02", "group": "157", "title": "A counterexample to the Colin de Verdière chromatic conjecture", "pdf": "preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026", "abstract_original": "We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3750, "coverage": "catalog_only", "read_detail": null, "pages": 131, "sha256": "97677c7a183884f099bff03396dacf857ae723f03aa9485819bfb1c52273d4be", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "157-03", "group": "157", "title": "A linear list-coloring bound in terms of the Hadwiger number", "pdf": "preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf", "folder": "preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026", "abstract_original": "We prove that every finite nonempty graph G satisfies $`\\chi_{\\mathrm{list}}(G)\\le C h(G)`$ for an absolute integer C , where $`h(G)`$ is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3759, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "1becff24453872c5c7973bdc72bc2b12120048580ac71fbac816b523f910163e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "158-01", "group": "158", "title": "The Euclidean plane is not five-colorable", "pdf": "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf", "folder": "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026", "abstract_original": "We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3775, "coverage": "catalog_only", "read_detail": null, "pages": 62, "sha256": "607c2601b5ae655c7c116272840e260f71f637aacf611453186e0a1c65111d36", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "159-01", "group": "159", "title": "Quasipolynomial Bounds for Arithmetic Progressions", "pdf": "preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf", "folder": "preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026", "abstract_original": "We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show $`\\displaystyle r_k(N)\\le C_kN\\exp\\bigl(-c_k(\\log N)^{\\varepsilon_k}\\bigr)`$ with $`C_k,c_k,\\varepsilon_k\\gt 0`$, where $`r_k(N)`$ is the largest size of a subset of $`\\{1,\\ldots,N\\}`$ with no nonconstant k -term arithmetic progression.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3791, "coverage": "catalog_only", "read_detail": null, "pages": 198, "sha256": "481eaf7730dc4d581e5270cc4d514c6ee2ec386d5af643cf914e20b4d00c59be", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "160-01", "group": "160", "title": "Quantitative Superexponential Bounds for van der Waerden Numbers", "pdf": "preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf", "folder": "preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026", "abstract_original": "We prove that there are absolute constants c > 0 and K 0 such that $`W_r(k)\\gt k^{ck\\lfloor\\log_2 r\\rfloor}`$ for every $`k\\ge K_0`$ and r ≥ 2. Consequently $`W_r(k)^{1/k}\\to\\infty`$ for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3811, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "042acea33a57b58d17600a9ec5e0d8d6f424787c5edf28b9d8d5f9bd9297bd18", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "161-01", "group": "161", "title": "A counterexample to Sidorenko's conjecture", "pdf": "preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026", "abstract_original": "We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3827, "coverage": "catalog_only", "read_detail": null, "pages": 51, "sha256": "0c0d7b5eb1ed43c00ed2f60d2347b7d5511bad2218efe12c28fd9968ee549b0f", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "162-01", "group": "162", "title": "Balanced counterexamples to Ryser's conjecture at prime orders", "pdf": "preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf", "folder": "preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026", "abstract_original": "For every sufficiently large prime q , we construct a finite intersecting $`(q+1)`$-partite $`(q+1)`$-uniform hypergraph with covering number $`q+1`$ and exactly $`q+1`$ nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3843, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "5be7cf2d1f382e802c7b3e738bc143fff8b48f9504bd579f74719dc754f7a469", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "162-02", "group": "162", "title": "A counterexample to Ryser's covering conjecture", "pdf": "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026", "abstract_original": "For every sufficiently large prime $`s\\equiv2\\pmod3`$ and every sufficiently large odd integer n , with the threshold depending on s , we construct an intersecting $`(s^n+1)`$-partite $`(s^n+1)`$-uniform hypergraph with covering number $`s^n+1`$. This disproves Ryser's covering conjecture in its intersecting case.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3852, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "ba76ca1f7af545c68090e0cfa1defba423af058cb4c0ac6087e563b5123b1eb7", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "164-01", "group": "164", "title": "Monochromatic finite sums and products in the positive integers", "pdf": "preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf", "folder": "preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026", "abstract_original": "We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k , there is a k -element set whose nonempty subset sums and nonempty subset products all have the same color.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3868, "coverage": "catalog_only", "read_detail": null, "pages": 70, "sha256": "ab835f6a98c10d6d4498b19e7446facd7b481fef85fee54232c76b45bc9e0888", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "165-01", "group": "165", "title": "The crossing number of complete graphs", "pdf": "preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf", "folder": "preprints/The-crossing-number-of-complete-graphs-September-23-2026", "abstract_original": "We prove the Harary–Hill conjecture: for every positive integer n , the ordinary crossing number of the complete graph K n is $`\\displaystyle \\frac14\\left\\lfloor\\frac n2\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor\\frac{n-2}{2}\\right\\rfloor \\left\\lfloor\\frac{n-3}{2}\\right\\rfloor.`$", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3884, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "8b4c78b04d710b7d0e60d18b3fd9e9ea7042b5201ca6644ee72f50c5adee310d", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "165-02", "group": "165", "title": "The crossing number of complete bipartite graphs", "pdf": "preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf", "folder": "preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026", "abstract_original": "We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m , n , the ordinary crossing number of the complete bipartite graph $`K_{m,n}`$ is $`\\displaystyle \\left\\lfloor\\frac m2\\right\\rfloor \\left\\lfloor\\frac{m-1}{2}\\right\\rfloor \\left\\lfloor\\frac n2\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor.`$", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3895, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "95b9fc254d730aa35a82585779dfcbe7e1e39840749ef17b329aecc55659f599", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "166-01", "group": "166", "title": "The higher-dimensional Erdős distinct-distances conjecture", "pdf": "preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf", "folder": "preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026", "abstract_original": "For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ d determines at least $`c_d n^{2/d}`$ distinct distances, where $`c_d\\gt 0`$ depends only on d . This resolves the higher-dimensional Erdős distinct-distances conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3913, "coverage": "catalog_only", "read_detail": null, "pages": 103, "sha256": "37f6355365a18510e75d897dc9ab6d29305d7ff132cf148845b6f16d080ee1bf", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "167-01", "group": "167", "title": "The weak pinned planar distance theorem", "pdf": "preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf", "folder": "preprints/The-weak-pinned-planar-distance-theorem-September-23-2026", "abstract_original": "We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but $`o(n)`$ points of any n -point planar set determine at least $`n^{1-\\varepsilon}`$ distinct nonzero distances.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3929, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "6350959db2b8310fdad1792ad04d4f8d64d6c030808cb6d00b29eb66513789f0", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "167-02", "group": "167", "title": "A power saving for planar unit distances", "pdf": "preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf", "folder": "preprints/A-power-saving-for-planar-unit-distances-September-23-2026", "abstract_original": "We prove a power saving for the planar unit-distance problem: for some absolute $`\\beta\\lt 4/3`$, every set of n points in the Euclidean plane determines $`O(n^\\beta)`$ unordered pairs at unit distance.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3938, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "72bfa9153772b2124385cf4e53a5557320ab92e29fe4bdcb35a05a34481aede0", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "168-01", "group": "168", "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "pdf": "preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf", "folder": "preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026", "abstract_original": "We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3954, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "7e1be2107a85e28d9ec339b4ee62d4efeb909968df6768368e2779710624e331", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "169-01", "group": "169", "title": "Elementary positivity of chromatic quasisymmetric functions", "pdf": "preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf", "folder": "preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026", "abstract_original": "We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over $`\\mathbb N[q]`$. This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3970, "coverage": "catalog_only", "read_detail": null, "pages": 51, "sha256": "9e1c6414a9ea8ce93b8bea6d40639da43f744f04c938e57c12c72b29f9d4c8ea", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "170-01", "group": "170", "title": "The sharp logarithmic exponent of r(5,t)", "pdf": "preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf", "folder": "preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026", "abstract_original": "We determine the sharp logarithmic exponent of the off-diagonal Ramsey number $`r(5,t)`$: $`\\displaystyle r(5,t)=\\frac{t^4}{(\\log t)^{3+o(1)}} \\qquad (t\\longrightarrow\\infty).`$", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3986, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "71adbadd9abac57ba7a19be3b6d27bd68fd1ee9b23d95955489cef9b80a3bec6", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "170-02", "group": "170", "title": "Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers", "pdf": "preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf", "folder": "preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026", "abstract_original": "For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: $`\\displaystyle r(s,t)=\\frac{t^{s-1}}{(\\log t)^{s-2+o(1)}} \\qquad (t\\longrightarrow\\infty).`$", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 3997, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "20ab597a1d3dea22bd24a35cefd3938c3046ae51c5ddb6ac072ef9439a1d7957", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "171-01", "group": "171", "title": "The hypercube Ramsey number has linear order", "pdf": "preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf", "folder": "preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026", "abstract_original": "We prove that the two-color Ramsey number of the n -dimensional binary cube is at most $`C2^n`$, where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4015, "coverage": "catalog_only", "read_detail": null, "pages": 172, "sha256": "7d4b26e897038267bcc6b7a409d803c33fbd8b3d4310d30ebe6bda6a883b1226", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "172-01", "group": "172", "title": "A classification of finite Euclidean Ramsey configurations", "pdf": "preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf", "folder": "preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026", "abstract_original": "We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4031, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "0900061bb0c703ae6d96506a7861a78089ddd83bf2496fd969fe9a5379283daa", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "173-01", "group": "173", "title": "A proof of Seymour’s second-neighborhood conjecture", "pdf": "preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026", "abstract_original": "We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4047, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "20be859799f83e3412dba6fd645a72a0d929cbe05ca546807c591f001f9e6e82", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "174-01", "group": "174", "title": "The strong thin tree conjecture", "pdf": "preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf", "folder": "preprints/The-strong-thin-tree-conjecture-September-23-2026", "abstract_original": "We prove that every finite loopless k -edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most $`C/k`$ times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4063, "coverage": "catalog_only", "read_detail": null, "pages": 52, "sha256": "a2b847ce72f66109dee2db00ea456bad74c13889b93bc7666e5ad9abefc23da4", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "174-02", "group": "174", "title": "A polynomial-time construction of strong thin trees", "pdf": "preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf", "folder": "preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026", "abstract_original": "We give a deterministic polynomial-time construction of strong thin trees. Given a finite k -edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a $`C/k`$ fraction of its edges, for a universal constant C . The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4072, "coverage": "catalog_only", "read_detail": null, "pages": 38, "sha256": "da0e75ba53ff393fd57e9028170fce25aed47ae02954131c9eb04513bdffa658", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "175-01", "group": "175", "title": "Graph Decompositions at the Integral Expectation Threshold", "pdf": "preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf", "folder": "preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026", "abstract_original": "We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4088, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "5c0b3a5fc1764ff357f2b791f559f02d6dccaad8f36726a53d106fb0b97873af", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "175-02", "group": "175", "title": "Integral and fractional expectation thresholds are equivalent", "pdf": "preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf", "folder": "preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026", "abstract_original": "We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4097, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "0c02b128253cb6e07f23931992bf377e8dd74c032572143878b1821cdcd32f39", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "175-03", "group": "175", "title": "Talagrand’s discrete-convexity conjecture", "pdf": "preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf", "folder": "preprints/Talagrands-discrete-convexity-conjecture-September-23-2026", "abstract_original": "We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least $`1-1/k`$, the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4106, "coverage": "catalog_only", "read_detail": null, "pages": 10, "sha256": "217afc1b8177a802016efa7fe14038b86fa593cb0da2ebbd679e09806422479e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "176-01", "group": "176", "title": "The second Kahn–Kalai conjecture", "pdf": "preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf", "folder": "preprints/The-second-Kahn-Kalai-conjecture-September-24-2026", "abstract_original": "We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for $`G(n,p)`$ to contain an ordinary copy of H is at most $`C p_{\\mathrm E}(n,H)(1+\\log_2 h)`$, where C is universal. Here $`p_{\\mathrm E}(n,H)`$ is the least density at which every subgraph of H has expected copy count at least one half.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4122, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "dd0c26b99de9e1d01d25f9b646a1fca1cd92fe34180ba7e5042ddcf91b61eb85", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "177-01", "group": "177", "title": "Bounded-degree coboundary expanders in every dimension", "pdf": "preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf", "folder": "preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026", "abstract_original": "For every integer d ≥ 3, we construct arbitrarily large finite d -dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽 2 coboundary expansion in every degree below d . Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽 2 coboundary expanders in every positive dimension.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4138, "coverage": "catalog_only", "read_detail": null, "pages": 14, "sha256": "a1d158719d5e92a2f98b118d03e11cfd699acc3600108c9dcdf838a2871e4530", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "178-01", "group": "178", "title": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "pdf": "preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf", "folder": "preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026", "abstract_original": "For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d -regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d . Every nonconstant adjacency eigenvalue lies strictly between $`-2\\sqrt{d-1}`$ and $`2\\sqrt{d-1}`$.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4154, "coverage": "catalog_only", "read_detail": null, "pages": 85, "sha256": "6d8ea4bb85aae750e6d9e3101e197669ca9f3f5a5e551ad718a2fc9cb1abbd9e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "179-01", "group": "179", "title": "The circulant Hadamard conjecture", "pdf": "preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf", "folder": "preprints/The-circulant-Hadamard-conjecture-September-23-2026", "abstract_original": "We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4170, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "68f1d360a86238733b5d94bc2d0d091749cbb867faa3e35fd2de79e070aa7022", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "180-01", "group": "180", "title": "Paired states and Hamiltonian cycles in cubic bipartite planar graphs", "pdf": "preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf", "folder": "preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026", "abstract_original": "We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4186, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "b519ff0c2b1f1554f34f01e644cd67651749826d43919862db6c42303701db18", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "181-01", "group": "181", "title": "A linear cycle-and-edge decomposition of every graph", "pdf": "preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf", "folder": "preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026", "abstract_original": "We prove that every finite simple undirected graph on n vertices has an edge partition into at most $`Cn`$ simple cycles and single edges, for an absolute constant C . This resolves the Erdős–Gallai cycle decomposition conjecture positively.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4202, "coverage": "catalog_only", "read_detail": null, "pages": 32, "sha256": "4aa44187578ea59b64c78258eb8c5dd3601ee979a1a402ca9ed26696b30b515f", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "182-01", "group": "182", "title": "A power saving for intersective polynomial differences with an exponent depending only on the degree", "pdf": "preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf", "folder": "preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026", "abstract_original": "An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent $`c_k\\gt 0`$ such that every set $`A\\subseteq\\{1,\\ldots,N\\}`$ whose differences avoid all nonzero values $`h(1),h(2),\\ldots`$, where h is an intersective polynomial of degree k with positive leading coefficient, satisfies $`|A|=O_h(N^{1-c_k})`$. The implied constant may depend on h , but the power-saving exponent depends only on its degree.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4218, "coverage": "catalog_only", "read_detail": null, "pages": 47, "sha256": "d69e417ac953581d7f373c9457e809e47ad4b534c1f79407ff5c0e13f382b605", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "182-02", "group": "182", "title": "A Power Saving for Polynomial Differences at Prime Arguments", "pdf": "preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf", "folder": "preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026", "abstract_original": "Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $`A\\subseteq\\{1,\\ldots,N\\}`$ whose differences avoid all nonzero values $`h(p)`$ at primes satisfies $`|A|\\le C_hN^{1-c_h}`$, where $`c_h\\gt 0`$ and $`C_h\\ge1`$ depend only on h . Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L -functions to obtain the required prime-distribution estimates.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4227, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "b42ac864305ade86faecfe7a599249cf07a47681d7ce2008b51f6c8a3591f819", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "182-03", "group": "182", "title": "A power saving for square-difference-free sets", "pdf": "preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf", "folder": "preprints/A-power-saving-for-square-difference-free-sets-September-24-2026", "abstract_original": "We prove that there are absolute constants c > 0 and C < ∞ such that every set $`A\\subseteq\\{1,\\ldots,N\\}`$ with no nonzero square difference satisfies $`|A|\\le C N^{1-c}`$. This answers the fixed-power question posed by Green and Sawhney.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4236, "coverage": "catalog_only", "read_detail": null, "pages": 48, "sha256": "5a8fe70cdef077a7a73e668e8248e60f8cf535daa53af80f5c9dedb355cb0369", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "183-01", "group": "183", "title": "A power saving for planar halving lines", "pdf": "preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf", "folder": "preprints/A-power-saving-for-planar-halving-lines-September-25-2026", "abstract_original": "There are absolute constants ε > 0 and C such that every sufficiently large even n -point set in the plane with no three collinear has at most $`Cn^{4/3-\\varepsilon}`$ unordered halving pairs. This gives a power saving over the classical $`O(n^{4/3})`$ bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4252, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "61d7aa1ffad4b7968e5882374ce0e542d3221cf8e7c48b51c52256e1331bb4d9", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "184-01", "group": "184", "title": "Correspondence coloring graphs with a forbidden clique", "pdf": "preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf", "folder": "preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026", "abstract_original": "For every fixed integer r ≥ 4, we prove that every K r -free graph of sufficiently large maximum degree Δ has correspondence chromatic number $`O_r(\\Delta/\\log\\Delta)`$. This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F , holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4268, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "8809bcac3ea81e7bc15d75c6a034b1256af8fe042453f96004236fbbee2eed0e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "184-02", "group": "184", "title": "A logarithmic independence bound for clique-free graphs", "pdf": "preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf", "folder": "preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026", "abstract_original": "For every fixed integer r ≥ 4, every K r -free graph on n vertices with average degree d ≥ 2 has an independent set of size at least $`c_r n\\log d/d`$, where $`c_r\\gt 0`$ depends only on r . This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4277, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "15717a7c82914df2f220fad05a3fd6de7fd15f2abe26a4d1454d2b005ef39606", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "185-01", "group": "185", "title": "A Counterexample to the Infinite Matroid Packing/Covering Conjecture", "pdf": "preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf", "folder": "preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026", "abstract_original": "We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4293, "coverage": "catalog_only", "read_detail": null, "pages": 20, "sha256": "320eb5d308aabe2d7ab5d206329aed63091745888f021a77a9ad97d625477b9e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "186-01", "group": "186", "title": "A uniform influence bound for hypergraph properties", "pdf": "preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf", "folder": "preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026", "abstract_original": "For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r -uniform hypergraphs on n vertices satisfies $`\\mathop{\\mathrm{Var}}\\nolimits _p(f)\\le C_r I_p(f)/(\\log n)^{r/(r-1)}`$. The constant depends only on r , and the bound holds uniformly for all $`0\\lt p\\lt 1`$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $`r/(r-1)`$, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4309, "coverage": "catalog_only", "read_detail": null, "pages": 8, "sha256": "64d690bef14f4214272df2f04ab9cf843646b472171b68fb7869d7889203cb99", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "186-02", "group": "186", "title": "A Sharp Threshold Bound for Monotone Graph Properties", "pdf": "preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf", "folder": "preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026", "abstract_original": "We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every $`0\\lt \\varepsilon\\lt 1/2`$, the edge probabilities at which its probability equals ε and $`1-\\varepsilon`$ differ by at most $`C\\log(1/(2\\varepsilon))/(\\log n)^2`$, for a universal constant C .", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4318, "coverage": "catalog_only", "read_detail": null, "pages": 10, "sha256": "0e6a660e755ef2b6a30755fa40c18256a4ad7f54fe1040f79e13458c1ecc290a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "187-01", "group": "187", "title": "Snaky in 21 Maker moves", "pdf": "preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf", "folder": "preprints/Snaky-in-21-Maker-moves-September-25-2026", "abstract_original": "We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a $`17\\times17`$ square; in fact, Maker can confine its claims to a fixed 251-cell board.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4334, "coverage": "catalog_only", "read_detail": null, "pages": 84, "sha256": "e5ce107fdc4ec77fbbe2d3eab6babe6198e7160b0904462ae3fddfab0c88634c", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "188-01", "group": "188", "title": "The sharp terminal leave in random triangle removal", "pdf": "preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf", "folder": "preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026", "abstract_original": "Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n 3/2 , converges in L 2 to $`1/(2\\sqrt2)`$. This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4350, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "85fbaba652137663487dc07af0cdecc75b908fed99df036768328ce2f4dc169e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "189-01", "group": "189", "title": "Cycle--clique Ramsey numbers", "pdf": "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf", "folder": "preprints/Cycle-clique-Ramsey-numbers-September-25-2026", "abstract_original": "We prove that $`R(C_m,K_n)=(m-1)(n-1)+1`$ for every pair of integers $`m\\ge n\\ge3`$ other than $`(m,n)=(3,3)`$, for which $`R(C_3,K_3)=6`$. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to $`3{,}099`$ finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4366, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "366fc1d94b08e498183f475b170db56c72c7c898a24f3d3f1106e637ef42209b", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "190-01", "group": "190", "title": "Polynomial removal fails for ordered binary matrices", "pdf": "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf", "folder": "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026", "abstract_original": "We construct a fixed $`66\\times66`$ binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4382, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "120f7749fe55baa4ae10b6c40963bc7d8ef31c37a8e0bfe6c17cd520146c0647", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "191-01", "group": "191", "title": "A power improvement in the Heilbronn triangle lower bound", "pdf": "preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf", "folder": "preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026", "abstract_original": "There are absolute constants $`\\eta,c_1\\gt 0`$ such that, for every sufficiently large integer n , one can choose n points in the unit square so that every triangle they determine has area at least $`c_1n^{-2+\\eta}`$. Thus the almost n −2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4398, "coverage": "catalog_only", "read_detail": null, "pages": 24, "sha256": "e43687a84b82e23fd73f8025d8d8c91e7fee91c8ceead9bc41502b0595f09f0a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "192-01", "group": "192", "title": "Unbounded Violations of the Square-Root Degree Bound", "pdf": "preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf", "folder": "preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026", "abstract_original": "We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function $`f:\\{-1,1\\}^n\\to\\{-1,1\\}`$ on a finite sign cube such that $`\\displaystyle \\sum_{i=1}^n \\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}.`$ Here $`\\widehat f(\\{i\\})`$ is the linear Fourier coefficient associated with the i th input, and $`\\deg(f)`$ is the degree of the real multilinear polynomial representing f .", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 4414, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "4b847d47e281e37df99e6c3c6561cfd1db55380e6504a73e30918ead56b74894", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "333-01", "group": "333", "title": "Smooth isometric immersions of closed surfaces into Euclidean four-space", "pdf": "preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf", "folder": "preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026", "abstract_original": "Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8127, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "1f0bf283be86c80bc4c41547dac66f49bdfe73768373554058b53c8681d6ba3a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "334-01", "group": "334", "title": "A Smooth Metric with No Local Isometric Immersion into Three-Space", "pdf": "preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf", "folder": "preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026", "abstract_original": "We construct a smooth positive-definite Riemannian metric on $`(-1,1)^2`$ that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8143, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "6c2cf05d89a23473c52fecb2aea37bcbda7279db50004cf0b212b000430bd374", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "335-01", "group": "335", "title": "An integral scalar curvature bound for real simplicial volume", "pdf": "preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf", "folder": "preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026", "abstract_original": "We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant $`a_n\\gt 0`$, depending only on n , such that $`\\displaystyle \\int_M(\\mathop{\\mathrm{Scal}}\\nolimits _g^-)^{n/2}\\,dV_g\\ge a_n\\|M\\|`$ for every closed connected oriented smooth n -manifold M and every smooth Riemannian metric g . Here $`\\mathop{\\mathrm{Scal}}\\nolimits _g^-:=\\max\\{0,-\\mathop{\\mathrm{Scal}}\\nolimits _g\\}`$, and $`\\|M\\|`$ is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8159, "coverage": "catalog_only", "read_detail": null, "pages": 84, "sha256": "d5ed8374cef6389d2d54c921422a4523ecc11216b65887ede13bdf25a38ea6f1", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "335-02", "group": "335", "title": "Positive scalar curvature forces rational inessentiality", "pdf": "preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf", "folder": "preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026", "abstract_original": "We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8172, "coverage": "catalog_only", "read_detail": null, "pages": 37, "sha256": "dae319c1cc2582086b82ff65f2c210a972975ad321b340cff4c9d5211dcb14dc", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "336-01", "group": "336", "title": "Spectral scalar curvature and uniform Urysohn width", "pdf": "preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf", "folder": "preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026", "abstract_original": "For every n ≥ 4, a complete connected smooth Riemannian n -manifold without boundary satisfying $`-4\\Delta+\\mathop{\\mathrm{Scal}}\\nolimits \\ge1`$ as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most $`n-2`$ whose entire fibers have diameter bounded only in terms of n . The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8188, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "1b55879b1d10b0f5b31aa41b220a84f101f44bd01347626293f3fc70c85d917d", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "336-02", "group": "336", "title": "Spectral scalar curvature and Urysohn width in dimension three", "pdf": "preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf", "folder": "preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026", "abstract_original": "Every connected complete smooth Riemannian three-manifold without boundary satisfying $`-4\\Delta+\\mathop{\\mathrm{Scal}}\\nolimits \\ge\\lambda\\gt 0`$ as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most $`500/\\sqrt\\lambda`$ in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8197, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "2a0852ff04d11752b6a39d3da257ef8d768e11d1dd269e44f76745a55afae99a", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "336-03", "group": "336", "title": "Positive scalar curvature and uniform codimension-two width", "pdf": "preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf", "folder": "preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026", "abstract_original": "We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n -manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most $`n-2`$ whose entire fibers have diameter bounded only in terms of n . We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8206, "coverage": "catalog_only", "read_detail": null, "pages": 99, "sha256": "f67c30579306bd4dd256ece4d24cdad18ba149930fa97f9ab31a1e6a768cef89", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "337-01", "group": "337", "title": "Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity", "pdf": "preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf", "folder": "preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026", "abstract_original": "We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ . For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8222, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "a7310d708fef81eaa14c96d4a420e39098858ee98e23201381ff5e86b584b29f", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "337-02", "group": "337", "title": "Sharp integral fillings in CAT(0) spaces", "pdf": "preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf", "folder": "preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026", "abstract_original": "Every compactly supported integral n -cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8231, "coverage": "catalog_only", "read_detail": null, "pages": 46, "sha256": "1a8212bf07eeb82de51663b4b63e9f6a326f4705adfa8dededf08b2378ca1c99", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "338-01", "group": "338", "title": "Uniformization of complete Kähler manifolds with positive bisectional curvature", "pdf": "preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf", "folder": "preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026", "abstract_original": "We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8247, "coverage": "catalog_only", "read_detail": null, "pages": 77, "sha256": "2960f958368ee0ab8b9b2dd15b236d634b64595f29812035e12e2c705b207cdb", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "339-01", "group": "339", "title": "Entropy equality and local symmetry in negative curvature", "pdf": "preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf", "folder": "preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026", "abstract_original": "For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8263, "coverage": "catalog_only", "read_detail": null, "pages": 66, "sha256": "8658ab8209a8aa0c0f472b6793ecb738a30ffeec74821915e82d08a185b0ca97", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "340-01", "group": "340", "title": "A counterexample to the nearby Lagrangian conjecture", "pdf": "preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026", "abstract_original": "We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N , we construct a closed exact smoothly embedded Lagrangian in $`T^*(S^9\\times S^{N-1})`$ that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8279, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "ce7a208ddd7ef666a814114a9c92bbb782eda8ea82e21502f7624709539a9a34", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "341-01", "group": "341", "title": "Deforming hypersymplectic four-manifolds to hyperkähler triples", "pdf": "preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf", "folder": "preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026", "abstract_original": "Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8295, "coverage": "catalog_only", "read_detail": null, "pages": 72, "sha256": "92c36602fb4968087ec043e7b3816473b15ab21d0072899f31154337676b2bec", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "342-01", "group": "342", "title": "Taming implies compatibility on four-manifolds", "pdf": "preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf", "folder": "preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026", "abstract_original": "We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8311, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "cbddc627efab3dd49a69a32b270b52b6ea4528846bd3b313038a696ed3281265", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "343-01", "group": "343", "title": "Symplectic Ball Packings in Higher Dimensions", "pdf": "preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf", "folder": "preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026", "abstract_original": "We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities $`R_1,\\ldots,R_k`$, the closed standard symplectic $`2n`$-balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if $`\\displaystyle \\sum_{i=1}^k R_i^n\\lt R^n, \\qquad R_i+R_j\\lt R\\quad(i\\ne j).`$ Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8327, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "54a390c7a58b2f003896a3ea5319b0e8376dbf69059b7c00645d0d55babdbdb2", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "344-01", "group": "344", "title": "The metric Blaschke theorem", "pdf": "preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf", "folder": "preprints/The-metric-Blaschke-theorem-September-23-2026", "abstract_original": "We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8347, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "c631e2d36940a81e3c5f5476a13bdfbf12d77b9debd6ce2f3852357627256257", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "345-01", "group": "345", "title": "Infinitely many closed geodesic images on every Riemannian sphere", "pdf": "preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf", "folder": "preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026", "abstract_original": "We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere S n , n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8363, "coverage": "catalog_only", "read_detail": null, "pages": 50, "sha256": "123ee5276415debb732951573b23d8dbe7ee63112ec7246478de53715fe699e2", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "346-01", "group": "346", "title": "A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold", "pdf": "preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf", "folder": "preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026", "abstract_original": "The singular set of every stationary integral m -varifold in an open Euclidean set has Hausdorff dimension at most $`m-1`$, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8379, "coverage": "catalog_only", "read_detail": null, "pages": 35, "sha256": "550b6c1b4bf40bda7958c13378471d3347d592e6d4711e7ef63ccf9e0edad044", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "346-02", "group": "346", "title": "Almost-everywhere regularity of stationary integral varifolds", "pdf": "preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf", "folder": "preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026", "abstract_original": "We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8388, "coverage": "catalog_only", "read_detail": null, "pages": 77, "sha256": "f12bd8f9ace9a350d5eba44a7cd57e27b1f955d0751e9c42a2731c43293dec75", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "347-01", "group": "347", "title": "Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two", "pdf": "preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf", "folder": "preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026", "abstract_original": "We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number $`80+1968m`$ and a Hamiltonian diffeomorphism with exactly $`80+1952m`$ fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8404, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "12e5b850a57973914df99d6e8686b3b8c1bf77bd738605fffc1cd3cc26787058", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "347-02", "group": "347", "title": "Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number", "pdf": "preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf", "folder": "preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026", "abstract_original": "We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly $`1\\,872\\,232`$ fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is $`1\\,872\\,264`$. Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8413, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "b9fec0a6b56d3a8dd7ffe7ce53bfd6304e3a6560f98d0ee7f73812862b9a1c02", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "347-03", "group": "347", "title": "Hamiltonian Fixed Points Below the Stable Morse Number", "pdf": "preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf", "folder": "preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026", "abstract_original": "We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8422, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "b482e3ec793c52e06a1a3f6cd92f13518a806511e2974cd4f9005eea116f5f0f", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "347-04", "group": "347", "title": "A nondegenerate counterexample to the Morse-number Arnold bound", "pdf": "preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf", "folder": "preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026", "abstract_original": "We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8431, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "f7980995971b08dfbdcf0a1dcc34d1669c2bcc24648dc3782bc111463af0ce9e", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "347-05", "group": "347", "title": "Three fixed points on the symplectic quadric threefold", "pdf": "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf", "folder": "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026", "abstract_original": "We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8440, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "bebf04613245ab3cd52f29724c9fe43e2943c07dfabd96fb21fd5394f175c474", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "348-01", "group": "348", "title": "Zero-Plane Rigidity for Einstein Four-Manifolds", "pdf": "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf", "folder": "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026", "abstract_original": "We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization $`\\mathop{\\mathrm{Ric}}\\nolimits =3g`$, both spheres have radius $`1/\\sqrt3`$. The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8456, "coverage": "catalog_only", "read_detail": null, "pages": 48, "sha256": "35a3ab1d3d2458b5d78f7b304e94e0c611c06af3f03ab8f9d89c1294683bfb31", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "348-02", "group": "348", "title": "An L² Einstein Gap for Nonnegatively Curved Four-Manifolds", "pdf": "preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf", "folder": "preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026", "abstract_original": "We prove a universal, scale-invariant L 2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S 4 , $`\\mathbb{CP}^2`$, or $`S^2\\times S^2`$. The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8465, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "3be17e2ac8ba5f5f2f8de7af7c5d8c47321d3366b0fe012d9b9771b1aaf772ef", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "348-03", "group": "348", "title": "Positively curved Einstein four-manifolds", "pdf": "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf", "folder": "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026", "abstract_original": "We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8474, "coverage": "catalog_only", "read_detail": null, "pages": 59, "sha256": "2eb05d3e177c39f3aaf6084352f658f5afcda414110db8417aec4d565d6ce407", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "349-01", "group": "349", "title": "The Solomon–Yau least-volume theorem", "pdf": "preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf", "folder": "preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026", "abstract_original": "We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m -dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8490, "coverage": "catalog_only", "read_detail": null, "pages": 32, "sha256": "b44def9b041e8f318f01e2fad404216fb52f5f1c3642cddb45df23ea8322cb96", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "350-01", "group": "350", "title": "Sharp nodal length on smooth surfaces", "pdf": "preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf", "folder": "preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026", "abstract_original": "We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most $`C\\sqrt\\lambda`$. Together with the known lower bound, this proves Yau's conjecture in this setting.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8506, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "1301138f44ca53efdafefec7a5f5f817ddf4cd341d46cc66717b61bc499948bf", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "350-03", "group": "350", "title": "Power-law violations of Yau's nodal upper bound", "pdf": "preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf", "folder": "preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026", "abstract_original": "We construct a smooth Riemannian metric on $`S^4\\times S^1`$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $`\\lambda^{1/2+\\epsilon_0}`$ for one fixed $`\\epsilon_0\\gt 0`$. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.", "archive": "math_research_02_of_08.zip", "exists": true, "source_line": 8524, "coverage": "catalog_only", "read_detail": null, "pages": 34, "sha256": "b8b6757d7a3a49d70e4cd367848cc017f777403acf72d18337c70cbfc2a410c7", "bundled_pdf": null, "local_href": "../math_sources/math_research_02_of_08/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" } ], "extra_folders": [] }