{ "archive": { "archive": "math_research_03_of_08.zip", "sha256": "3ca84563769b2e4f4276b6084989f9b0184d6fdd5ddb6edb4c5f4fc709e7d3ec", "bytes": 70248777, "crc_ok": true, "files": 1666, "physical_work_folders": 109, "preprint_pdfs": 115, "groups": [ "143", "144", "145", "146", "147", "148", "149", "150", "151", "152", "153", "154", "240", "241", "242", "243", "244", "245", "246", "247", "248", "249", "250", "251", "252", "253", "254", "255", "256", "257", "258", "259", "260", "261", "262", "263", "264", "322", "323", "324", "325", "326", "327", "328", "329", "330", "331", "332", "350", "351", "352", "353", "354", "355", "356", "357", "358", "359", "360", "361" ], "current_manuscripts": 109 }, "reading_policy": "Archive catalog parsed; PDFs opened structurally; only explicit read_detail entries represent actual reading.", "works": [ { "id": "143-01", "group": "143", "title": "Uniform bounds for planar polynomial limit cycles", "pdf": "preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf", "folder": "preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026", "abstract_original": "For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3450, "coverage": "catalog_only", "read_detail": null, "pages": 160, "sha256": "8ff8e55372a94e4f494f3412539a3dfb05cb7f36b66e7b94d9be8f99603fbe89", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "143-02", "group": "143", "title": "Two limit cycles for quintic Liénard systems", "pdf": "preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf", "folder": "preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026", "abstract_original": "Every classical Liénard system $`\\dot x=y-F(x)`$, $`\\dot y=-x`$, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3459, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "33941e46486c5fef246371cc3db57c18edb1935e49bc9e71dce6c30d8afbe71a", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "144-01", "group": "144", "title": "A smooth three-torus diffeomorphism with simple Lebesgue spectrum", "pdf": "preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf", "folder": "preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026", "abstract_original": "We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C ∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L 2 space.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3475, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "0a6684b31440fc6b221df4abae9cf68ff989d6e4978cc53fa599fb4517b035c1", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "145-01", "group": "145", "title": "Rokhlin's multiple-mixing problem for one transformation", "pdf": "preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf", "folder": "preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026", "abstract_original": "Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3491, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "45000cec6bcec95daede6af976803b7fc2ea995c5953cc2448ec48218db2b86d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "146-01", "group": "146", "title": "Positive Metric Entropy for the Standard Map at Large Parameters", "pdf": "preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf", "folder": "preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026", "abstract_original": "We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3507, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "35559ad1965dc13fa41d6ac4a39d6cd0dcd9e1b58b60f10b730dff9f8c51e2ac", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "147-01", "group": "147", "title": "Continuous Phase Foliations Create Analytic Caustic Collars", "pdf": "preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf", "folder": "preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026", "abstract_original": "For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3523, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "357beeb0c0854e2643802e176bcd394c7875954f1f954f2c0d96a351ec0b31a6", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "147-02", "group": "147", "title": "Rigidity of Smooth Billiards with a Continuous Caustic Collar", "pdf": "preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf", "folder": "preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026", "abstract_original": "We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3532, "coverage": "catalog_only", "read_detail": null, "pages": 146, "sha256": "0fefc18481d52acb6118fa389d84d4d62b8e5e08cc3fd278cb29c040a0b41150", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "148-01", "group": "148", "title": "The entropy-rate dimension formula for self-similar measures on the line", "pdf": "preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf", "folder": "preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026", "abstract_original": "We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3548, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "a09f48f43c41539211024d8e079580fe0712dc3dfd8035a16f7057d9b8b5d198", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "149-01", "group": "149", "title": "Uniform Permanence in Weakly Reversible Mass-Action Systems", "pdf": "preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf", "folder": "preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026", "abstract_original": "We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3564, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "69edba6a8946a5517aae4b2add648ff9782069abe6409f7ecd31a82428853f07", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "149-02", "group": "149", "title": "Boundedness and persistence of weakly reversible mass-action systems", "pdf": "preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf", "folder": "preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026", "abstract_original": "We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3573, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "f953bd681c2ddb7d2f502f4712841e48e044f1a73ff2e4d00f875338b4ade507", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "150-01", "group": "150", "title": "Weak mixing of triangular billiards with an irrational angle", "pdf": "preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf", "folder": "preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026", "abstract_original": "We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3589, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "2d03664d20b89a527c73f2eb1fa746bee055ff436ae8e0a274131b8cdb153492", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "150-02", "group": "150", "title": "Ergodicity of triangular billiards with an irrational angle", "pdf": "preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf", "folder": "preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026", "abstract_original": "We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3598, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "200dd093e6de75c66f774cccabb1f2176c800c697752a88aa63ab63b73f0e0cc", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "151-01", "group": "151", "title": "A C^1 Counterexample to the Entropy Conjecture", "pdf": "preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf", "folder": "preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026", "abstract_original": "We disprove the general C 1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C 1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3614, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "b94336b4fbbd3b5589bd97ffa954ce7cd46d4c9599c7d4b24fe3cd00ad7ad8a1", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "152-01", "group": "152", "title": "A zero-entropy system without a smooth positive-volume model", "pdf": "preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf", "folder": "preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026", "abstract_original": "We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C ∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3630, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "67d6dd8619de6c6fc11eb88edd3791b180a2c3c9bc6d5a6a68f272a5ab445687", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "153-01", "group": "153", "title": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "pdf": "preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf", "folder": "preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026", "abstract_original": "We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter $`\\lambda\\in(0,1)`$. The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm. We also establish singular examples beyond reciprocals of Pisot numbers: singularity holds at the reciprocal of every quartic Salem number in $`(1,2)`$ and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3646, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "d7c67b840e4422308a315ddebfa30f7eaddf8dde07affa3954afa84ac63034a2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "154-01", "group": "154", "title": "Pointwise Multiple Ergodic Averages for Mixing Transformations", "pdf": "preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf", "folder": "preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026", "abstract_original": "Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3662, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "b9d8fcffac455475a2d846d1e28904f920e10b73dff498b718783023524f7d77", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "154-02", "group": "154", "title": "Pointwise convergence of fourfold ergodic averages for mixing transformations", "pdf": "preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf", "folder": "preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026", "abstract_original": "We prove that fourfold ergodic averages along the times $`n,2n,3n,4n`$ converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3671, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "dbb86e1a3da44a8bcccb1b8ab6bd0b8083ebfbebd64c67f880db59f8ddbada7e", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "154-03", "group": "154", "title": "Triple ergodic averages with distinct integer slopes", "pdf": "preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf", "folder": "preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026", "abstract_original": "For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3680, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "883a45bc7b95cfafd7049d00ebd2b26ca233a089e6aa03e19c0c16b25bfda74e", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "154-04", "group": "154", "title": "Pointwise convergence of triple ergodic averages for mixing transformations", "pdf": "preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf", "folder": "preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026", "abstract_original": "We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $`(X,\\mathcal F,\\mu)`$, with measurable inverse. For every triple of bounded measurable functions $`f_1,f_2,f_3:X\\to\\mathbb C`$, $`\\displaystyle \\frac1N\\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) \\longrightarrow \\prod_{j=1}^3\\int_X f_j\\,d\\mu`$ for μ -almost every x as $`N\\to\\infty`$ through all positive integers.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 3689, "coverage": "catalog_only", "read_detail": null, "pages": 61, "sha256": "e76fd7732f10b2501961f4c0e1febe00c3ed29ec553aefd4fb97b31c5473091d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "240-01", "group": "240", "title": "A CH obstruction to a prescribed categoricity threshold", "pdf": "preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf", "folder": "preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026", "abstract_original": "Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ 0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality $`\\beth_{\\omega_2}`$. Thus categoricity does not transfer down to the proposed bound $`\\beth_{(2^{\\aleph_0})^+}`$, which equals $`\\beth_{\\omega_2}`$ under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 5987, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "328d494bf2f1938f3a6c61fd892863f95b04e5e182666d5ef708edae49cf1ecb", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "240-02", "group": "240", "title": "Eventual categoricity for abstract elementary classes", "pdf": "preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf", "folder": "preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026", "abstract_original": "We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 5996, "coverage": "catalog_only", "read_detail": null, "pages": 121, "sha256": "dcb7401285f5ac78750c1651b59176a1ce390e545e29048b0fdd06f7de44c839", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "241-01", "group": "241", "title": "Rigidity of the Turing degrees", "pdf": "preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf", "folder": "preprints/Rigidity-of-the-Turing-degrees-September-24-2026", "abstract_original": "We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6012, "coverage": "catalog_only", "read_detail": null, "pages": 10, "sha256": "70adb04239703baa0764da565c161b0900c82a708a39602b68a550c374e96631", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "242-01", "group": "242", "title": "Single-fold Diophantine representations", "pdf": "preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf", "folder": "preprints/Single-fold-Diophantine-representations-September-24-2026", "abstract_original": "Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6028, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "0fa6bf81ce22ef3a220e0d05ef9dde5cd3af01fb2eb2c493b43dbcd5d79077f5", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "243-01", "group": "243", "title": "Choiceless polynomial time with counting does not capture polynomial time", "pdf": "preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf", "folder": "preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026", "abstract_original": "We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽 3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6044, "coverage": "catalog_only", "read_detail": null, "pages": 34, "sha256": "3e16a83bb7707c83952e2068363ee9b0c8e990f258405058686ee686eccbf10f", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "243-02", "group": "243", "title": "Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting", "pdf": "preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf", "folder": "preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026", "abstract_original": "We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6053, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "d648a0d83b5f41b13ec8fd546b97119e52d525013b6d4a4d00fcaa3c386ddb9f", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "244-01", "group": "244", "title": "The Partition Principle does not imply Choice", "pdf": "preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf", "folder": "preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026", "abstract_original": "We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6069, "coverage": "catalog_only", "read_detail": null, "pages": 44, "sha256": "7101fb4828d8d2ba26e000ee813f467bc38fb974f00ed38d8da37926e4386005", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "245-01", "group": "245", "title": "Weak and strong normalization in pure type systems", "pdf": "preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf", "folder": "preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026", "abstract_original": "We prove that every weakly β -normalizing pure type system is strongly β -normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β -Barendregt–Geuvers–Klop conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6085, "coverage": "catalog_only", "read_detail": null, "pages": 72, "sha256": "9db295a439634e3169700af595760553afafc9c5db979de9d2ac27a534af63ef", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "246-01", "group": "246", "title": "A Modulus Proof of Cannon’s Conjecture", "pdf": "preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026", "abstract_original": "We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6101, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "650697e8e95aa8cbf86c4f3898d399d75d3f1551d683f9db6e254b370aea080b", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "247-01", "group": "247", "title": "An infinite finitely presented residually finite 2-group", "pdf": "preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf", "folder": "preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026", "abstract_original": "We prove that the infinite, ordinarily finitely presented periodic Steinberg group $`\\Gamma=\\mathop{\\mathrm{St}}\\nolimits _{12}(R)`$ of a companion paper is residually finite. Its finite-index subgroup $`G=\\ker(\\Gamma\\to\\mathop{\\mathrm{St}}\\nolimits _{12}(\\mathbb F_2))`$ is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6117, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "756b9ed247d6d50a90e8d4e2bf138126847cecb25fdfa1147592863ca014de23", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "247-02", "group": "247", "title": "An infinite finitely presented periodic group", "pdf": "preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf", "folder": "preprints/An-infinite-finitely-presented-periodic-group-September-23-2026", "abstract_original": "We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽 2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh's algebraic finiteness question negatively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6126, "coverage": "catalog_only", "read_detail": null, "pages": 48, "sha256": "df870e288cb454596de6d9fce348203ca573e2c2d22d8b4c728d367fb372e7f8", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "248-01", "group": "248", "title": "Thompson's group F is nonamenable", "pdf": "preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf", "folder": "preprints/Thompsons-group-F-is-nonamenable-September-23-2026", "abstract_original": "We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F .", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6142, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "a474b04a7e94e172d4260fc42c3643f1f9a4924d7383e12448f4ba8d636175e5", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "249-01", "group": "249", "title": "A finitely generated counterexample to the Eilenberg–Ganea conjecture", "pdf": "preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf", "folder": "preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026", "abstract_original": "We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6158, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "bbc781ac6f1739388fb537680e6597ddcfdce94978578b2e413666e592d1875d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "250-01", "group": "250", "title": "Finite algebraic envelopes and the Boone–Higman conjecture", "pdf": "preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf", "folder": "preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026", "abstract_original": "We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6174, "coverage": "catalog_only", "read_detail": null, "pages": 35, "sha256": "ff461d9e2465f187f52a0479a0931b473ea9e5bc2c397ebed7ed93bacc60ea46", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "250-02", "group": "250", "title": "Simple F∞ overgroups of groups with decidable word problem", "pdf": "preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf", "folder": "preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026", "abstract_original": "Every finitely generated group with decidable word problem embeds in a simple group of type F ∞ . This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6183, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "2e99bf33c152d2e248c4feeccddfe64a8d83cb6ad61d822bcc427b60ae5e918d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "250-03", "group": "250", "title": "A universal group of type F∞", "pdf": "preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf", "folder": "preprints/A-universal-group-of-type-F-infinity-September-23-2026", "abstract_original": "We construct a single group of type F ∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F ∞ form of the higher-dimensional Higman embedding question.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6192, "coverage": "catalog_only", "read_detail": null, "pages": 18, "sha256": "67e7907dc8afc00512ff80ba196072860e0d237e5703d9f013af4dbfde835733", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "251-01", "group": "251", "title": "Unitarizability implies amenability for discrete groups", "pdf": "preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf", "folder": "preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026", "abstract_original": "We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6208, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "deec58813ffe560edaa6ffa48b13569ed8017145c7165f4da8a50b8b9ba6e9e6", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "251-02", "group": "251", "title": "Strong Ulam Stability Characterizes Amenability", "pdf": "preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf", "folder": "preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026", "abstract_original": "A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6217, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "b28d280b0c673c6904f2459168a8a5dba32ff7363158f7dc101abb1ed3cb3911", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "252-01", "group": "252", "title": "A torsion-free hyperbolic group that is not residually finite", "pdf": "preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf", "folder": "preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026", "abstract_original": "We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6233, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "8c67817e29f63b66daf5697bb4bab7913ee7d65c5f656e6be54a6a2e856705f9", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "253-01", "group": "253", "title": "An infinite finitely presented simple amenable group", "pdf": "preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf", "folder": "preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026", "abstract_original": "We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6249, "coverage": "catalog_only", "read_detail": null, "pages": 58, "sha256": "810ff1d8e0901a21d4cb9075a3dc4716716ebfd75f2279583b335bdfdf43f5f0", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "254-01", "group": "254", "title": "Harmonic heights and the Artin K(pi,1) conjecture", "pdf": "preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf", "folder": "preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026", "abstract_original": "We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin $`K(\\pi,1)`$ conjecture in finite rank.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6265, "coverage": "catalog_only", "read_detail": null, "pages": 62, "sha256": "08da57cf59c88b26c0af15bc72e375ff6e316c8ebe50057e79ad7d6e14f19039", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "254-02", "group": "254", "title": "An Artin group with no geometric CAT(0) action", "pdf": "preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf", "folder": "preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026", "abstract_original": "We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6274, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "42a55e77bec64b365f0a37d92c7f0dd020a65a3418483cfc8b824d7d13ae8549", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "254-03", "group": "254", "title": "Parabolic intersections in Artin groups", "pdf": "preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf", "folder": "preprints/Parabolic-intersections-in-Artin-groups-September-23-2026", "abstract_original": "We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6283, "coverage": "catalog_only", "read_detail": null, "pages": 55, "sha256": "4cd7fa316371447c2cb8a99d36bf9f48934c2366550609917210574e3f4f7e2e", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "255-01", "group": "255", "title": "Quasi-isometric recognition of virtually polycyclic groups", "pdf": "preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf", "folder": "preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026", "abstract_original": "We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6299, "coverage": "catalog_only", "read_detail": null, "pages": 73, "sha256": "c7261a742d7ca88303c7c38071e7e69fc2a53d953a1d7890fbcc5aeb04d137b2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "256-01", "group": "256", "title": "Nonsingular systems of equations over arbitrary groups", "pdf": "preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf", "folder": "preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026", "abstract_original": "Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6315, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "732c93a33db2c8c7da84a2bc8505ac9e00f7fe345731cf9a4bfe00b82c17cea2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "256-02", "group": "256", "title": "The Kervaire theorem for groups", "pdf": "preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf", "folder": "preprints/The-Kervaire-Theorem-for-Groups-September-24-2026", "abstract_original": "We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6324, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "f43dad0504bcbe962f20a9e9a6f9eb016a21f5abff1e03cbc2c062ecd8f04449", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "257-01", "group": "257", "title": "A hyperbolic group with no geometric CAT(0) action", "pdf": "preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf", "folder": "preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026", "abstract_original": "We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6340, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "124f55b0821cc7f18c1b64600b5e5157d939bbc4a6f66294dc1456fc633eb264", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "258-01", "group": "258", "title": "Baumslag-Solitar-free one-relator groups are hyperbolic", "pdf": "preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf", "folder": "preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026", "abstract_original": "We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group $`\\mathop{\\mathrm{BS}}\\nolimits (m,n)`$, for nonzero integers m , n , is word-hyperbolic.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6356, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "b57883368214111506c2a050d54f486630c66b36ec467fe9cf746ed133557b93", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "258-02", "group": "258", "title": "Virtual compact specialness of hyperbolic one-relator groups", "pdf": "preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf", "folder": "preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026", "abstract_original": "We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6365, "coverage": "catalog_only", "read_detail": null, "pages": 145, "sha256": "a6c45541e6943005cb7dd13535773d9acbbce746b730c1771bf7681e957f2ad6", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "259-01", "group": "259", "title": "A group without fixed price", "pdf": "preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf", "folder": "preprints/A-group-without-fixed-price-October-5-2026", "abstract_original": "We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6381, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "aa2ee4de0db26fc8aa62f95ec695e2ef11db6c9b326d2d354465aa50bee4c078", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-01", "group": "260", "title": "The spacetime Penrose inequality with charge and original-data rigidity", "pdf": "preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf", "folder": "preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026", "abstract_original": "Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and r A for the minimum-enclosing-area radius, the bound is m ≥ Q and $`r_A\\le m+\\sqrt{m^2-Q^2}`$. The polynomial mass bound $`m\\ge(r_A+Q^2/r_A)/2`$ is asserted only when $`r_A\\gt Q`$. When m > Q , equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses identifies the original data as a smooth spacelike slice of dyonic Reissner–Nordström, including smooth attachment at the future horizon.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6397, "coverage": "catalog_only", "read_detail": null, "pages": 91, "sha256": "84ca5bc05bc907a1da73b742544fe87e6fb9d22ce263c2fbe8f84752f9e5151f", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-02", "group": "260", "title": "A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four", "pdf": "preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf", "folder": "preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026", "abstract_original": "Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4 for one-ended charged data satisfying the charged dominant energy condition and $`\\mathop{\\mathrm{div}}\\nolimits _g E=0`$, with the specified decay, integrability, and finite-flux assumptions. The data may have arbitrary interior topology and ADM momentum, with a future or past trapping sign chosen independently on each boundary component. Positive enclosing area and strict ADM timelikeness are conclusions. The polynomial mass rearrangement is asserted only when the $`(n-2)`$nd power of the enclosing-area radius exceeds $`|Q|`$. When $`m\\gt |Q|`$, equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses recovers the original metric, second fundamental form, and electric field as a global spacelike slice of a Reissner–Nordström–Tangherlini exterior. The equality classification includes only slices whose induced magnetic two-form vanishes.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6406, "coverage": "catalog_only", "read_detail": null, "pages": 101, "sha256": "3e5e445d305051a15f2f9925df4f38372f1fc0892ba516501d6e37795f1ba837", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-03", "group": "260", "title": "Spacetime Penrose inequalities: enclosing area, charge, and rigidity", "pdf": "preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf", "folder": "preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026", "abstract_original": "We prove sharp spacetime Penrose inequalities for invariant ADM mass and minimum enclosing area in three and four spatial dimensions. Under their respective designated-end conventions, the neutral numerical results allow arbitrary second fundamental form, nonzero momentum, disconnected weakly future trapped boundary, and finitely many ends. Under the stated horizon hypotheses, equality for the neutral inequalities reconstructs the original exterior data as spacelike slices of Schwarzschild spacetime in three dimensions and Schwarzschild–Tangherlini spacetime in four; the four-dimensional equality theorem concerns a connected, one-ended exterior. We also prove a three-dimensional electric–magnetic charged upper-area inequality for one-ended data with nonzero momentum, with separate purely electric rest-frame corollaries allowing finitely many ends. Finally, for each fixed transverse-traceless seed and prescribed decaying solution branch, we prove a local Schwarzschild–anti-de Sitter inequality.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6415, "coverage": "catalog_only", "read_detail": null, "pages": 262, "sha256": "794bf524061c4b36d238e5fd7a60165b275476010425528d5a2a4e18c8bf6886", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-04", "group": "260", "title": "The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors", "pdf": "preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf", "folder": "preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026", "abstract_original": "We prove the sharp Kerr–Newman Penrose inequality $`\\displaystyle m^2\\ge \\frac{A}{16\\pi}+\\frac{Q^2}{2} +\\frac{\\pi(Q^4+4J^2)}{A}.`$ Here $`Q^2=Q_e^2+Q_b^2`$, and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition $`A\\ge4\\pi\\sqrt{Q^4+4J^2}`$. No maximality assumption is made. On the strict area branch, equality within this class characterizes the original data as an admissible spacelike exterior slice of a subextremal dyonic Kerr–Newman spacetime, with the boundary mapped smoothly to a future-horizon cross-section or the bifurcation sphere.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6424, "coverage": "catalog_only", "read_detail": null, "pages": 88, "sha256": "ff3098c8faf29685970c83683b711fa03522f2ae7ac50f80f06d884da6cd71b3", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-05", "group": "260", "title": "Electromagnetic tails and the Kerr–Newman Penrose inequality", "pdf": "preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf", "folder": "preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026", "abstract_original": "We construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only $`O(r^{-2})`$ decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero. They have a connected outermost, outer-area-minimizing future marginally outer trapped boundary and lie strictly on the physical area branch. We also obtain equality examples whose original data admit no Kerr–Newman spacelike realization with the corresponding mass, angular momentum, and charges. These counterexamples do not refute formulations using conserved total angular momentum with its electromagnetic correction or stronger Coulomb asymptotics.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6437, "coverage": "catalog_only", "read_detail": null, "pages": 17, "sha256": "ffdf93f60ad3ab232fba8a47ed21619410901366a3da8fcd53812c0acdfb91d3", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-06", "group": "260", "title": "The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity", "pdf": "preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf", "folder": "preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026", "abstract_original": "We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts. On the connected outermost, outer area-minimizing horizon subclass, equality characterizes the original data as a spacelike hypersurface in Schwarzschild–anti-de Sitter spacetime with matching metric mass. No maximality, evolution, or auxiliary solvability assumption is used.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6446, "coverage": "catalog_only", "read_detail": null, "pages": 153, "sha256": "9e2b9ba2768918dcc5b8abd44ad47e4662fe0b62662c9b049e014ffc09861065", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-07", "group": "260", "title": "The Penrose inequality for maximal asymptotically hyperbolic initial data", "pdf": "preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf", "folder": "preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026", "abstract_original": "We prove the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary. The boundary may be disconnected, and no restriction is imposed on the compact topology.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6455, "coverage": "catalog_only", "read_detail": null, "pages": 88, "sha256": "641d4aa3892adbca1c14c7bca0f165edc6b508e88f6b18880e3494a5495c39c2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-08", "group": "260", "title": "A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data", "pdf": "preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf", "folder": "preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026", "abstract_original": "We prove the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions. The bound uses the area of the marginally outer trapped boundary itself; it requires neither outermostness nor outer area-minimization, and no bulk-versus-boundary domination condition. For sufficiently small positive parameters, equality holds for radial seeds and the inequality is strict otherwise.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6464, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "624b287b6675a96256bf93ce5eefca9e08e04d6655f6808fd8529e6001b75012", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-09", "group": "260", "title": "The spacetime Penrose inequality and enclosing area", "pdf": "preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf", "folder": "preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026", "abstract_original": "We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric. In dimensions three and four, two metric derivatives and one tensor derivative suffice, and both positive enclosing area and future timelikeness follow from the hypotheses.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6473, "coverage": "catalog_only", "read_detail": null, "pages": 100, "sha256": "9bb0c350914915db56195840fc40dc31c8ee792628d488148a66f86940d1e35d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-10", "group": "260", "title": "Conformal flow and the Riemannian Penrose inequality with minimizing frontiers", "pdf": "preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf", "folder": "preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026", "abstract_original": "We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6482, "coverage": "catalog_only", "read_detail": null, "pages": 71, "sha256": "d49778ad5b338b573424e49f07cc0f73af92d5bff4c9ffa16ae34d3ccf20853c", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-11", "group": "260", "title": "Equality and rigidity in the spacetime Penrose inequality", "pdf": "preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "folder": "preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026", "abstract_original": "Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6491, "coverage": "catalog_only", "read_detail": null, "pages": 160, "sha256": "4f1aab59ba484eb8a903daed582b30ffe17bbd42392e233213a30abb3aadc710", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-12", "group": "260", "title": "Boundary graph deformations for the spacetime Penrose inequality", "pdf": "preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "folder": "preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026", "abstract_original": "We construct graph and conformal deformations of trapped initial-data exteriors in three and four spatial dimensions. The resulting metrics have nonnegative scalar curvature, strictly negative inner mean curvature, a lower bound protecting every enclosing cut, and an arbitrarily small upper error in ADM energy. We also give a direct four-dimensional maximal vacuum construction that retains a noncompact decaying second fundamental form. These constructions give boundary routes to the corresponding numerical Penrose inequalities.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6500, "coverage": "catalog_only", "read_detail": null, "pages": 158, "sha256": "4205f51d8f1b130b487c27fa7243ad9f64747dfbfc1c7081ecb22f4030281379", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "260-13", "group": "260", "title": "Area-controlled end replacement and the Bondi Penrose inequality in the CKS class", "pdf": "preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf", "folder": "preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026", "abstract_original": "We replace a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by an asymptotically flat end while preserving the dominant energy condition and each fixed compact interior. For strictly future-timelike CKS initial-data charge, the replacements lose asymptotically no enclosing area and their ADM masses tend to the invariant Bondi mass. Combining this construction with the companion numerical spacetime Penrose theorem yields the sharp Bondi bound for possibly disconnected weakly future trapped boundaries in this class. Horizon-regular Schwarzschild exteriors attain equality at every positive mass.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6509, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "dbf670183435f2189936378a40363c6fa2f4ce746937ea430c465709ade4cb43", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "261-01", "group": "261", "title": "Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three", "pdf": "preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf", "folder": "preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026", "abstract_original": "For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤ d with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6525, "coverage": "catalog_only", "read_detail": null, "pages": 93, "sha256": "0d971c56f6ba164196838b56aa8ccac2bf0ce7fce5f5ba4cd39124b54bf918b5", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "261-02", "group": "261", "title": "Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder", "pdf": "preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf", "folder": "preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026", "abstract_original": "For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6534, "coverage": "catalog_only", "read_detail": null, "pages": 78, "sha256": "17487e3a2720b36fd7970cb6027726d8a21843bded4e1156ba998902ed96c098", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "262-01", "group": "262", "title": "Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality", "pdf": "preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf", "folder": "preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026", "abstract_original": "We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $`1/2\\lt \\gamma\\lt 3/2`$. For measurable Hermitian positive semidefinite potentials W with $`\\int_\\mathbb R\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty`$, equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6550, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "e88f3118d513e3133e672b809b58c6c7be43fcf09e802302ac0b569eeccd3298", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "262-02", "group": "262", "title": "Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials", "pdf": "preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf", "folder": "preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026", "abstract_original": "We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $`1/2\\lt \\gamma\\lt 3/2`$. The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying $`\\int_\\mathbb R\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty`$. The matrices may have arbitrary rank and need not commute at different points.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6559, "coverage": "catalog_only", "read_detail": null, "pages": 24, "sha256": "cca7663c4920764a4150cb69ec72cd164137f895bfdf7ca11a3a9ac3e33509a5", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "262-03", "group": "262", "title": "Sharp one-dimensional Lieb–Thirring constants", "pdf": "preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf", "folder": "preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026", "abstract_original": "We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $`\\frac12\\lt \\gamma\\lt \\frac32`$, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in $`L^{\\gamma+1/2}(\\mathbb R)`$, with all negative eigenvalues included.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6568, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "58446aaaaebf0ed0beca386eef559b48fee5f5cd8a4f97304e79a81f7613b23e", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "263-01", "group": "263", "title": "Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms", "pdf": "preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf", "folder": "preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026", "abstract_original": "We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most $`Z+CM`$ electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6584, "coverage": "catalog_only", "read_detail": null, "pages": 84, "sha256": "5abedb3de1f8ed28c500a0bb91ddf1cc1f31d8ac4ae4aecc6b28c5cef4aeeacf", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "263-02", "group": "263", "title": "Generalized ionization energies for full Coulomb atoms", "pdf": "preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf", "folder": "preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026", "abstract_original": "We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to $`a_{\\mathrm{TF}}m^{7/3}`$ whenever $`m\\to\\infty`$ and $`Z/m\\to\\infty`$, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6593, "coverage": "catalog_only", "read_detail": null, "pages": 33, "sha256": "1450c11be0743629e3d65fc8318c005fee278d61019f5021d2383b697ac787e8", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "263-03", "group": "263", "title": "Generalized outer-electron radii of neutral Coulomb atoms", "pdf": "preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf", "folder": "preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026", "abstract_original": "We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to $`(81\\pi^2/2)^{1/3}m^{-1/3}`$ as m tends to infinity.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6602, "coverage": "catalog_only", "read_detail": null, "pages": 17, "sha256": "15737301853393ce7fad8549c68f0b53e5a90667258b7caa0bf47bf7f87b7bd9", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "264-02", "group": "264", "title": "Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes", "pdf": "preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf", "folder": "preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026", "abstract_original": "For each fixed Kerr spacetime with mass M > 0 and rotation $`0\\lt \\mathfrak a\\lt M`$, we prove that a dense G δ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C 1 extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 6627, "coverage": "catalog_only", "read_detail": null, "pages": 68, "sha256": "702eacef09cb61650fbd1733c826160c548ca6c09202f4a937e58875b7738763", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "322-01", "group": "322", "title": "A positive solution to Tingley’s problem", "pdf": "preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf", "folder": "preprints/A-positive-solution-to-Tingleys-problem-September-23-2026", "abstract_original": "Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7879, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "6b4ae2372e8f56539f55cd0678357248520fc7b7b095b1812690968f68e6f6ae", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "323-01", "group": "323", "title": "Relative independence of the separable quotient problem", "pdf": "preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf", "folder": "preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026", "abstract_original": "The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ 1 , we obtain independence relative to the consistency of ZFC alone.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7895, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "77cc28d21bfb07f729a6e0e4ae85ff22fd32e7e2a24831e79a15e6db4acf54ad", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "324-01", "group": "324", "title": "Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic", "pdf": "preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf", "folder": "preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026", "abstract_original": "There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7911, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "cc670229303a1ed9044a6bb04c397dd5b09a5c4046f08aa637ec8b112d6fa879", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "324-02", "group": "324", "title": "Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0", "pdf": "preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf", "folder": "preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026", "abstract_original": "We construct a separable real Banach space Z that contains no linear copy of c 0 , yet is bi-Lipschitz equivalent to $`Z\\oplus_\\infty c_0`$. The same space contains a bi-Lipschitz image of every separable metric space.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7920, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "4779ddc34774d95aedfb77742849d2b9b4e9078dff503d82cef542539731d1dc", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "325-01", "group": "325", "title": "A direct proof of the complete Crouzeix inequality", "pdf": "preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf", "folder": "preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026", "abstract_original": "We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7936, "coverage": "intro_theorems", "read_detail": { "work": "325-01", "pages": "1–3", "scope": "Главная теорема, предшественники, аналитическая схема" }, "pages": 12, "sha256": "d9aa4ef2dccb1f044fdf78afd7a4c93f64e71bd60e53d0f0460329596f199bb6", "bundled_pdf": "sources/papers/325-01/paper.pdf", "local_href": "../math_sources/math_research_03_of_08/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "325-02", "group": "325", "title": "The complete Crouzeix theorem: optimal similarity and a common positive boundary representation", "pdf": "preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf", "folder": "preprints/The-complete-Crouzeix-theorem-September-23-2026", "abstract_original": "We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with condition number at most two. For the similar matrix, one continuous positive boundary density of mass the identity represents the evaluation of every matrix-valued function holomorphic near the closed domain.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7945, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "b5e37fffcae2be84577cdc73f8d0ae459ecc85da994dcf605c38f69539cc962b", "bundled_pdf": "sources/papers/325-02/paper.pdf", "local_href": "../math_sources/math_research_03_of_08/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "326-01", "group": "326", "title": "The cotype–cotype conjecture under the approximation property", "pdf": "preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf", "folder": "preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026", "abstract_original": "We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K -convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7961, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "c2f55fe9562e2ee729cee24795789294679cea7160218c585b71bf15896be4b8", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "327-01", "group": "327", "title": "Nontrivial Markov Type Forces Superreflexivity", "pdf": "preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf", "folder": "preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026", "abstract_original": "We prove that every real Banach space with Markov type p > 1 is superreflexive. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. This answers Naor's question: every real Banach space with nontrivial Markov type admits an equivalent uniformly smooth norm.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7977, "coverage": "catalog_only", "read_detail": null, "pages": 14, "sha256": "5df09ebb518ff529acd50e9325765d8f7301a2fa83891056681af7960be29709", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "328-01", "group": "328", "title": "Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces", "pdf": "preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf", "folder": "preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026", "abstract_original": "Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 7993, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "0e1524096dc221e727fb67d508473d394aaf0cc9a5a1c75b6c3b5831fcddf06a", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "329-01", "group": "329", "title": "Counterexamples to the duality conjecture for metric entropy", "pdf": "preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf", "folder": "preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026", "abstract_original": "We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8009, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "6d6f9d0518a6e50da9bb3c026ec384b22edd12a2f112eec67832c76b62c9c8f7", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "330-01", "group": "330", "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "pdf": "preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf", "folder": "preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026", "abstract_original": "We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8025, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "aa4ce066ba72c12f5984b7f401355a7030c1cecc9d9148278113a29e89914083", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-01", "group": "331", "title": "Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space", "pdf": "preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf", "folder": "preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026", "abstract_original": "We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least $`\\sqrt{1+t^2/12}-1`$, and the countably branching diamond of depth k has distortion at least $`\\sqrt{1+k/12}`$ in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8041, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "1e08111eaf334a068d25387a687d1236aa74d5775890fa2d6f978b7a3bde5d66", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-02", "group": "331", "title": "Midpoint lenses in segment spaces", "pdf": "preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf", "folder": "preprints/Midpoint-lenses-in-segment-spaces-September-27-2026", "abstract_original": "For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by $`2\\sqrt{R^2-\\|x\\|^2}`$, where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8050, "coverage": "catalog_only", "read_detail": null, "pages": 74, "sha256": "e5fa3ffa1ac9778a6f3c639304abed4104d9e905f1407b7f9d78e2eebb42f2eb", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-03", "group": "331", "title": "Distortion of countably branching diamonds from midpoint and tree energies", "pdf": "preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf", "folder": "preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026", "abstract_original": "We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion- D embedding of the depth- k diamond satisfies $`D^2\\ge1+k/4`$. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8059, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "d71f7eafe03d0d83cf972e144eaf726573ca2e8e8dc17e37f520295951e85f5e", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-04", "group": "331", "title": "Exact asymptotic moduli in a Daugavet subspace of L1", "pdf": "preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf", "folder": "preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026", "abstract_original": "For an infinite-dimensional real subspace of L 1 whose unit ball is totally bounded in measure and whose norm has the Daugavet property, we compute the averaged midpoint and one-sided asymptotic moduli at every unit center. They are $`\\max\\{t/2,t-1\\}`$ and $`\\max\\{0,t-2\\}`$, respectively. The same weak-neighborhood geometry excludes every equivalent asymptotically uniformly convex norm. A quantitative realization of the Kadets–Werner construction supplies a space with both hypotheses.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8068, "coverage": "catalog_only", "read_detail": null, "pages": 13, "sha256": "92b3cc3a790bb7fe171866f3e185a5416409cdb56689c890cc53f1fa9b27a903", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-05", "group": "331", "title": "Midpoint convexity from bounded tree potentials and path costs", "pdf": "preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf", "folder": "preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026", "abstract_original": "Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound $`\\widehat\\delta(t)\\ge\\sqrt{1+t^2/4}-1`$ for $`0\\lt t\\lt 1`$, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8077, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "965c8f078bcaef0396003b94e27549eb15318b32ec27b7f467f26746af6c3f2c", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-06", "group": "331", "title": "Independent products in real L1: asymptotic midpoint convexity without AUC renormings", "pdf": "preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf", "folder": "preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026", "abstract_original": "For a countably branching tree, the closed real L 1 spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8086, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "40d5f05445bd8d39f9bcbc21fc8cc082ea246498e77bcc7e72a5560adc68018c", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "331-07", "group": "331", "title": "Midpoint convexity from two recursive potentials", "pdf": "preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf", "folder": "preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026", "abstract_original": "We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least $`t^3/128`$ for $`0\\lt t\\lt 1`$, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8095, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "649d9f5632eb0ab801cd8aeea8293633e37ec2b58b37ec280527b2e1794f9b2a", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "332-01", "group": "332", "title": "Metric Markov Cotype Two of ℓ 1", "pdf": "preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf", "folder": "preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026", "abstract_original": "We prove that the real Banach space ℓ 1 has metric Markov cotype two, answering a question of Mendel and Naor. As a consequence, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ 1 extends to the whole Hilbert space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8111, "coverage": "catalog_only", "read_detail": null, "pages": 9, "sha256": "d1977f73e5a70137791d0739b58b049fa30e3ad9caa4dfeb8353142db37ab60f", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "350-02", "group": "350", "title": "Smooth counterexamples to Yau's nodal upper bound in dimensions three and four", "pdf": "preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf", "folder": "preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026", "abstract_original": "We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $`S^2\\times\\mathbb T^2`$ for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8515, "coverage": "catalog_only", "read_detail": null, "pages": 60, "sha256": "8e4c129bdb0f2f45d12278030946f76af0dccb5c373cac9797933dccee1bee80", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "351-01", "group": "351", "title": "A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup", "pdf": "preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf", "folder": "preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026", "abstract_original": "We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8540, "coverage": "catalog_only", "read_detail": null, "pages": 20, "sha256": "88b1d8270cb2bc5fb315b3421f984f9db84a4431d4780f89b1bcf6504839510b", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "351-02", "group": "351", "title": "Bounded scalar curvature and smooth extension of four-dimensional Ricci flow", "pdf": "preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf", "folder": "preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026", "abstract_original": "We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8549, "coverage": "catalog_only", "read_detail": null, "pages": 58, "sha256": "95655c4053c94f2bf129eddfda0c10b9d2c007f45a10548a02cd1c5624567bca", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "351-03", "group": "351", "title": "Path selection and an elliptic inequality on degenerating Ricci-flat trees", "pdf": "preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf", "folder": "preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026", "abstract_original": "We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies $`E=o(M)`$. The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8558, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "a49f3dc92b22b888e356c7d8e02620683b9e1c00ca1b28212cf800dd4d692327", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "352-01", "group": "352", "title": "A finite-time singularity of Calabi flow on projective space", "pdf": "preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf", "folder": "preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026", "abstract_original": "We construct a smooth Kähler metric in the Fubini–Study class of $`\\mathbb{CP}^{10}`$ whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8574, "coverage": "catalog_only", "read_detail": null, "pages": 54, "sha256": "486e9e5ff47e92a4db90c7904eb5842085de78f02f825f5b68bcc5bd2719f25d", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "353-01", "group": "353", "title": "A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten", "pdf": "preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf", "folder": "preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026", "abstract_original": "We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8590, "coverage": "catalog_only", "read_detail": null, "pages": 7, "sha256": "f7851c8489621a2df0e520d7aebe9c4dca5e6d3e68c9086d914b479dc9983da9", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "353-02", "group": "353", "title": "The affine Bernstein theorem in dimensions three through nine", "pdf": "preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf", "folder": "preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026", "abstract_original": "We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8599, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "f028b02be7ed31404334d03f5874bd368ad32b2a5b4d5820fc075bac0872efb3", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "354-01", "group": "354", "title": "The Isoperimetric Conjecture for the Cubic Flat Three-Torus", "pdf": "preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf", "folder": "preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026", "abstract_original": "We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes $`4\\pi/81`$ and $`1/\\pi`$. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8615, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "ce66c455780d2efc19b18a9b767a7453d2e951d55f4b58d5167ec2716b3974c2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "355-01", "group": "355", "title": "Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow", "pdf": "preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf", "folder": "preprints/Tangent-flow-uniqueness-2026-09-24", "abstract_original": "We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ 3 , all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8631, "coverage": "catalog_only", "read_detail": null, "pages": 100, "sha256": "19b2f77a4cbfb0d5180a6456312c545da5be942f849ca7f7fad31b1af2566a8a", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "356-01", "group": "356", "title": "Gigli’s distributional curvature characterization of Alexandrov spaces", "pdf": "preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf", "folder": "preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026", "abstract_original": "For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n -dimensional Alexandrov space of curvature at least κ if and only if, with reference measure $`\\mathcal H^n`$, it is a full-support $`\\mathrm{RCD}((n-1)\\kappa,n)`$ space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8647, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "e104c43128bea2923c4f6d33f911a519f0c0fd2ca6904a2bcdad807ed7a42fd6", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "356-02", "group": "356", "title": "Weak Hessian bounds along every geodesic in RCD spaces", "pdf": "preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf", "folder": "preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026", "abstract_original": "On a full-support $`\\mathrm{RCD}(K,N)`$ space with $`1\\lt N\\lt \\infty`$, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8656, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "eb34f06c23632a5745015f9e4ef363b9074540a400f95ed3de7aa9dc10aaecc2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "357-01", "group": "357", "title": "Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces", "pdf": "preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf", "folder": "preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026", "abstract_original": "We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K , every regular point of a noncollapsed $`\\mathrm{RCD}(K,n)`$ space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝ n . The bi-Lipschitz constant depends only on n , and the neighborhood uses the restricted ambient distance.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8672, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "5500abd2391d238779793572caf1df5270a1ac488f6d88cc897d5cf51ac33093", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "358-01", "group": "358", "title": "A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric", "pdf": "preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf", "folder": "preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026", "abstract_original": "We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8688, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "0ac6d8b58c960941ae1b45338abf08664ca3f61e5bcfa0e28d91758fea63c5bd", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "359-01", "group": "359", "title": "A negatively pinched Kähler threefold without bounded holomorphic coordinates", "pdf": "preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf", "folder": "preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026", "abstract_original": "We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ 3 with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8704, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "59e7bcf19b96d5d90e787622473764011de3be9d93854b39fc065f019667e022", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "359-02", "group": "359", "title": "One-sided negative sectional curvature and the holomorphic Liouville property", "pdf": "preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf", "folder": "preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026", "abstract_original": "We construct, in some sufficiently large fixed finite complex dimension m , a domain in ℂ m diffeomorphic to $`\\mathbb R^{2m}`$ that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8713, "coverage": "catalog_only", "read_detail": null, "pages": 58, "sha256": "dedbbd2f24f8281c384e8927c78a54fd71d894113da70e5f702b8deb1f5bc046", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "360-01", "group": "360", "title": "Global Support and Convex Injectivity Domains under Weak MTW", "pdf": "preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf", "folder": "preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026", "abstract_original": "We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8729, "coverage": "catalog_only", "read_detail": null, "pages": 24, "sha256": "e16087753fc93a840bd357fe203ffd1d7a511245db91f5e629132e91db440724", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "360-02", "group": "360", "title": "Uniform Bi-Holder Transport from Weak MTW", "pdf": "preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf", "folder": "preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026", "abstract_original": "We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8738, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "1e1a412dff1dd2634c638857ed802fa444b2cd044346277297f6415ba1889b10", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "361-01", "group": "361", "title": "A counterexample to integer-degree harmonic dimension comparison", "pdf": "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf", "folder": "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026", "abstract_original": "For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝ n with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8754, "coverage": "catalog_only", "read_detail": null, "pages": 35, "sha256": "c6adbdbf2331bf0fd02ee79836e71bd1cd8dae988f68a28d05d09c06f52bed32", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "361-02", "group": "361", "title": "A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison", "pdf": "preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf", "folder": "preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026", "abstract_original": "For every $`4/9\\lt v\\lt 1`$ and $`1\\lt c\\lt 9v/4`$, all sufficiently large integers k admit a complete smooth metric on ℝ 3 with nonnegative Ricci curvature, asymptotic volume ratio v , and at least $`c(k+1)^2`$ linearly independent real harmonic functions of pointwise growth at most k . This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each $`1\\lt c\\lt 9/4`$, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k .", "archive": "math_research_03_of_08.zip", "exists": true, "source_line": 8763, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "6c7b86f0f59fcdb9da790ae15c943dc881e1ca4f758033611e84eca8dd93e4a2", "bundled_pdf": null, "local_href": "../math_sources/math_research_03_of_08/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" } ], "extra_folders": [] }