{ "archive": { "archive": "math_research_06_of_08.zip", "sha256": "e19b2af39873d8224d505e96914b0b2b8c6592288f7ae8acf5744aa88d9b90ac", "bytes": 68141572, "crc_ok": true, "files": 1483, "physical_work_folders": 83, "preprint_pdfs": 85, "groups": [ "071", "072", "073", "074", "075", "076", "077", "220", "221", "222", "223", "224", "225", "226", "227", "228", "229", "230", "231", "232", "233", "234", "235", "236", "237", "238", "239" ], "current_manuscripts": 83 }, "reading_policy": "Archive catalog parsed; PDFs opened structurally; only explicit read_detail entries represent actual reading.", "works": [ { "id": "071-01", "group": "071", "title": "Removable Boundaries and Rigidity of Circle Domains", "pdf": "preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf", "folder": "preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026", "abstract_original": "We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1788, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "2793106e2bc200b83a2aac02de379412ffc6355ba8c80647c5c01dfffe537a52", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "071-02", "group": "071", "title": "Koebe's Circle-Domain Conjecture", "pdf": "preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf", "folder": "preprints/Koebes-Circle-Domain-Conjecture-September-23-2026", "abstract_original": "We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1797, "coverage": "catalog_only", "read_detail": null, "pages": 64, "sha256": "f0080bd36e38b2cd844e872509f286af431d8c061de1954bc2a27b18c22b1b4d", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "072-01", "group": "072", "title": "Brennan's conjecture and sharp inverse-square integral means", "pdf": "preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf", "folder": "preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026", "abstract_original": "We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then $`|\\varphi'|^s`$ is area-integrable for every $`4/3\\lt s\\lt 4`$. We also prove the sharp inverse-square integral-means exponent $`B_{\\mathcal S}(-2)=1`$ for the normalized schlicht class $`\\mathcal S`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1813, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "27a773e887282fd97d9de50d45c1a28b335713a4d2707d613f54a6a8ebc593bd", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "072-02", "group": "072", "title": "A strict inverse-first-power bound for univalent functions", "pdf": "preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf", "folder": "preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026", "abstract_original": "We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies $`B_b(-1)\\lt 1/4`$, disproving Kraetzer's conjectured spectrum at p = −1.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1822, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "b258aff1bca50ec32ec39d26b3e501e3b31df5e4e6db2807d3073c15826c196f", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "073-01", "group": "073", "title": "The Falconer distance conjecture in all dimensions", "pdf": "preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf", "folder": "preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026", "abstract_original": "We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝ d with Hausdorff dimension greater than $`d/2`$ determines a set of Euclidean distances of positive Lebesgue measure.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1838, "coverage": "catalog_only", "read_detail": null, "pages": 64, "sha256": "98a7e721547308b9cec673140ed1d86ecef44d73c56995b68a2c46fd8d4beb23", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "074-01", "group": "074", "title": "The Kakeya maximal conjecture in three dimensions", "pdf": "preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf", "folder": "preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026", "abstract_original": "We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps $`L^3(\\mathbb R^3)`$ to $`L^3(S^2)`$ with norm at most $`C_\\varepsilon\\delta^{-\\varepsilon}`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1854, "coverage": "catalog_only", "read_detail": null, "pages": 97, "sha256": "49cc52aa3de89f0e91d181b00ee8050fb29635ee0da9f29145d52318e7168204", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "074-02", "group": "074", "title": "Every four-dimensional Kakeya set has full Hausdorff dimension", "pdf": "preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf", "folder": "preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026", "abstract_original": "We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ 4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1863, "coverage": "catalog_only", "read_detail": null, "pages": 175, "sha256": "30d0c186aa9c697d02ab9bf736968028f1ce8a2f164f1d7291398d530035ccde", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "075-01", "group": "075", "title": "Almost-everywhere Fourier convergence in L log L", "pdf": "preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf", "folder": "preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026", "abstract_original": "We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $`L\\log L`$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $`L\\log L`$ sufficiency conjecture.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1879, "coverage": "catalog_only", "read_detail": null, "pages": 76, "sha256": "9a77d41e5b9b9bebd18d958c58bbbd02facb910ede62e549b8887c877b3d6492", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "076-01", "group": "076", "title": "Ultraflat real Littlewood polynomials", "pdf": "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf", "folder": "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026", "abstract_original": "For every $`\\varepsilon\\in(0,1)`$ and every sufficiently large integer N , there is a polynomial of length N with coefficients in $`\\{-1,1\\}`$ whose modulus lies between $`(1-\\varepsilon)\\sqrt N`$ and $`(1+\\varepsilon)\\sqrt N`$ everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1895, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "89e4fd2beb7f4dabc5123c83eb291aaf463000e4a27652456f76363932ac0d60", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "076-02", "group": "076", "title": "Nearly minimal maxima and positive minima of Littlewood polynomials", "pdf": "preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf", "folder": "preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026", "abstract_original": "For every η > 0 and every sufficiently large integer N , there is a polynomial with N consecutive coefficients in $`\\{-1,1\\}`$ whose modulus lies between $`\\sqrt N/16`$ and $`(1+\\eta)\\sqrt N`$ everywhere on the unit circle.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1904, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "639fe69dbb6e6e93441af292dde643b1a508658a43692bee514229433d706969", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "076-03", "group": "076", "title": "Asymptotically minimal maxima of real Littlewood polynomials", "pdf": "preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf", "folder": "preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026", "abstract_original": "We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in $`\\{-1,1\\}`$ is $`(1+o(1))\\sqrt N`$, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1913, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "3f6f94e6343c03b7a85c25f9fe8a8dd2f50c57f4a49cd073dc92c2fa3d54bfa2", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "077-01", "group": "077", "title": "Elliptic capacity propagation and Fourier restriction to the sphere", "pdf": "preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf", "folder": "preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026", "abstract_original": "We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps $`L^\\infty(S^2)`$ boundedly into $`L^p(\\mathbb R^3)`$ for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1929, "coverage": "catalog_only", "read_detail": null, "pages": 103, "sha256": "669efdbef915014402795e07bfbcb958ae534b6910f05a8737a6fce0c216d394", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "077-02", "group": "077", "title": "Diagonal Fourier extension for positively curved surfaces in three dimensions", "pdf": "preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf", "folder": "preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026", "abstract_original": "We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces $`\\Sigma\\subset\\mathbb R^3`$, including surfaces with smooth boundary. For every $`3\\lt p\\lt \\infty`$, the extension operator maps $`L^p(\\Sigma)`$ boundedly into $`L^p(\\mathbb R^3)`$. The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every $`3\\lt p\\lt \\infty`$ and Sobolev order $`s\\gt 2-6/p`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 1938, "coverage": "catalog_only", "read_detail": null, "pages": 25, "sha256": "098a6acc53131ae5e7ad0405fe19a121f60a31ebd1de49290dc2733167f0cd63", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "220-03", "group": "220", "title": "Directional transience implies ballisticity", "pdf": "preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf", "folder": "preprints/Directional-transience-implies-ballisticity-September-23-2026", "abstract_original": "We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤ d , d ≥ 2. This resolves the ballisticity conjecture positively.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5211, "coverage": "catalog_only", "read_detail": null, "pages": 58, "sha256": "8b305d1fbb11b2a62d76f7decbc45accf5d1c02362937d0af5c4108dd0cc1411", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "221-01", "group": "221", "title": "The Mézard–Parisi formula for diluted spin glasses", "pdf": "preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf", "folder": "preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026", "abstract_original": "We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5227, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "f415bacf0f91bf5aa188cb80f95e00fac1142996fe70d3d8c5c1372b3262a8e5", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "222-01", "group": "222", "title": "The free energy of the Ising random perceptron", "pdf": "preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf", "folder": "preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026", "abstract_original": "We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5243, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "4d848ece0b215a2b147e282fbf0ba44df3501470b8c2d78b6c6052ca3b00090d", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "222-02", "group": "222", "title": "Microscopic jamming in the negative spherical perceptron", "pdf": "preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf", "folder": "preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026", "abstract_original": "We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances $`1-\\varepsilon`$ and $`1+\\varepsilon`$, for every sufficiently small fixed ε . The contact-removed gap cumulative law and the mean-one force cumulative law satisfy $`\\displaystyle G_J(u)=u^{1-\\gamma+o(1)},\\qquad F_J(s)=s^{1+\\theta+o(1)},`$ as $`u\\downarrow0`$ and $`s\\downarrow0`$, with $`\\gamma=(2+\\theta)^{-1}`$, $`0.4126930\\lt \\gamma\\lt 0.4126934`$, and $`0.4231063\\lt \\theta\\lt 0.4231088`$. A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5252, "coverage": "catalog_only", "read_detail": null, "pages": 133, "sha256": "8d94e973e689e60bdd0443e0f466711519993188fb478dee573993afaf8ff896", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "222-03", "group": "222", "title": "The spherical perceptron with bi-orthogonally invariant disorder", "pdf": "preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf", "folder": "preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026", "abstract_original": "We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5265, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "9c819fab2884b68bb15c28e88172a972331559ec4a0cb5c7c839ee44505b859c", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "222-04", "group": "222", "title": "The free energy of the spherical random perceptron", "pdf": "preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf", "folder": "preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026", "abstract_original": "We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5274, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "f7aaf46c793f8a8e81f524092432be10b469cea2536f5ebc2b0c70689f3c6756", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-01", "group": "223", "title": "Square-lattice FK interfaces and nested loops for 1 <= q < 4", "pdf": "preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf", "folder": "preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026", "abstract_original": "For every fixed $`1\\le q\\lt 4`$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $`\\mathop{\\mathrm{SLE}}\\nolimits _{\\kappa(q)}`$, where $`\\kappa(q)=4\\pi/\\arccos(-\\sqrt q/2)`$, confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane $`\\mathop{\\mathrm{CLE}}\\nolimits _{\\kappa(q)}`$ in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5290, "coverage": "catalog_only", "read_detail": null, "pages": 110, "sha256": "d2ffa94ece599e88cd64e2dddab2ec80ca09f8ecba01a5933ccc6c3796875766", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-02", "group": "223", "title": "Self-dual random-cluster interfaces below one", "pdf": "preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf", "folder": "preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026", "abstract_original": "For every fixed $`0\\lt q\\lt 1`$, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLE κ , where $`\\kappa=4\\pi/\\arccos(-\\sqrt q/2)\\in(6,8)`$. The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5299, "coverage": "catalog_only", "read_detail": null, "pages": 207, "sha256": "bf15a86507a1018464d2dc7a5d8ad3c02921dc9b0275dd424ba846e87dd15210", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-03", "group": "223", "title": "Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces", "pdf": "preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf", "folder": "preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026", "abstract_original": "We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE 16/3 . The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5308, "coverage": "catalog_only", "read_detail": null, "pages": 49, "sha256": "c054f7856b32267e12bd228b3a510739f9d8c7a00eb1bd7e549093745866c45d", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-04", "group": "223", "title": "Thermal FK–Ising interfaces and massive SLE", "pdf": "preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf", "folder": "preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026", "abstract_original": "We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE$`_{16/3}`$ law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5317, "coverage": "catalog_only", "read_detail": null, "pages": 50, "sha256": "c2a2a4a8b4213752e486fdc5efc675ddabfe6fd2360fd88fac0b91860619bd49", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-05", "group": "223", "title": "Natural Occupation Measures for Critical Square-Lattice FK Interfaces", "pdf": "preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf", "folder": "preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026", "abstract_original": "We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight $`1\\le q\\lt 4`$. A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5326, "coverage": "catalog_only", "read_detail": null, "pages": 59, "sha256": "d0fd0c30ada6a1f0d1548054d84aa89cda9cfab54672020e2675302ba548c376", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "223-06", "group": "223", "title": "Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces", "pdf": "preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf", "folder": "preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026", "abstract_original": "For every fixed $`1\\le q\\le4`$, critical square-lattice random-cluster Dobrushin interfaces converge to chordal $`\\mathop{\\mathrm{SLE}}\\nolimits _{\\kappa(q)}`$, where $`\\kappa(q)=4\\pi/\\arccos(-\\sqrt q/2)`$, and the complete nested plane loop collections converge to whole-plane $`\\mathop{\\mathrm{CLE}}\\nolimits _{\\kappa(q)}`$. The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5335, "coverage": "catalog_only", "read_detail": null, "pages": 127, "sha256": "b14b09c2088272ae1f3b135cf349f6dbf05aa81e9cb66fc629396a8ac518d10b", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "224-01", "group": "224", "title": "From critical crossings to quenched near-critical universality in Voronoi percolation", "pdf": "preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf", "folder": "preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026", "abstract_original": "Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5351, "coverage": "catalog_only", "read_detail": null, "pages": 61, "sha256": "164f1043f61eb5e74b7aabbc0a21d9c58783af253dc37b6a76288b7844d1a922", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "224-02", "group": "224", "title": "A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula", "pdf": "preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf", "folder": "preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026", "abstract_original": "Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times $`\\varepsilon ^{-3/4}`$, where the point intensity is $`\\varepsilon ^{-2}`$. No rate of convergence in Cardy's formula is required.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5360, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "50c54704fe210d89ed392432d8b3e61c9d950c6a1d6dd171169362a41ce5d305", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "224-03", "group": "224", "title": "Cardy’s formula for critical Poisson–Voronoi percolation", "pdf": "preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf", "folder": "preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026", "abstract_original": "We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5369, "coverage": "catalog_only", "read_detail": null, "pages": 75, "sha256": "1334302b31272ae0ed36b4b42f1e93e63de1b7bc32f1015a21fdf1d22e30eafb", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "225-01", "group": "225", "title": "The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)", "pdf": "preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf", "folder": "preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026", "abstract_original": "We prove that the height function of the square-lattice six-vertex model with weights $`a=b=1`$ and $`0\\lt c\\le2`$, in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel $`-(2\\pi)^{-1}\\log|x-y|`$, the squared multiplier is $`1/\\arcsin(c/2)`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5385, "coverage": "catalog_only", "read_detail": null, "pages": 93, "sha256": "26deb84be9ead6846c9144bab5260a03659cc94d615866a5b3717e632444170f", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "226-01", "group": "226", "title": "The curve scaling limit of half-plane double dimers", "pdf": "preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf", "folder": "preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026", "abstract_original": "We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE 4 . The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE 4 scaling-limit conjecture.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5401, "coverage": "catalog_only", "read_detail": null, "pages": 33, "sha256": "2e1bcd39145db94c77ef81c50c59e09edaae5859270ab0fad1767407f95deb5e", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-01", "group": "227", "title": "Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model", "pdf": "preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf", "folder": "preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026", "abstract_original": "We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n 1/3 , and waiting times and lags are measured in units n 2/3 . The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5417, "coverage": "catalog_only", "read_detail": null, "pages": 112, "sha256": "12d511db42d09b8caeec0651174c81b13ec93d419422031a423344659347a757", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-02", "group": "227", "title": "Functional universality of critical SK autocorrelations", "pdf": "preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf", "folder": "preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026", "abstract_original": "We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n 2/3 and the mean spin autocorrelation is multiplied by n 1/3 . The functions converge in law uniformly on compact positive-time intervals. Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5426, "coverage": "catalog_only", "read_detail": null, "pages": 70, "sha256": "f38734b35e639f883e767d3fa2cfe34a4298d09a81b12df1e48bff465cc8c950", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-03", "group": "227", "title": "A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase", "pdf": "preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf", "folder": "preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026", "abstract_original": "For every fixed inverse temperature $`0\\lt \\beta\\lt 1`$, we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5435, "coverage": "catalog_only", "read_detail": null, "pages": 66, "sha256": "d1eccdc9781a9cf3ea3ff54891d8288f8ab77fad5be4f539c6ddae1d67a98bb6", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-04", "group": "227", "title": "Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase", "pdf": "preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf", "folder": "preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026", "abstract_original": "We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature $`0\\leq\\beta\\lt 1`$. With rate-one refresh at each spin, the cutoff location is $`\\log n/(2\\lambda(\\beta))`$ for a positive deterministic rate $`\\lambda(\\beta)`$. The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5444, "coverage": "catalog_only", "read_detail": null, "pages": 158, "sha256": "d99d391c8c3ed79f0dc70c9db4135300006e83dc4e09f9f3d7c5f71cda6230d5", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-05", "group": "227", "title": "Critical slowing down in the Sherrington–Kirkpatrick model", "pdf": "preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf", "folder": "preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026", "abstract_original": "At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence $`t_n=o(n^{2/3})`$ in rate-one-per-site time, the Gibbs mass of initial states whose time- t n total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence $`k_n=o(n^{5/3})`$ of uniform-site update attempts.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5453, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "20041aa4df40c551ba66fd9629f70a2efbc7778d8ea4e85c8bcfca9519826623", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-06", "group": "227", "title": "Stretched-exponential barriers for typical SK initial states", "pdf": "preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf", "folder": "preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026", "abstract_original": "At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time $`\\exp(n^{1/10000})`$ exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5462, "coverage": "catalog_only", "read_detail": null, "pages": 63, "sha256": "7bc4d5a64b6465feb0a32de1a322204f0c8f1a1def6e5e4830ba71608b5ebfda", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-07", "group": "227", "title": "A typical-start upper bound for low-temperature SK Glauber dynamics", "pdf": "preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf", "folder": "preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026", "abstract_original": "For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time $`\\exp(n^{1-1/40000000})`$, with joint probability tending to one over the disorder and the sampled state.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5471, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "a4a9ccfe1526c056978ba2bf9bd482e78438115e8116c2a525082f2027cfa690", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "227-08", "group": "227", "title": "Critical mixing in the Sherrington–Kirkpatrick model", "pdf": "preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf", "folder": "preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026", "abstract_original": "At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is $`n^{2/3+o(1)}`$ when every site has rate one, or $`n^{5/3+o(1)}`$ attempted uniform-site updates, in probability over the disorder.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5480, "coverage": "catalog_only", "read_detail": null, "pages": 190, "sha256": "3ac6296cc238bae9bab78ede1147cf4cf465d69e3fbe17c192f0f9059a9ff8a6", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "228-01", "group": "228", "title": "A continuum temperature singularity for a radial pair potential", "pdf": "preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf", "folder": "preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026", "abstract_original": "We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying $`\\phi(r)=o(r^{-3})`$. Its free-cube canonical free energy is finite at every positive inverse temperature and density.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5496, "coverage": "catalog_only", "read_detail": null, "pages": 31, "sha256": "d3c00f7387c84c306cde835b975962752138b0aca201fce5b8d5fc2a2021b0c4", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "228-02", "group": "228", "title": "A radial continuum phase transition with algebraic decay", "pdf": "preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf", "folder": "preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026", "abstract_original": "We construct a bounded, continuous, stable radial pair potential in three dimensions with $`|\\phi(r)|\\le Cr^{-3-1/32}`$ for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5505, "coverage": "catalog_only", "read_detail": null, "pages": 33, "sha256": "d53cafa68c322a072098e8a3cb2e3b7e059b4239ba95466d089632b8ce368587", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "229-01", "group": "229", "title": "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model", "pdf": "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf", "folder": "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026", "abstract_original": "We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d -ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when $`d\\lambda^2\\gt 1`$; we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5521, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "a8e3ff75c819ee3419e05b7eeca3ac5d69e46baa02fe77745f93e4c9a55adb01", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "229-02", "group": "229", "title": "A Capacity Criterion for Four-State Potts Reconstruction on Trees", "pdf": "preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf", "folder": "preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026", "abstract_original": "For the ferromagnetic four-state broadcast model with $`0\\lt \\lambda\\lt 1`$, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L 3 capacity with edge resistances $`\\lambda^{-2|e|}`$ is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5530, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "6c725aee122ce7659ea4b30bbb9805e23b41666eb9922d6bc8778635f566de34", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "229-03", "group": "229", "title": "The exact reconstruction threshold for the three-state symmetric channel", "pdf": "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf", "folder": "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026", "abstract_original": "We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b -ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when $`d\\lambda^2\\gt 1`$, with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates $`a,b\\gt 0`$, and mean degree $`(a+2b)/3\\gt 1`$: recovery is possible exactly when $`(a-b)^2\\gt 3(a+2b)`$. Above this threshold it is achievable in $`O(n\\log n)`$ time; at or below it, weak recovery is information-theoretically impossible.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5539, "coverage": "catalog_only", "read_detail": null, "pages": 60, "sha256": "e87d0854f29320d5b8ddd3015f3f89964e32231eec54a609aa1a9567d582f71a", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "230-01", "group": "230", "title": "An exact Hausdorff gauge for SLE", "pdf": "preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf", "folder": "preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026", "abstract_original": "For each $`0\\lt \\kappa\\lt 8`$, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5555, "coverage": "catalog_only", "read_detail": null, "pages": 37, "sha256": "513ca9aedce0f3421f979721c2274be13c3e9f03a754e8a4ff582f528c280a53", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "230-02", "group": "230", "title": "An explicit exact Hausdorff gauge for SLE", "pdf": "preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf", "folder": "preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026", "abstract_original": "For each fixed $`0\\lt \\kappa\\lt 8`$, let $`d=1+\\kappa/8`$. The gauge $`h(r)=r^d(\\log\\log(1/r))^{(2-d)/2}`$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE κ . This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5564, "coverage": "catalog_only", "read_detail": null, "pages": 49, "sha256": "c6e7116ea33f29eeb0628a91c4dcf391bfb53696890c1379f05bd74fb3e7bb85", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "231-01", "group": "231", "title": "The free uniform spanning forest is a factor of IID", "pdf": "preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf", "folder": "preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026", "abstract_original": "We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5580, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "478f5137621e0b3b41f2b9ab0bd45133d6d03f2c4e68a1eecc8fbc60344aac84", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "232-01", "group": "232", "title": "Gaussian free-field limits of weighted integer Lipschitz heights", "pdf": "preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf", "folder": "preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026", "abstract_original": "We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed $`x\\in[1/\\sqrt2,1]`$. The convergence holds as a random distribution on every bounded C 2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5596, "coverage": "catalog_only", "read_detail": null, "pages": 85, "sha256": "3dae0156982e5d205484a60ebb25c02591cf8a67cb7ce0ea0abf5486bdaaa61b", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "232-02", "group": "232", "title": "The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data", "pdf": "preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf", "folder": "preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026", "abstract_original": "We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5605, "coverage": "catalog_only", "read_detail": null, "pages": 64, "sha256": "f0047faf78711628ea4ee59901c37478c00840e166a4ac43504106ca1b385abe", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "232-03", "group": "232", "title": "Uniform real Lipschitz surfaces on the triangular lattice", "pdf": "preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf", "folder": "preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026", "abstract_original": "We prove the Gaussian free field and SLE$`_4`$ scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE$`_4`$. We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5614, "coverage": "catalog_only", "read_detail": null, "pages": 112, "sha256": "dce5ae39087f446aa1a721983682065be578bc7da2d5f25efc8c00960b685cbc", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "233-01", "group": "233", "title": "The joint scaling limit of critical Ashkin-Teller currents", "pdf": "preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf", "folder": "preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026", "abstract_original": "For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5630, "coverage": "catalog_only", "read_detail": null, "pages": 105, "sha256": "cec129cca8c2a0a61462b1ad8173132dc0c3f444fb25dd90e183c68f0aaeb748", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "234-01", "group": "234", "title": "All-temperature pressure for orthogonally invariant Ising spin glasses", "pdf": "preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf", "folder": "preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026", "abstract_original": "We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5646, "coverage": "catalog_only", "read_detail": null, "pages": 59, "sha256": "ecaeb31108feca6fa7f9768de75adf73990407ccdd2bb70e9416bb51c5762d57", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "235-01", "group": "235", "title": "A Limiting Satisfiability Threshold for Every Fixed Clause Size", "pdf": "preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf", "folder": "preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026", "abstract_original": "For every fixed integer k ≥ 3, random k -SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5662, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "5318fe3d0e3db4e4c22f79dc8b342ae38fc1ddd60172ba78a00ee612c1576365", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "235-02", "group": "235", "title": "Linear Variance of the Random 3-SAT Hitting Time", "pdf": "preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf", "folder": "preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026", "abstract_original": "For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance $`\\Theta(n)`$. The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5671, "coverage": "catalog_only", "read_detail": null, "pages": 10, "sha256": "01f912a6aaba76c36cb38913ae3d902329696e1e5502feed9b71903d42b8bc08", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "235-03", "group": "235", "title": "Variance of the Random k-SAT Hitting Time", "pdf": "preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf", "folder": "preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026", "abstract_original": "Let H n be the index of the first unsatisfiable prefix in random k -SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove $`\\mathop{\\mathrm{Var}}\\nolimits (H_n)=\\Theta_k(n)`$. For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of $`n\\log n`$; the companion paper on random 3-SAT sharpens this to $`\\Theta(n)`$. The same upper bounds proved here hold after clipping at any fixed positive multiple of n .", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5680, "coverage": "catalog_only", "read_detail": null, "pages": 37, "sha256": "9e47a7c16a835f9111435037d4103a0dd974111512c62c0f0e9c96ea14bc2d56", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "235-04", "group": "235", "title": "Computing the Random 3-SAT Threshold", "pdf": "preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf", "folder": "preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026", "abstract_original": "The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5689, "coverage": "catalog_only", "read_detail": null, "pages": 48, "sha256": "be7bde6cd4e513e134ffa36b94506ddd9fd0c13fd9f740d2ebac8494d249c530", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "236-01", "group": "236", "title": "The sharp factor-of-IID threshold for the free Ising model on regular trees", "pdf": "preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf", "folder": "preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026", "abstract_original": "For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d -regular tree is a factor of IID exactly when $`\\tanh\\beta\\le(d-1)^{-1/2}`$. We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5705, "coverage": "catalog_only", "read_detail": null, "pages": 50, "sha256": "aabb8cc15b9072c5d5f468d933ef929b19b654ea065b911445567d36570eabdd", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-01", "group": "237", "title": "Radial transfer estimates and polygon length laws for honeycomb walks", "pdf": "preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf", "folder": "preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026", "abstract_original": "We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter $`n^{3/4+o(1)}`$ in probability under either weight.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5721, "coverage": "catalog_only", "read_detail": null, "pages": 100, "sha256": "02d2e8df0dcf370b889087c3eceb5a2ed7af24b40d9606731b9a6fee75ea54ff", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-02", "group": "237", "title": "Critical honeycomb chords with prescribed boundary endpoints", "pdf": "preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf", "folder": "preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026", "abstract_original": "Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length $`R^{4/3+o(1)}`$ in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length $`R^{4/3+o(1)}`$. A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5730, "coverage": "catalog_only", "read_detail": null, "pages": 44, "sha256": "6b7adfc0aa278ae3ab692114466887bbf4656cb6adb2677ac16dd467b9d39fe1", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-03", "group": "237", "title": "Cylinder loop weights and planar nesting", "pdf": "preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf", "folder": "preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026", "abstract_original": "We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5739, "coverage": "catalog_only", "read_detail": null, "pages": 94, "sha256": "2e5effc6ded133601d01d0608f8031145b1fdc6519949b424c92bad191a4b209", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-04", "group": "237", "title": "Mass and covering exponents for fixed-length honeycomb walks", "pdf": "preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf", "folder": "preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026", "abstract_original": "For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5748, "coverage": "catalog_only", "read_detail": null, "pages": 66, "sha256": "58011713f8bb7cd5510768168545cc0a4a0948ce51c91aeab93aa9f293aa9da5", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-05", "group": "237", "title": "Signed cylinder propagation and marked polygons on the honeycomb lattice", "pdf": "preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf", "folder": "preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026", "abstract_original": "At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R , with both boundary ports summed and diameter at least $`R/10`$, the finite critical partition is $`R^{3/4+o(1)}`$ and the normalized mean length is $`R^{4/3+o(1)}`$. A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H , modulo translations, by $`H^{2/3+o(1)}`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5757, "coverage": "catalog_only", "read_detail": null, "pages": 128, "sha256": "2dda2109318f92bb397aafce68063a7cc71b8db4a4e540ab7533992d3ca0609c", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-06", "group": "237", "title": "Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice", "pdf": "preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf", "folder": "preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026", "abstract_original": "We prove a logarithmic window of critical honeycomb bridge lengths: summed through height $`2h(\\log h)^{1/64}`$, bridges of lengths between $`h^{4/3}(\\log h)^{-1/8}`$ and $`2h^{4/3}(\\log h)^{1/2}`$ have mass at least $`h^{3/4}(\\log h)^{-C}`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5766, "coverage": "catalog_only", "read_detail": null, "pages": 96, "sha256": "de92d884cd5298c4760a1b1a94424f51ea4ceb796b286174fac6682cf502544b", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-07", "group": "237", "title": "Marked polygon correlations and one-arc bounds", "pdf": "preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf", "folder": "preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026", "abstract_original": "At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H , counted modulo translations, is at most $`H^{2/3+o(1)}`$. We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5775, "coverage": "catalog_only", "read_detail": null, "pages": 61, "sha256": "acb3276f993c6d8ec5a2d9cb52b8f3f48c78b3cc4080cc8510f24e2b0c84305a", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-08", "group": "237", "title": "Disk transfer representations and confined bridge mass", "pdf": "preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf", "folder": "preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026", "abstract_original": "We prove that critical honeycomb bridges crossing a strip of width R have mean length $`R^{4/3+o(1)}`$, with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R −1/4 ; the half-plane arch kernel at endpoint separation m is comparable to m −5/4 .", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5784, "coverage": "catalog_only", "read_detail": null, "pages": 81, "sha256": "d64dc634fe49d994dec4bfed9d5d4656791b4679f7e62ccc84f4434752ed5d45", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-09", "group": "237", "title": "Polynomial vacuum representations and bridge mass for honeycomb walks", "pdf": "preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf", "folder": "preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026", "abstract_original": "We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length $`h^{4/3+o(1)}`$. We also obtain central visit probability $`R^{-2/3+o(1)}`$ and mean length $`R^{4/3+o(1)}`$ for macroscopic free-boundary chords in a regular hexagon.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5793, "coverage": "catalog_only", "read_detail": null, "pages": 69, "sha256": "69885ae7308d1f03e44116c1c26114005825913669e398ee388a5d06c2e579b4", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-10", "group": "237", "title": "Renewal and changes of law for critical honeycomb walks", "pdf": "preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf", "folder": "preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026", "abstract_original": "For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5802, "coverage": "catalog_only", "read_detail": null, "pages": 98, "sha256": "d179fa55e4b82fed6e226b4d9b9894f4ea28c990283e0b55173dc98b10c42356", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-11", "group": "237", "title": "Critical strip-crossing mass on the honeycomb lattice", "pdf": "preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf", "folder": "preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026", "abstract_original": "At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N −1/4 . The first horizontal-displacement moment of return paths is comparable to N 3/4 .", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5811, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "33e92f50d83f53439caa07ff0f71472694dc67d6d7c573e28cad30294b3eb1c0", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-12", "group": "237", "title": "Uniform marked-polygon estimates and sharp finite bridge moments", "pdf": "preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf", "folder": "preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026", "abstract_original": "We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h −1/4 , first-length mass at most $`Ch^{13/12}`$, and mass at least $`ch^{-1/4}`$ on bridges with length at least $`ch^{4/3}`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5820, "coverage": "catalog_only", "read_detail": null, "pages": 109, "sha256": "e9b7a0252f329287bc2cd0360c7602a65f1a99aec92328ba2a755825b980c5e5", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "237-13", "group": "237", "title": "Cap-selected amplitudes and triangle chords for honeycomb walks", "pdf": "preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf", "folder": "preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026", "abstract_original": "At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R , summed over both boundary endpoints and restricted to diameter at least $`R/100`$, have partition sum comparable to R 3/4 and mean length $`R^{4/3+o(1)}`$. We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as $`N^{1/6+o(1)}`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5829, "coverage": "catalog_only", "read_detail": null, "pages": 92, "sha256": "7fd6eb0adc06ad82c6a5dba6286d3c49096692b9d967ac3d117e753fdc09db8f", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-01", "group": "238", "title": "Optimal-order mixing of the Thorp shuffle", "pdf": "preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf", "folder": "preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026", "abstract_original": "We prove that the Thorp shuffle on $`2^d`$ cards mixes in $`\\Theta(d)`$ complete shuffles. The full permutation law after $`1600d`$ shuffles converges to uniform in total variation as $`d\\to\\infty`$, uniformly over the initial deck, while a support count gives a lower bound of $`2d-O(1)`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5845, "coverage": "catalog_only", "read_detail": null, "pages": 14, "sha256": "9b02c5b13a61d6c071b9a12849d0177a4ece970f3b182893a1c3d14b660c5984", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-02", "group": "238", "title": "Random coordinate frames and partial permutation laws", "pdf": "preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf", "folder": "preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026", "abstract_original": "For the Thorp shuffle on $`2^d`$ cards, we prove that the worst-start total-variation distance after $`32800d`$ physical shuffles tends to zero as $`d\\to\\infty`$. Combining our fixed-list estimate with the companion Fourier transfer improves this bound to $`512d`$ shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5854, "coverage": "catalog_only", "read_detail": null, "pages": 72, "sha256": "0a520da2d4a92a176e8833ef81fd1d4ecf19c117954c9606577f9d819be5d394", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-03", "group": "238", "title": "From partial permutation information to Fourier bounds", "pdf": "preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf", "folder": "preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026", "abstract_original": "We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen $`7n/8`$ labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on S n .", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5863, "coverage": "catalog_only", "read_detail": null, "pages": 66, "sha256": "88926b5e48099ed836f49f15c3ef759c14f7aabb140a14e9005ca7fe09727a8b", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-04", "group": "238", "title": "Conditional information under deterministic coordinate sweeps", "pdf": "preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf", "folder": "preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026", "abstract_original": "We bound the information remaining after paths of specified cards in the Thorp shuffle on $`n=2^d`$ cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after $`2048d`$ physical shuffles.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5872, "coverage": "catalog_only", "read_detail": null, "pages": 81, "sha256": "ad4125b0a60530a3f364c0bb4e61d34eb1277b2477f50b3383411cecadd092d0", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-05", "group": "238", "title": "Conditional permutations in a revealed switching environment", "pdf": "preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf", "folder": "preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026", "abstract_original": "Fix half the labels in a Thorp shuffle on $`2^d`$ positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as $`d\\to\\infty`$, uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5881, "coverage": "catalog_only", "read_detail": null, "pages": 32, "sha256": "59a17db01074b8bb4124cad3b8ea3ecd70cc009b339b78bde5366a4825bc87e3", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-06", "group": "238", "title": "Routing densities and representation contraction for Thorp sweeps", "pdf": "preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf", "folder": "preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026", "abstract_original": "For $`N=2^d`$ cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order $`\\Theta(\\log N)`$ in physical shuffles.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5890, "coverage": "catalog_only", "read_detail": null, "pages": 86, "sha256": "4e5fbcff0dec65fd5a5d929e38c255f03e89038866b2e3cce044eee9e04cd862", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-07", "group": "238", "title": "Row–column symmetry and contraction of coordinate sweeps", "pdf": "preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf", "folder": "preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026", "abstract_original": "We prove that the Thorp shuffle on $`N=2^d`$ labeled cards has full-permutation total-variation mixing time $`\\Theta(\\log N)`$ in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5899, "coverage": "catalog_only", "read_detail": null, "pages": 38, "sha256": "d27697f3427868fdda8bfad6c2540fb5daec635fa2eeb69aa0bac0bd2deb7de0", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-08", "group": "238", "title": "Random-subspace tests and trace smoothing for coordinate sweeps", "pdf": "preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf", "folder": "preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026", "abstract_original": "We prove that an absolute number of coordinate sweeps of the Thorp shuffle on $`n=2^d`$ positions brings the full permutation to total-variation distance at most $`\\frac12n^{-5}`$ from uniform, uniformly over the initial deck. Thus $`O(d)`$ physical shuffles suffice, which is optimal in order.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5908, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "9e1c693d0e6175b9f2ff38ffc881725f9214571f35a583883a0cd44b43e71c2f", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-09", "group": "238", "title": "Compatibility entropy and the spectrum of a Thorp sweep", "pdf": "preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf", "folder": "preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026", "abstract_original": "We prove that the Thorp shuffle on $`n=2^d`$ cards mixes in $`\\Theta(d)`$ physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as $`d\\to\\infty`$.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5917, "coverage": "catalog_only", "read_detail": null, "pages": 128, "sha256": "dae8f5ca5b99fbecbbafd5644faea7f6d147b1b16493b31383ebd4c1e2e8fec1", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-10", "group": "238", "title": "Signed tensor densities and diagram budgets for the Thorp shuffle", "pdf": "preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf", "folder": "preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026", "abstract_original": "We prove that a fixed number of coordinate sweeps of the Thorp shuffle on $`2^d`$ cards makes the squared L 2 distance of the full permutation density from uniform tend to zero as $`d\\to\\infty`$. In particular, the total-variation mixing time is $`\\Theta(d)`$ physical shuffles.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5926, "coverage": "catalog_only", "read_detail": null, "pages": 109, "sha256": "69f29b044ee825efaf2717129649aee4bb47c8917d804842a0f66db546b630a8", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-11", "group": "238", "title": "Conditional coordinate sweeps and analytic transfer", "pdf": "preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf", "folder": "preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026", "abstract_original": "For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on $`2^d`$ cards in $`O(d)`$ physical steps, matching the order of the support lower bound.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5935, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "50b927834abd484c8602aa21111ad7f2ff161035c8309c41fc39484481cc8adb", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "238-12", "group": "238", "title": "A strict four-row permanent inequality and permutation moments", "pdf": "preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf", "folder": "preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026", "abstract_original": "We prove a permanent inequality with exponent strictly below two for laws on S 4 sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on $`N=2^d`$ cards to total-variation distance tending to zero from uniform as $`N\\to\\infty`$. Thus the mixing time has the optimal order $`O(\\log N)`$ in physical shuffles.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5944, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "1fc78f640d9422a5ea8b626e5cbcbb0edf7b24845f0a0057790f10a866dc74e3", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "239-01", "group": "239", "title": "The sharp exponential rate of singularity for symmetric Bernoulli matrices", "pdf": "preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf", "folder": "preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026", "abstract_original": "Let A n be a symmetric $`n\\times n`$ matrix whose entries on and above the diagonal are independent uniform signs. We prove $`\\displaystyle \\Pr(\\det A_n=0)=\\left(\\frac12+o(1)\\right)^n.`$", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5960, "coverage": "catalog_only", "read_detail": null, "pages": 67, "sha256": "8dfe5c0b8a0ca85cdad9b7194ed70217b1efa8618de7b80ec1fbe4db7b33674c", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "239-02", "group": "239", "title": "The sharp singularity rate for biased symmetric sign matrices", "pdf": "preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf", "folder": "preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026", "abstract_original": "Let A n be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability $`p\\in(0,1)\\setminus\\{1/2\\}`$. We prove that $`\\mathbb P(\\det A_n=0)=(p^2+(1-p)^2+o(1))^n`$. The rate is attained by the event that two rows agree.", "archive": "math_research_06_of_08.zip", "exists": true, "source_line": 5971, "coverage": "catalog_only", "read_detail": null, "pages": 34, "sha256": "62c6a85b5437d9da99317ed5cb27ac6789316ee154f72863d923dd4c75389b6d", "bundled_pdf": null, "local_href": "../math_sources/math_research_06_of_08/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" } ], "extra_folders": [] }