{ "archive": { "archive": "math_research_04_of_08.zip", "sha256": "f63163e51eff12d62acae20421c7e382f0cca78ff33fcf062b6217fa1731420d", "bytes": 67474379, "crc_ok": true, "files": 1693, "physical_work_folders": 94, "preprint_pdfs": 97, "groups": [ "001", "002", "003", "004", "005", "006", "007", "008", "009", "010", "011", "012", "013", "014", "015", "016", "017", "018", "019", "020", "021", "022", "023", "024", "025", "026", "027", "028", "029", "030", "264", "265", "266", "267", "268", "269", "270", "271", "272", "273", "274", "275", "276", "277", "278", "279", "280", "281", "282", "283", "284" ], "current_manuscripts": 94 }, "reading_policy": "Archive catalog parsed; PDFs opened structurally; only explicit read_detail entries represent actual reading.", "works": [ { "id": "001-01", "group": "001", "title": "Milne's rationality conjecture for abelian varieties", "pdf": "preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf", "folder": "preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026", "abstract_original": "We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to- p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 23, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "640701a0ea8d8026d2f019e571e052f2d106cad3f5216cf6f7fca0d643873234", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "002-01", "group": "002", "title": "Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank", "pdf": "preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf", "folder": "preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026", "abstract_original": "We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q -power Selmer group has corank zero or one at some prime q . The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 39, "coverage": "catalog_only", "read_detail": null, "pages": 94, "sha256": "64b46603e174462e5bb32d242852eb7f0afe6e519f26534e42fd3f178a16b885", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "002-02", "group": "002", "title": "The Selmer converse for elliptic curves at every prime", "pdf": "preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf", "folder": "preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026", "abstract_original": "We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p : if the full p -power Selmer group has ℤ p -corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal r , and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 48, "coverage": "catalog_only", "read_detail": null, "pages": 82, "sha256": "bdb2f21b345a4f0fc9943a4e39bee27fedae5cf1e6842c40b3a77bd528ddea01", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "002-03", "group": "002", "title": "The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one", "pdf": "preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf", "folder": "preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026", "abstract_original": "We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 57, "coverage": "catalog_only", "read_detail": null, "pages": 164, "sha256": "734e4050dc8416c0f9ea7c9e15a29c6a8627feb53073e40b456d118f68d4dc5a", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "003-01", "group": "003", "title": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$", "pdf": "preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "folder": "preprints/The-Quasi-Riemann-Hypothesis-September-30-2026", "abstract_original": "We prove that all finite-order Hecke L -functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet L -functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 73, "coverage": "intro_theorems", "read_detail": { "work": "003-01", "pages": "4–6", "scope": "Главная теорема и обзор двух стадий; не полный текст" }, "pages": 199, "sha256": "8fe93046f8cf5ef1ba5969c89addc02d76311adc4ee907509ff9cd96f7ec99e7", "bundled_pdf": "sources/papers/003-01/paper.pdf", "local_href": "../math_sources/math_research_04_of_08/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "003-02", "group": "003", "title": "The Quasi-Riemann Hypothesis (alternate 11/12 proof)", "pdf": "preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf", "folder": "preprints/The-Quasi-Riemann-Hypothesis-October-5-2026", "abstract_original": "We establish the quasi-Riemann hypothesis by proving that every Dirichlet L -function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke L -function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 82, "coverage": "intro_theorems", "read_detail": { "work": "003-02", "pages": "2–3", "scope": "Главное утверждение и первые следствия" }, "pages": 49, "sha256": "f919b57829b178c8e60e7c17b018cf773e7907cf642ef5a3347d8a826e8dbf18", "bundled_pdf": "sources/papers/003-02/paper2.pdf", "local_href": "../math_sources/math_research_04_of_08/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "003-03", "group": "003", "title": "Uniform exclusion of Landau–Siegel zeros", "pdf": "preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf", "folder": "preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026", "abstract_original": "We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet L -function of conductor q ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 91, "coverage": "catalog_only", "read_detail": null, "pages": 9, "sha256": "fab9f6e84ece2a37d834b2c031d8eb14e81413d9f6270791e7b5468b295214ef", "bundled_pdf": "sources/papers/003-03/paper.pdf", "local_href": "../math_sources/math_research_04_of_08/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "004-01", "group": "004", "title": "Hilbert’s tenth problem over the rational numbers", "pdf": "preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf", "folder": "preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026", "abstract_original": "We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 107, "coverage": "catalog_only", "read_detail": null, "pages": 62, "sha256": "f79236c22b981fe66aaf3ba494a466bf26db212117db47d95a31bf0315af2b30", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "004-02", "group": "004", "title": "A pointwise 2-converse for elliptic curves with rational two-torsion", "pdf": "preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf", "folder": "preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026", "abstract_original": "We prove a pointwise 2-converse for elliptic curves over $`\\mathbf Q`$ with nonzero rational two-torsion: if the $`2^\\infty`$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 116, "coverage": "catalog_only", "read_detail": null, "pages": 91, "sha256": "df42f33f68fbef7766e5679b298e68e910157f78f2266889fb757824b9e44939", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "005-01", "group": "005", "title": "Catalan's constant is irrational", "pdf": "preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf", "folder": "preprints/Catalans-constant-is-irrational-September-24-2026", "abstract_original": "We prove that Catalan's constant $`G=\\sum_{j\\geq0}(-1)^j/(2j+1)^2`$ is irrational.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 132, "coverage": "catalog_only", "read_detail": null, "pages": 44, "sha256": "4023d0344abceb79dc46a16380dac3a662193f9d4f23e6c340b2162068b04c0e", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "006-01", "group": "006", "title": "Goldfeld's analytic density conjecture and the 2-converse for elliptic curves", "pdf": "preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf", "folder": "preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026", "abstract_original": "We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $`2^\\infty`$-Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 148, "coverage": "catalog_only", "read_detail": null, "pages": 130, "sha256": "c417858c9fe566daacb7738741dc66e211eea148b3da587eb5ee68eaad208d64", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "006-02", "group": "006", "title": "The mean analytic rank of quadratic twists of elliptic curves", "pdf": "preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf", "folder": "preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026", "abstract_original": "For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 157, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "718fcfce4014a878685365edfb8a285a611512dea362dc5a9e1dc93d023cd7b1", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "007-01", "group": "007", "title": "Ordinary two-point correlations of multiplicative functions", "pdf": "preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf", "folder": "preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026", "abstract_original": "We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 173, "coverage": "catalog_only", "read_detail": null, "pages": 87, "sha256": "ccb6f339a7505db7ab158dae4b739164ed8dda32be4012b0310198dd32f5c70a", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "008-01", "group": "008", "title": "The Deligne-Drinfeld conjecture", "pdf": "preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf", "folder": "preprints/The-Deligne-Drinfeld-conjecture-September-23-2026", "abstract_original": "We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 189, "coverage": "catalog_only", "read_detail": null, "pages": 41, "sha256": "5ae63f326803a634e7c6dc5684ce7062ea0ac6940a88a75c7052875382dbc762", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "009-01", "group": "009", "title": "Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory", "pdf": "preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf", "folder": "preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026", "abstract_original": "We prove a mod- ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ . The groups $`K^{\\mathrm M}_1/\\ell`$ and $`K^{\\mathrm M}_2/\\ell`$, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $`\\mathbb F_\\ell^\\times`$, with only Frobenius ambiguity in the field isomorphism in positive characteristic.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 205, "coverage": "catalog_only", "read_detail": null, "pages": 36, "sha256": "fafdf71eafc4700ada91598eba0620dc3263dd48b5879aa42af806254124bb2f", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "009-02", "group": "009", "title": "Reconstruction from Milnor K-theory modulo the characteristic", "pdf": "preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf", "folder": "preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026", "abstract_original": "Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p . We prove that every isomorphism of their first Milnor K -groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of p th powers by an elementary calculation with derivations.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 214, "coverage": "catalog_only", "read_detail": null, "pages": 11, "sha256": "df8a6b4a54b121c692f7221e7b606be0dde89351fcb9ab5bb3853f7c2da4da90", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "009-03", "group": "009", "title": "The Bogomolov-Pop reconstruction theorem", "pdf": "preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf", "folder": "preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026", "abstract_original": "We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ . The pro- ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ -adic unit ambiguities.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 223, "coverage": "catalog_only", "read_detail": null, "pages": 26, "sha256": "4d805627401fffef78e1f0d8e21a0585f6c733e8be90643d0065cd27ffaa24f2", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "010-01", "group": "010", "title": "Unrestricted pro-modularity at the prime two", "pdf": "preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf", "folder": "preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026", "abstract_original": "Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of $`G_{\\mathbb Q}`$ that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 239, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "ef74b48557930663cab287cabd5280854f059de298e3016cc2172106eba1a1db", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "010-02", "group": "010", "title": "The Dimension of the Two-Adic Hecke Algebra at Odd Level", "pdf": "preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf", "folder": "preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026", "abstract_original": "For every odd positive integer N , every irreducible component of the full two-adic Hecke algebra of level $`\\Gamma_1(N)`$ has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 248, "coverage": "catalog_only", "read_detail": null, "pages": 12, "sha256": "b15081ee4a06c6c609eee447ee2dffd69578ce3550cbcd923e7206c7443333c5", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "010-03", "group": "010", "title": "Fontaine–Mazur modularity at the prime 2", "pdf": "preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf", "folder": "preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026", "abstract_original": "We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of $`G_{\\mathbb Q}`$, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 257, "coverage": "catalog_only", "read_detail": null, "pages": 72, "sha256": "3c8797c7d468cdcccbfedf719c9a758a083954ba3c3005975736eb911d26a636", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "011-01", "group": "011", "title": "Weighted dilation graphs, smooth shifted primes and totient fibers", "pdf": "preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf", "folder": "preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026", "abstract_original": "We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than $`n^{1-\\varepsilon }`$ preimages. We also show that, for every fixed δ > 0, there are at least $`x^{1-o(1)}`$ primes p in $`2x\\lt p\\le5x`$ whose predecessors have no prime factor exceeding x δ .", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 273, "coverage": "catalog_only", "read_detail": null, "pages": 68, "sha256": "351692885d112277e969901e75fe56eeee0d263d5d5847d0b980fe2c8c7744c0", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "011-02", "group": "011", "title": "The Poisson-Dirichlet law for prime predecessors", "pdf": "preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf", "folder": "preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026", "abstract_original": "For a prime p chosen uniformly from $`3\\le p\\le x`$, list the prime factors of $`p-1`$ in decreasing order, with multiplicity. As $`x\\to\\infty`$, their logarithms, divided by $`\\log(p-1)`$, converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 282, "coverage": "catalog_only", "read_detail": null, "pages": 74, "sha256": "4b54d35506a1928022ac21e6d270d30f6533423ddc5990975270a1e39a87bce7", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "011-03", "group": "011", "title": "Prime Predecessors with an Even Number of Prime Factors", "pdf": "preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf", "folder": "preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026", "abstract_original": "We prove that there are infinitely many primes p for which $`p-1`$ is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with $`\\mu(p-1)=1`$.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 291, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "8ee2d8f79198dd5f09c5fbf71e309a63493d1c089772499614ee58dc8575eae9", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "012-01", "group": "012", "title": "The joint Dickman law for consecutive integers", "pdf": "preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf", "folder": "preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026", "abstract_original": "Let $`P^+(n)`$ denote the largest prime factor of n . We prove that $`\\log P^+(n)/\\log n`$ and $`\\log P^+(n+1)/\\log n`$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering $`P^+(n)\\lt P^+(n+1)`$ has natural density 1/2.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 307, "coverage": "catalog_only", "read_detail": null, "pages": 84, "sha256": "e66b4e8a1fe197d3bb2c195548cd3bdd3fba8595db1356e571205a6baab43cb3", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "013-01", "group": "013", "title": "The additive indecomposability of the primes", "pdf": "preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf", "folder": "preprints/the-additive-indecomposability-of-the-primes-September-24-2026", "abstract_original": "We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as $`A+B`$, where A and B are sets of nonnegative integers with at least two elements each.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 323, "coverage": "catalog_only", "read_detail": null, "pages": 80, "sha256": "2c4ec07457be59763b37d2197852341a7e6615e419982f9f3705c5f3a94c2c3f", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-01", "group": "014", "title": "Global Arthur Enhancements of Cuspidal Excursion Parameters", "pdf": "preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf", "folder": "preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026", "abstract_original": "Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL 2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 339, "coverage": "catalog_only", "read_detail": null, "pages": 69, "sha256": "98c8561d3049c5b9ec55f39e980e1ec3a3b8b76adc1392ec4e5c3e71e83a50d4", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-02", "group": "014", "title": "Rationality of the Canonical Unramified Arthur Filtration", "pdf": "preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf", "folder": "preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026", "abstract_original": "Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 348, "coverage": "catalog_only", "read_detail": null, "pages": 43, "sha256": "eaf8e38720714e2319c52a2c62e33e11f4383614e894012624d6580c4d3af850", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-03", "group": "014", "title": "Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level", "pdf": "preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf", "folder": "preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026", "abstract_original": "We prove rational and $`\\overline{\\mathbb Q}_\\ell`$ decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin. For split connected adjoint absolutely simple groups, we also prove that a cuspidal automorphic representation with one generic unramified local component is tempered at every unramified place. Thus globally generic cuspidal representations in this scope satisfy the unramified part of the generalized Ramanujan conjecture.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 357, "coverage": "catalog_only", "read_detail": null, "pages": 64, "sha256": "b4b0cfe23007d159e20a561d64584a5704b3701639fdba7a0f33204433bcb4e9", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-04", "group": "014", "title": "Temperedness at ramified places for globally generic exceptional groups", "pdf": "preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf", "folder": "preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026", "abstract_original": "We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 366, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "101a2a51cda87af8cd868366d60b3ced23c99fc1995c6e872e034b15873c6ad5", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-05", "group": "014", "title": "The Restricted Geometric Langlands Equivalence in Positive Characteristic", "pdf": "preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf", "folder": "preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026", "abstract_original": "We prove the $`\\overline{\\mathbb Q}_\\ell`$-linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p , in two regimes. Over $`\\overline{\\mathbb F}_q`$, we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order. This proves Gaitsgory–Raskin's full-support conjecture [[12, Conjecture 1.3.10]](https://arxiv.org/abs/2508.02237v1) in these regimes.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 375, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "52c37d4f70f505c0685f98e590d93f629bb686b0436b645a820d5d3efeb67ea0", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-06", "group": "014", "title": "Constructible tame Hecke eigensheaves in positive characteristic", "pdf": "preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf", "folder": "preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026", "abstract_original": "We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ -adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms are compatible with tensor products, permutations, and fusion.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 384, "coverage": "catalog_only", "read_detail": null, "pages": 46, "sha256": "73db2ee4a6d9f8adf7236ba7ab8cf4891a899aa74fe9a822c9d53bbf60ab1ece", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-07", "group": "014", "title": "Tame Hecke Eigensheaves with Several Marked Points", "pdf": "preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf", "folder": "preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026", "abstract_original": "For SL n in characteristic p > n , we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ -adic PGL n -local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms retain the full tensor and fusion structure.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 393, "coverage": "catalog_only", "read_detail": null, "pages": 40, "sha256": "fb3f3a7a4115f5ec312f24d5f4020b68acb41fd5c672bdc9b5802ad2a86fb259", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "014-08", "group": "014", "title": "Frobenius Structures on Tame Hecke Eigensheaves", "pdf": "preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf", "folder": "preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026", "abstract_original": "We construct nonzero perverse Weil Hecke eigensheaves for SL n with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGL n -local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n . The full multi-leg eigenstructure is Frobenius compatible with the prescribed arithmetic eigenvalue, and the geometric sheaf has parabolic nilpotent singular support.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 402, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "b0540eb39aad89c2323551cbb707dd0df06fd8dbb7ec4a6d026f3f13502f6f8c", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "015-01", "group": "015", "title": "Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type", "pdf": "preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf", "folder": "preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026", "abstract_original": "We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 418, "coverage": "catalog_only", "read_detail": null, "pages": 16, "sha256": "7e6a0188abd45ae4520d4592c773d264e029751e3ef6fb2623388aee8b0f56c0", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "015-02", "group": "015", "title": "Equidistribution of primitive quartic torus packets for arbitrary orders", "pdf": "preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf", "folder": "preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026", "abstract_original": "Let K range over totally real quartic fields with no proper intermediate field, and let $`\\mathcal O`$ be any order in K . We prove that the packets of periodic diagonal orbits attached to invertible $`\\mathcal O`$-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $`|\\mathop{\\mathrm{Disc}}\\nolimits (\\mathcal O)|`$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 427, "coverage": "catalog_only", "read_detail": null, "pages": 70, "sha256": "95c2a937da74097afe5d155a412fa89a2c4560a8231f10c4230244a8c6bed0f5", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "015-03", "group": "015", "title": "Equidistribution of Primitive Sextic Torus Packets", "pdf": "preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf", "folder": "preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026", "abstract_original": "We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 436, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "3d96820596471fc3099f9d73a92d1bce10b991416925374a9d57b36597e41828", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "016-01", "group": "016", "title": "The abelian Zilber–Pink conjecture", "pdf": "preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf", "folder": "preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026", "abstract_original": "We prove the abelian Zilber–Pink conjecture over $`\\overline{\\mathbb Q}`$. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 452, "coverage": "catalog_only", "read_detail": null, "pages": 44, "sha256": "53e8f20495ac61681db3f19f43358f621c90f53ca6c18cc4d6f83f32e9959014", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "016-02", "group": "016", "title": "The E×CM component of Zilber–Pink for curves in A2", "pdf": "preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "folder": "preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026", "abstract_original": "We prove the $`E\\times\\mathrm{CM}`$ component of Zilber–Pink for Hodge-generic algebraic curves in $`\\mathcal A_2`$ over $`\\overline{\\mathbb Q}`$. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 461, "coverage": "catalog_only", "read_detail": null, "pages": 15, "sha256": "fd6298d9c106fea90c945dd51c2358fcebe61dae44c7a297f43e1385f71e32df", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "016-03", "group": "016", "title": "Quaternionic division points on curves in the Siegel threefold", "pdf": "preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf", "folder": "preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026", "abstract_original": "We prove the quaternionic-division component of Zilber–Pink for curves in $`\\mathcal A_2`$ over $`\\overline{\\mathbb Q}`$. A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 470, "coverage": "catalog_only", "read_detail": null, "pages": 46, "sha256": "e11b4b068502ced9e8fd3901a60f81a95d6cc5c1c8eb70cf691ea3bd73bbba35", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "016-04", "group": "016", "title": "Elliptic squares and Zilber–Pink for curves in A2", "pdf": "preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "folder": "preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026", "abstract_original": "We prove that every Hodge-generic algebraic curve in $`\\mathcal A_2`$ over $`\\overline{\\mathbb Q}`$ contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion $`E\\times\\mathrm{CM}`$ and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in $`\\mathcal A_2`$ over $`\\overline{\\mathbb Q}`$, without a boundary hypothesis.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 479, "coverage": "catalog_only", "read_detail": null, "pages": 65, "sha256": "d3ecf115570d0b89e5d45869f555e16660d057171bda8163a4ff16f3ff24c977", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "017-01", "group": "017", "title": "The irrationality exponent of pi is 2", "pdf": "preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf", "folder": "preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026", "abstract_original": "We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series $`\\sum_{n\\ge1}1/(n^3\\sin^2 n)`$ converges, with angles in radians.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 495, "coverage": "catalog_only", "read_detail": null, "pages": 23, "sha256": "26eec5de7ac9dc49d28183fdea458e0858bc3323b5ba93c3d41bb105d8e3eab6", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "018-01", "group": "018", "title": "The Margulis–Platonov conjecture over global function fields", "pdf": "preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf", "folder": "preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026", "abstract_original": "We prove the Margulis–Platonov conjecture over every global function field, including characteristic two. For an absolutely almost simple simply connected algebraic group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 511, "coverage": "catalog_only", "read_detail": null, "pages": 81, "sha256": "7e90f88648f7bafcc58a6fa895e2b94c2e4ed81a5510c9237488e18416ccc498", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "018-02", "group": "018", "title": "The Margulis–Platonov conjecture over number fields", "pdf": "preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf", "folder": "preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026", "abstract_original": "We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 520, "coverage": "catalog_only", "read_detail": null, "pages": 78, "sha256": "545dc0b02e847a4cc226ba5959a13396f2aa3be2e4f710ca07004a9c1a1a4600", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "019-01", "group": "019", "title": "Étale covers with a prescribed exterior sheet", "pdf": "preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf", "folder": "preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026", "abstract_original": "Let X be a smooth proper hyperbolic curve over an algebraic closure of a p -adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 536, "coverage": "catalog_only", "read_detail": null, "pages": 7, "sha256": "713430a73f1530f5d89cd38050c45a6bc2ccc458878a191917bb37b451b2f26a", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "019-02", "group": "019", "title": "The p-adic section conjecture", "pdf": "preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf", "folder": "preprints/The-p-adic-section-conjecture-October-6-2026", "abstract_original": "We prove the local p -adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ p . Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes $`X_0(N)`$ and $`X_1(N)`$ of genus at least two over ℚ.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 545, "coverage": "catalog_only", "read_detail": null, "pages": 24, "sha256": "f3c8d6f784d6bae48305f6959c44c7cc337f9c2b6078d93a5595d317b3a82f40", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "020-01", "group": "020", "title": "Squarefree values of quartics and power-free values of polynomials", "pdf": "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf", "folder": "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026", "abstract_original": "We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding $`(d-2)`$-power-free density in degrees $`4\\le d\\le8`$. The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the $`(d-2)`$-power-free density for every d ≥ 4.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 561, "coverage": "catalog_only", "read_detail": null, "pages": 55, "sha256": "c0320233615387079f999f6d955807230607689d7df0e6988893b2a947c06ba1", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "021-01", "group": "021", "title": "A quadratic bound for Jacobsthal's function", "pdf": "preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf", "folder": "preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026", "abstract_original": "Let $`h(k)`$ be the least integer such that every interval of $`h(k)`$ consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove $`h(k)\\ll k^2/(\\log\\log(3k))^2`$, giving an affirmative answer to Jacobsthal's quadratic-bound question.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 577, "coverage": "catalog_only", "read_detail": null, "pages": 83, "sha256": "f80267eebc82a9b8905057b45b35ef8734485b2b0b7b809a3051c06240cf0b1e", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "022-01", "group": "022", "title": "The Weak Inhomogeneous Duffin–Schaeffer Conjecture", "pdf": "preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf", "folder": "preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026", "abstract_original": "We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $`\\psi:\\mathbb N\\to[0,\\infty)`$, divergence of $`\\sum_{q\\ge1}\\phi(q)\\psi(q)/q`$ implies $`\\|qx-\\gamma\\|\\lt \\psi(q)`$ for infinitely many q , for almost every x . Here ϕ is Euler's totient and $`\\|\\cdot\\|`$ is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 593, "coverage": "catalog_only", "read_detail": null, "pages": 87, "sha256": "d91ccb7b68c5b0e2e081710c3c9012d0d790992703fc3609853953c538bc7203", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "023-01", "group": "023", "title": "An unconditional first moment for cubic Gauss sums", "pdf": "preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf", "folder": "preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026", "abstract_original": "We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is $`(6/5)c_*X^{5/6}/\\log X`$, where $`c_*=(2\\pi)^{2/3}/(3\\Gamma(2/3))`$. For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 609, "coverage": "catalog_only", "read_detail": null, "pages": 52, "sha256": "0af3e675f8a6a5b3e47da9ea5ca7434a825f48f7186f79594b1ee5d54d443a37", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "024-01", "group": "024", "title": "An asymptotic formula for the number of totients", "pdf": "preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf", "folder": "preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026", "abstract_original": "Let $`V(x)`$ count the distinct values of Euler's totient function up to x . We give an explicit asymptotic equivalent for $`V(x)`$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $`V(cx)/V(x)\\to c`$ as $`x\\to\\infty`$ for every fixed c > 0, answering a question of Erdős and Hall.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 625, "coverage": "catalog_only", "read_detail": null, "pages": 39, "sha256": "5dda4f860a38357f5c4e3a912a422a13ed2886720fde18f54d97e2947f0e5669", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "025-01", "group": "025", "title": "Short Egyptian fractions", "pdf": "preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf", "folder": "preprints/Short-Egyptian-fractions-September-25-2026", "abstract_original": "We prove a conjecture of Erdős: for every sufficiently large integer b , every rational number $`a/b`$ with $`1\\le a\\lt b`$ is a sum of $`O(\\log\\log b)`$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k : their double logarithms have order k .", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 641, "coverage": "catalog_only", "read_detail": null, "pages": 33, "sha256": "25eebdc1f077f2451bada4d3f8301b59f8158dcdb9019425ce555c5e9c5a2646", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "026-01", "group": "026", "title": "Positive lower density of large prime gaps", "pdf": "preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf", "folder": "preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026", "abstract_original": "For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $`C\\log p`$, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C . It follows that the indices at which $`p_n/n`$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 657, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "7a4a37fa92b30f4e9749ed16a6e8d715eb98ab0c0332cc5061ac6b2a067009ed", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "027-01", "group": "027", "title": "Integral points on character varieties of curves", "pdf": "preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf", "folder": "preprints/Integral-points-on-character-varieties-of-curves-September-25-2026", "abstract_original": "We prove potential Zariski density of integral points on SL r -character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 673, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "cacb926d1f09eeb701c5a98ff7f4b7849039a2cf838005d3687db122fe8c0e8c", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "028-01", "group": "028", "title": "Bounded-Step Walks on Gaussian Primes", "pdf": "preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf", "folder": "preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026", "abstract_original": "We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 689, "coverage": "catalog_only", "read_detail": null, "pages": 29, "sha256": "a032c3f94f247e90de8077457439a0aad38325ae6f2557c2d08bfb63bfae78d4", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "029-01", "group": "029", "title": "Primitive roots for every admissible integer base", "pdf": "preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf", "folder": "preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026", "abstract_original": "We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least $`c_a x/(\\log x)^2`$ primes in $`(x,2x)`$ with primitive root a , for some $`c_a\\gt 0`$ and every sufficiently large x .", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 705, "coverage": "catalog_only", "read_detail": null, "pages": 92, "sha256": "671ac4b3ece879f614ed8f4b782e61a07805e533f385a694fe4d7afbda3c9365", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "029-02", "group": "029", "title": "Simultaneous primitive roots: a conditional lower bound for prime bases", "pdf": "preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf", "folder": "preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026", "abstract_original": "For every fixed finite set of distinct positive primes, we prove that at least $`cx/(\\log x)^2`$ primes in $`(x,2x)`$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x . The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 714, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "0f36acf08bd34758b209b05f11193c27df9da38173058c8f96731972d27e3716", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "030-01", "group": "030", "title": "Modularity of elliptic curves over imaginary quadratic fields", "pdf": "preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf", "folder": "preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026", "abstract_original": "We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 730, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "c74cc37755f500be0c22ccb4d08ce81df309404a7175a7fb6cf7321330b5187c", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "264-01", "group": "264", "title": "Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime", "pdf": "preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf", "folder": "preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026", "abstract_original": "For each fixed rotating subextremal Kerr background with mass M > 0 and rotation $`0\\lt |\\mathfrak a|\\lt M`$, we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future $`C^0\\cap W^{1,2}_{\\mathrm{loc}}`$ extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6618, "coverage": "catalog_only", "read_detail": null, "pages": 59, "sha256": "1518c8e1e86bb5328b3cc1168867e4dce5562f4f09b34b439418f0d5b724abad", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "264-03", "group": "264", "title": "Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility", "pdf": "preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf", "folder": "preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026", "abstract_original": "For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C 2 future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6636, "coverage": "catalog_only", "read_detail": null, "pages": 151, "sha256": "cd38cc55e3c203a3e5e080c4e3dbdf50302395537960563ae87276e7859b6fba", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "265-01", "group": "265", "title": "A two-dimensional area law from a global spectral gap", "pdf": "preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf", "folder": "preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026", "abstract_original": "We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6652, "coverage": "catalog_only", "read_detail": null, "pages": 90, "sha256": "91e272a2245ad0bd922be670e023ee76d9afa2564c8e309915070a2f42479153", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "265-02", "group": "265", "title": "Polynomial PEPS approximation of gapped square-grid ground states", "pdf": "preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf", "folder": "preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026", "abstract_original": "We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an $`L\\times L`$ square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L −1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6661, "coverage": "catalog_only", "read_detail": null, "pages": 56, "sha256": "0b55ba0b87ad04680d7ec5be5873deb20b20060dfd84193edc10dfa939a7e756", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "266-01", "group": "266", "title": "The maximum number of mutually unbiased bases in dimension six", "pdf": "preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf", "folder": "preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026", "abstract_original": "We prove that the maximum number of mutually unbiased orthonormal bases in ℂ 6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6677, "coverage": "catalog_only", "read_detail": null, "pages": 76, "sha256": "b4cc74c840c902494d589b51c8e9706490f37cf2735e69a0486122131c6b7446", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "266-02", "group": "266", "title": "Exact Fourier certificates for complex Hadamard matrices of order six", "pdf": "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf", "folder": "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026", "abstract_original": "We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of $`(1,1,1,-1,-1,-1)`$. We also give an exact certificate excluding seven mutually unbiased bases in ℂ 6 ; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6686, "coverage": "catalog_only", "read_detail": null, "pages": 35, "sha256": "6bc2448ff13557610112e91ae6e713d0b469588cd9e1e3f9b44c04fc422fd612", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "267-01", "group": "267", "title": "Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas", "pdf": "preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf", "folder": "preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026", "abstract_original": "We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6702, "coverage": "catalog_only", "read_detail": null, "pages": 56, "sha256": "262afaeb37eec4bcdfe82b0db97ed6e3042944565abf8eacffffa892cf8dcefb", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "267-02", "group": "267", "title": "Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas", "pdf": "preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf", "folder": "preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026", "abstract_original": "We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass $`8/(3\\sqrt\\pi)`$, giving the depletion fraction $`\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}+o(\\sqrt{\\rho a^3})`$ for density ρ and hard-sphere exclusion distance a .", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6711, "coverage": "catalog_only", "read_detail": null, "pages": 79, "sha256": "9e8a9907a3ac08117a26db2543c06ee2d1a232319ad4d12ae579f4fd46396133", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "267-03", "group": "267", "title": "Quantum Depletion for Fixed Bounded Repulsive Potentials", "pdf": "preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf", "folder": "preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026", "abstract_original": "We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a . The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is $`\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}+o(\\sqrt{\\rho a^3})`$ as $`\\rho\\downarrow0`$. The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6720, "coverage": "catalog_only", "read_detail": null, "pages": 49, "sha256": "808cf0e274a8852cd16dd3048da2cdbb62feeacaca6c2d332140ea6a95d6e0a9", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "267-04", "group": "267", "title": "A density-uniform condensate bound for dilute Bose gases", "pdf": "preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf", "folder": "preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026", "abstract_original": "For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6729, "coverage": "catalog_only", "read_detail": null, "pages": 42, "sha256": "5dceb5d5d293d6dcbc651560e65a2475219bf7ed9489ad55d9b5e49e75b1b0b0", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "267-05", "group": "267", "title": "Ground-state condensation in the dilute hard-sphere gas", "pdf": "preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf", "folder": "preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026", "abstract_original": "We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6738, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "3f861f3b57ec1d6ad17e2f01f1924a80bb1c48e4416471756b2257b90d2472f7", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "268-01", "group": "268", "title": "The periodic spin-one Haldane gap", "pdf": "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf", "folder": "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026", "abstract_original": "We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6754, "coverage": "catalog_only", "read_detail": null, "pages": 30, "sha256": "989f14096403897c693467f4007eb8997899648fd50029d5bee50ca9f8471ba6", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "268-02", "group": "268", "title": "A boundary-field gap for the spin-one Heisenberg chain", "pdf": "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf", "folder": "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026", "abstract_original": "We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field $`h=3/5`$ gives a gap greater than $`\\log(10)/392`$ on every chain of $`2L+1`$ sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6763, "coverage": "catalog_only", "read_detail": null, "pages": 28, "sha256": "7fb53ff5eb7f82060d746192174c1165fa4d5b994ffbcb5e489b4f2534ccc1a7", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "269-01", "group": "269", "title": "Uniform Stability of the Spherical Laughlin Gap", "pdf": "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf", "folder": "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026", "abstract_original": "We prove that the fermionic Laughlin V 1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux $`q=3(N-1)`$, the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6779, "coverage": "catalog_only", "read_detail": null, "pages": 69, "sha256": "b1bb20744f2bfa517cf927d9b7f48cfb2ca923ab73fd7fc5cbad077af30aa548", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "269-02", "group": "269", "title": "A Fock-space inequality and the Laughlin spectral gap", "pdf": "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf", "folder": "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026", "abstract_original": "We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V 1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, $`H_Q^2\\ge\\gamma H_Q`$ for some fixed $`\\gamma\\gt 1/25`$ on the entire lowest-Landau-level Fock space for all sufficiently large flux Q , independently of particle number.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6788, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "46a8f75298d339060d7a8ed72fbb7bd52e7b85e8b39f0438be55f19ae217ec0b", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "270-01", "group": "270", "title": "The unique threshold bound state of the SU(N) BFSS model", "pdf": "preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf", "folder": "preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026", "abstract_original": "For every finite N ≥ 2, we prove that undeformed $`\\mathop{\\mathrm{SU}}\\nolimits (N)`$ BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6804, "coverage": "catalog_only", "read_detail": null, "pages": 48, "sha256": "7a4b12bd2a7e8902c19fbda1fd0a9e73c063d8a7a0353667bceaf4194bffadef", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "270-02", "group": "270", "title": "Positive eigenvalues of the relative SU(2) BFSS Hamiltonian", "pdf": "preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf", "folder": "preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026", "abstract_original": "We prove that the relative $`\\mathop{\\mathrm{SU}}\\nolimits (2)`$ BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6813, "coverage": "catalog_only", "read_detail": null, "pages": 9, "sha256": "3501f8a562e65fc545fbc82070b0d5260ee0df6893cf232db3777015be1f7164", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "271-01", "group": "271", "title": "Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions", "pdf": "preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf", "folder": "preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026", "abstract_original": "We prove Bloch's T 3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ 3 , including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6829, "coverage": "catalog_only", "read_detail": null, "pages": 38, "sha256": "4884bc0984dfd0f680c5faf0fe6fb0121f923f2aed6d33267960e887eebde860", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "271-02", "group": "271", "title": "The first lattice correction to Bloch's law", "pdf": "preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf", "folder": "preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026", "abstract_original": "We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin $`S=\\tfrac12,1,\\tfrac32,\\ldots`$. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to $`o(\\beta^{-5/2})`$. In addition to the leading Bloch term, this gives the correction $`3\\zeta(5/2)(\\beta S)^{-5/2}/(128\\pi^{3/2})`$. The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6838, "coverage": "catalog_only", "read_detail": null, "pages": 37, "sha256": "4c19762cfd46fbb074ee35290bfd6200bc9827def6fa58c08f471dad7b2f5456", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "271-03", "group": "271", "title": "The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet", "pdf": "preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf", "folder": "preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026", "abstract_original": "We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6847, "coverage": "catalog_only", "read_detail": null, "pages": 47, "sha256": "07b4a3f530849ff7b470b878aef0fd90c21322532ee2b4f47b0acf6cfd749d2f", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "271-04", "group": "271", "title": "Spontaneous magnetization in the quantum Heisenberg ferromagnet", "pdf": "preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf", "folder": "preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026", "abstract_original": "For every dimension d ≥ 3 and every spin $`S\\in\\{\\frac12,1,\\frac32,\\ldots\\}`$, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least $`S/4`$. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6856, "coverage": "catalog_only", "read_detail": null, "pages": 54, "sha256": "8c9083e5eb7a9d6f877ef0695ec48ccbebb02c53d06b6e4af2719e24a3fdf70b", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "272-01", "group": "272", "title": "Entanglement with zero distillable secret key in local dimension ten", "pdf": "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf", "folder": "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026", "abstract_original": "We construct an entangled state on $`\\mathbb C^{10}\\otimes\\mathbb C^{10}`$ with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on $`M_{21}(\\mathbb C)`$ has a square that is not entanglement breaking.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6872, "coverage": "catalog_only", "read_detail": null, "pages": 53, "sha256": "be18f63c5a1d11a2d28822b8ed3cb5bffb96c46b16722918d0b2107542450bb6", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "273-01", "group": "273", "title": "The entropy photon-number inequality", "pdf": "preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf", "folder": "preprints/The-entropy-photon-number-inequality-September-24-2026", "abstract_original": "We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6888, "coverage": "catalog_only", "read_detail": null, "pages": 32, "sha256": "d826f8ebe18596419871100e79611868dff2a45a4aa157433c58e6f49a234d82", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "274-01", "group": "274", "title": "Product-projection localization and the QAC0 parity lower bound", "pdf": "preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf", "folder": "preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026", "abstract_original": "We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6904, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "12393c9f997aa808d71792f8afbd0cd2e65cf2892725f621ef845cc87b61efcf", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "274-02", "group": "274", "title": "Regular trajectories, pruning and quantum parity", "pdf": "preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf", "folder": "preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026", "abstract_original": "We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to $`|0\\rangle`$, one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6913, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "de3f7a04ba13787664d79ef15d5fc8d0a97f068d36af9c06ecd9a6004c3f34d8", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "275-01", "group": "275", "title": "Continuum Coulomb hardness with binary nuclear charges", "pdf": "preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf", "folder": "preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026", "abstract_original": "We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6929, "coverage": "catalog_only", "read_detail": null, "pages": 47, "sha256": "290f40f845dc3e0d56dfab4572a4f260c089df2f31e2677e23faa5e464d57190", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "275-02", "group": "275", "title": "QMA-hardness of continuum Coulomb energy with unit nuclear charges", "pdf": "preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf", "folder": "preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026", "abstract_original": "We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6938, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "d817c071169eb58cf10d5223ce89b9072342a6d8673ef3b09f1c73cf413b8b98", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "276-01", "group": "276", "title": "Classical capacity and entropy inequalities for generalized amplitude damping", "pdf": "preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf", "folder": "preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026", "abstract_original": "We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6954, "coverage": "catalog_only", "read_detail": null, "pages": 20, "sha256": "5656514823b55d5a4dbb15b284a2637952b957a451aef2e6428f3efbb88f6442", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "277-01", "group": "277", "title": "Threshold parallel repetition for finite-dimensional entangled games", "pdf": "preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf", "folder": "preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026", "abstract_original": "For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a $`v+\\delta`$ fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every $`0\\lt \\delta\\lt 1-v`$ and k ≥ 1. Its universal rate is proportional to $`\\delta^5/(1+\\log(|\\mathcal A||\\mathcal B|))`$, where $`\\mathcal A,\\mathcal B`$ are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ .", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6970, "coverage": "catalog_only", "read_detail": null, "pages": 22, "sha256": "ad2fcb38fca09ed70f48af4696d709617d424769ec1fe81a8e3313b2441b8234", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "278-01", "group": "278", "title": "A Coulomb ground-state density without Kohn-Sham ensemble representation", "pdf": "preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf", "folder": "preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026", "abstract_original": "We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in $`L^{3/2}(\\mathbf R^3)+L^\\infty(\\mathbf R^3)`$. The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 6986, "coverage": "catalog_only", "read_detail": null, "pages": 45, "sha256": "5a9d714fd656a2c9b5e50d13e696e07d8277ac97a3ca689ff11129a13d97fb1d", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "279-01", "group": "279", "title": "Exact quantum factoring over a fixed finite gate set", "pdf": "preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf", "folder": "preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026", "abstract_original": "We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7002, "coverage": "catalog_only", "read_detail": null, "pages": 24, "sha256": "35e89279253b6c8ae5ae7c941283b4b9de517243564f87e9b323ffeae4a85699", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "280-01", "group": "280", "title": "Strongly rational unitary vertex operator algebras and conformal nets", "pdf": "preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf", "folder": "preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026", "abstract_original": "Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net. Gui's extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7018, "coverage": "catalog_only", "read_detail": null, "pages": 47, "sha256": "b5f09142bda3ec3d351fe0e35483c07905de6a0f8573681cb5c191cbd3e43d2e", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "281-01", "group": "281", "title": "QAOA attains the SK ground-state energy in the thermodynamic-first limit", "pdf": "preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf", "folder": "preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026", "abstract_original": "We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7034, "coverage": "catalog_only", "read_detail": null, "pages": 70, "sha256": "4d07ebe181b03157398deef72b23eff8046183f67855d30f9e48e222550a44ba", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "281-02", "group": "281", "title": "Full support of the zero-temperature Sherrington-Kirkpatrick order parameter", "pdf": "preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf", "folder": "preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026", "abstract_original": "We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $`[0,1)`$. Thus its support has no gaps at any overlap scale below one. We use the covariance normalization $`\\xi(t)=t^2/2`$.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7043, "coverage": "catalog_only", "read_detail": null, "pages": 27, "sha256": "86256ac3e995c880c1f9075b3e2410d6cb5ca37e4f00ef6303c7605e87e1d3d8", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "282-01", "group": "282", "title": "Scale and conformal symmetry in four-dimensional operational quantum field theory", "pdf": "preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf", "folder": "preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026", "abstract_original": "We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support. The original fields and their adjoints have compatible affiliated realizations in a causally commuting bounded local net, and a physical local dilatation current generates scale transformations through its local Ward identities. The operational framework admits sharply localized reconstructed fields after their joint products and compatible affiliated realizations in the same net have been established. Under these hypotheses, the stress tensor admits a symmetric, conserved, traceless improvement with unchanged translation and Lorentz charges. The corresponding conformal currents satisfy unbroken local Ward identities on the physical field algebra. The conclusion is local; integration to a global conformal action on the whole bounded net remains a separate question.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7059, "coverage": "catalog_only", "read_detail": null, "pages": 112, "sha256": "485a44682f95ef0dd756e73644dc348f86a6e51b859ef8442b90b4abd378b89b", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "283-01", "group": "283", "title": "Polynomial-Time Unitary Synthesis from a Boolean Oracle", "pdf": "preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf", "folder": "preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026", "abstract_original": "We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n , a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n -qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates $`H,T,T^\\dagger,\\mathrm{CNOT}`$, the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary; the theorem does not give an efficient classical procedure for constructing it.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7075, "coverage": "catalog_only", "read_detail": null, "pages": 21, "sha256": "198849e472386b7a14e8cd7ff52e8ec305c1b6ebc596ec846bcd947bb04b101c", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" }, { "id": "284-01", "group": "284", "title": "A Nearly Quartic Separation Between Randomized and Quantum Query Complexity", "pdf": "preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf", "folder": "preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026", "abstract_original": "We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as $`\\mathrm R(f)`$ and $`\\mathrm Q(f)`$, the examples rule out every universal bound $`\\mathrm R(f)=O((1+\\mathrm Q(f))^\\alpha)`$ with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.", "archive": "math_research_04_of_08.zip", "exists": true, "source_line": 7091, "coverage": "catalog_only", "read_detail": null, "pages": 19, "sha256": "a99f1c38d086a33591b6204ae3957fe3f25c8f31d97c48087325a18fd66daf3e", "bundled_pdf": null, "local_href": "../math_sources/math_research_04_of_08/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf", "online_url": "https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf", "source_catalog": "sources/context/CONTENTS.md", "independent_verification": "not_identified_in_limited_search" } ], "extra_folders": [] }