sample_id,family_id,paper,key,prior_classification,direct_record_classification,classification,classification_basis,assessment,raw,bibliography_line,citing_source_url,original_evidence,evidence_urls,secondary_evidence,supplementary_notes,coverage_note
R001,015,Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,EKL2006,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{EKL2006,
  author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
  title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture},
  journal = {Annals of Mathematics},
  volume = {164}, number = {2}, year = {2006}, pages = {513--560},
  doi = {10.4007/annals.2006.164.513}
}",34,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib#L34,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2006.164.513"", ""title"": [""Invariant measures and the set of exceptions to Littlewood’s conjecture""], ""authors"": [{""given"": ""Manfred"", ""family"": ""Einsiedler"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Anatole"", ""family"": ""Katok"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Elon"", ""family"": ""Lindenstrauss"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2006, 9, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2006.164.513"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2006.164.513""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2006.164.513""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2102029381"", ""doi"": ""10.4007/annals.2006.164.513"", ""pmid"": null, ""title"": ""Invariant measures and the set of exceptions to Littlewood’s conjecture"", ""publication_year"": 2006, ""publication_date"": ""2006-09-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5009126028"", ""name"": ""Manfred Einsiedler"", ""orcid"": ""https://orcid.org/0009-0001-2372-5894"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Princeton University""]}, {""author_id"": ""A5108479328"", ""name"": ""Anatole Katok"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Pennsylvania State University""]}, {""author_id"": ""A5052232750"", ""name"": ""Elon Lindenstrauss"", ""orcid"": ""https://orcid.org/0000-0002-4757-6758"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Princeton University""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""164"", ""issue"": ""2"", ""first_page"": ""513"", ""last_page"": ""560""}, ""cited_by_count"": 185, ""fwci"": 14.7474, ""referenced_works_count"": 53, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v164-n2-p04.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v164-n2-p04.pdf"", ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""invariant measures"", ""ergodic measures"", ""Hausdorff dimension"", ""Diophantine approximation"", ""homogeneous dynamics""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2006.164.513"", ""referenced_works"": [""W653864992"", ""W1495027777"", ""W1522020530"", ""W1535953077"", ""W1546990169"", ""W1580077494"", ""W1677684899"", ""W1905444936"", ""W1967678020"", ""W1969863916"", ""W1989372125"", ""W1992791278"", ""W1998025851"", ""W2024216988"", ""W2039890329"", ""W2059059474"", ""W2059735611"", ""W2073200499"", ""W2074319183"", ""W2078778864"", ""W2079007018"", ""W2083309269"", ""W2088544526"", ""W2089575867"", ""W2090467845"", ""W2094101369"", ""W2098449382"", ""W2099683177"", ""W2114189018"", ""W2130426877"", ""W2140059063"", ""W2141471286"", ""W2156831121"", ""W2164710482"", ""W2169190331"", ""W2184075809"", ""W2217269573"", ""W2316225159"", ""W2324647620"", ""W2334384374"", ""W2574665036"", ""W2578193136"", ""W2809504227"", ""W2993508114"", ""W3006332256"", ""W3109381375"", ""W3130601852"", ""W3152478022"", ""W3212292137"", ""W4230097895"", ""W6607520151"", ""W6720792767"", ""W6774869894""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 8}, {""year"": 2025, ""cited_by_count"": 7}, {""year"": 2024, ""cited_by_count"": 5}, {""year"": 2023, ""cited_by_count"": 8}, {""year"": 2022, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 8}, {""year"": 2020, ""cited_by_count"": 10}, {""year"": 2019, ""cited_by_count"": 8}, {""year"": 2018, ""cited_by_count"": 9}, {""year"": 2017, ""cited_by_count"": 4}, {""year"": 2016, ""cited_by_count"": 13}, {""year"": 2015, ""cited_by_count"": 9}, {""year"": 2014, ""cited_by_count"": 7}, {""year"": 2013, ""cited_by_count"": 8}, {""year"": 2012, ""cited_by_count"": 4}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R002,015,Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,ELMV2011,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{ELMV2011,
  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
  title = {Distribution of periodic torus orbits and {Duke}'s theorem for cubic fields},
  journal = {Annals of Mathematics},
  volume = {173}, number = {2}, year = {2011}, pages = {815--885},
  doi = {10.4007/annals.2011.173.2.5}
}",18,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib#L18,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2011.173.2.5"", ""title"": [""Distribution of periodic torus orbits and Duke's theorem for cubic fields""], ""authors"": [{""given"": ""Manfred"", ""family"": ""Einsiedler"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Elon"", ""family"": ""Lindenstrauss"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Philippe"", ""family"": ""Michel"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Akshay"", ""family"": ""Venkatesh"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2011, 3, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2011.173.2.5"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2011.173.2.5""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2011.173.2.5""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2115929736"", ""doi"": ""10.4007/annals.2011.173.2.5"", ""pmid"": null, ""title"": ""Distribution of periodic torus orbits and Duke's theorem for cubic fields"", ""publication_year"": 2011, ""publication_date"": ""2011-03-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5009126028"", ""name"": ""Manfred Einsiedler"", ""orcid"": ""https://orcid.org/0009-0001-2372-5894"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""ETH Zurich""]}, {""author_id"": ""A5052232750"", ""name"": ""Elon Lindenstrauss"", ""orcid"": ""https://orcid.org/0000-0002-4757-6758"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Hebrew University of Jerusalem"", ""Princeton University""]}, {""author_id"": ""A5051368744"", ""name"": ""Philippe Michel"", ""orcid"": ""https://orcid.org/0000-0003-2419-3073"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""École Polytechnique Fédérale de Lausanne""]}, {""author_id"": ""A5066089199"", ""name"": ""Akshay Venkatesh"", ""orcid"": ""https://orcid.org/0000-0003-1648-484X"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Stanford University""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""173"", ""issue"": ""2"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 65, ""fwci"": 6.1035, ""referenced_works_count"": 72, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n2-p05-p.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n2-p05-p.pdf"", ""primary_topic"": {""id"": ""T11680"", ""display_name"": ""Advanced Algebra and Geometry"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""lattices"", ""cubic fields"", ""totally real fields"", ""equidistribution"", ""closed geodesics""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2011.173.2.5"", ""referenced_works"": [""W39931892"", ""W62210103"", ""W84318622"", ""W86209192"", ""W599328202"", ""W1488387456"", ""W1489895501"", ""W1492798315"", ""W1564577416"", ""W1584131488"", ""W1590252182"", ""W1639953915"", ""W1831906308"", ""W1850307075"", ""W1905444936"", ""W1969338287"", ""W1974099924"", ""W1979558683"", ""W1991150663"", ""W1991174092"", ""W1992416585"", ""W2001029715"", ""W2001729983"", ""W2002611642"", ""W2002625855"", ""W2005376004"", ""W2016713393"", ""W2017195856"", ""W2021362928"", ""W2021409800"", ""W2023554282"", ""W2036556379"", ""W2039971622"", ""W2044394599"", ""W2059735611"", ""W2063670237"", ""W2071750534"", ""W2083946017"", ""W2088446959"", ""W2088544526"", ""W2089408657"", ""W2090467845"", ""W2090897065"", ""W2098449382"", ""W2101109099"", ""W2102029381"", ""W2121902984"", ""W2125825082"", ""W2141471286"", ""W2142309744"", ""W2156948039"", ""W2163219038"", ""W2238455553"", ""W2244987905"", ""W2312285859"", ""W2333176438"", ""W2506168267"", ""W2597215152"", ""W2951914514"", ""W2952693472"", ""W2964152170"", ""W2988221775"", ""W3037513664"", ""W3140515965"", ""W4213035716"", ""W4231684077"", ""W4244080937"", ""W4244758685"", ""W4246077597"", ""W4285719527"", ""W4298724296"", ""W4300990447""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 4}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 9}, {""year"": 2022, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 5}, {""year"": 2020, ""cited_by_count"": 2}, {""year"": 2019, ""cited_by_count"": 7}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 3}, {""year"": 2016, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 4}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R003,015,Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,Khayutin2019,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Khayutin2019,
  author = {Khayutin, Ilya},
  title = {Arithmetic of double torus quotients and the distribution of periodic torus orbits},
  journal = {Duke Mathematical Journal},
  volume = {168}, number = {12}, year = {2019}, pages = {2365--2432},
  doi = {10.1215/00127094-2019-0016},
  eprint = {1510.08481}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}",51,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib#L51,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1215/00127094-2019-0016"", ""title"": [""Arithmetic of double torus quotients and the distribution of periodic torus orbits""], ""authors"": [{""given"": ""Ilya"", ""family"": ""Khayutin"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Duke University Press"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2019, 9, 1]]}, ""container_title"": [""Duke Mathematical Journal""], ""url"": ""https://doi.org/10.1215/00127094-2019-0016"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1215%2F00127094-2019-0016""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1510.08481"", ""version"": 2, ""id_versioned"": ""1510.08481v2"", ""title"": ""Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits"", ""abstract"": ""We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates and homogeneous dynamics."", ""authors"": [""Ilya Khayutin""], ""published"": ""2015-10-28T20:45:09Z"", ""updated"": ""2019-02-25T21:25:42Z"", ""primary_category"": ""math.NT"", ""categories"": [""math.NT"", ""math.DS""], ""doi"": ""10.1215/00127094-2019-0016"", ""journal_ref"": ""Duke Math. J. 168, no. 12 (2019), 2365-2432"", ""comment"": ""To appear in Duke Math. J. Updated according to referee comments. 61 pages"", ""abs_url"": ""http://arxiv.org/abs/1510.08481v2"", ""pdf_url"": ""https://arxiv.org/pdf/1510.08481v2""}}]","[""https://api.crossref.org/works/10.1215%2F00127094-2019-0016"", ""http://arxiv.org/abs/1510.08481v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2214473080"", ""doi"": ""10.1215/00127094-2019-0016"", ""pmid"": null, ""title"": ""Arithmetic of double torus quotients and the distribution of periodic torus orbits"", ""publication_year"": 2019, ""publication_date"": ""2019-08-24"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5061370140"", ""name"": ""Ilya Khayutin"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S171170845"", ""display_name"": ""Duke Mathematical Journal"", ""issn_l"": ""0012-7094"", ""type"": ""journal""}, ""biblio"": {""volume"": ""168"", ""issue"": ""12"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 3, ""fwci"": 0.6281, ""referenced_works_count"": 45, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://projecteuclid.org/journals/duke-mathematical-journal/volume-168/issue-12/Arithmetic-of-double-torus-quotients-and-the-distribution-of-periodic/10.1215/00127094-2019-0016.pdf""}, ""best_oa_pdf_url"": ""https://projecteuclid.org/journals/duke-mathematical-journal/volume-168/issue-12/Arithmetic-of-double-torus-quotients-and-the-distribution-of-periodic/10.1215/00127094-2019-0016.pdf"", ""primary_topic"": {""id"": ""T11680"", ""display_name"": ""Advanced Algebra and Geometry"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""arithmetic invariants"", ""S-arithmetic groups"", ""maximal torus"", ""equidistribution"", ""homogeneous dynamics"", ""geometric invariant theory"", ""Galois action""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1215/00127094-2019-0016"", ""referenced_works"": [""W62210103"", ""W599328202"", ""W616606718"", ""W1485511365"", ""W1488052761"", ""W1521322992"", ""W1572341022"", ""W1792107936"", ""W1922550957"", ""W1968275987"", ""W1979815727"", ""W1991174092"", ""W2011376265"", ""W2012605734"", ""W2016713393"", ""W2023554282"", ""W2026853928"", ""W2037272179"", ""W2053050060"", ""W2059059474"", ""W2102029381"", ""W2115929736"", ""W2125986605"", ""W2131330444"", ""W2131506390"", ""W2163219038"", ""W2164710482"", ""W2165694723"", ""W2239670832"", ""W2244987905"", ""W2597215152"", ""W2962792389"", ""W2963567239"", ""W2963860365"", ""W2964028636"", ""W2964325193"", ""W2964329865"", ""W3105786697"", ""W3125438820"", ""W4211073658"", ""W4212965887"", ""W4245371476"", ""W4251324578"", ""W4293472173"", ""W4300779862""], ""counts_by_year"": [{""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R004,015,Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,Stark1974,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Stark1974,
  author = {Stark, H. M.},
  title = {Some effective cases of the {Brauer--Siegel} theorem},
  journal = {Inventiones Mathematicae},
  volume = {23}, number = {2}, year = {1974}, pages = {135--152},
  doi = {10.1007/BF01405166}
}",60,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib#L60,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01405166"", ""title"": [""Some effective cases of the Brauer-Siegel Theorem""], ""authors"": [{""given"": ""H. M."", ""family"": ""Stark"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1974, 6]]}, ""container_title"": [""Inventiones Mathematicae""], ""url"": ""https://doi.org/10.1007/bf01405166"", ""license"": [{""start"": {""date-parts"": [[1974, 6, 1]], ""date-time"": ""1974-06-01T00:00:00Z"", ""timestamp"": 139276800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01405166""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01405166""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1992416585"", ""doi"": ""10.1007/bf01405166"", ""pmid"": null, ""title"": ""Some effective cases of the Brauer-Siegel Theorem"", ""publication_year"": 1974, ""publication_date"": ""1974-06-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""name"": ""H. M. Stark"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Massachusetts Institute of Technology""]}], ""source"": {""source_id"": ""S98347115"", ""display_name"": ""Inventiones mathematicae"", ""issn_l"": ""0020-9910"", ""type"": ""journal""}, ""biblio"": {""volume"": ""23"", ""issue"": ""2"", ""first_page"": ""135"", ""last_page"": ""152""}, ""cited_by_count"": 253, ""fwci"": 2.5682, ""referenced_works_count"": 8, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11166"", ""display_name"": ""Analytic Number Theory Research"", ""field"": ""Mathematics"", ""subfield"": ""Algebra and Number Theory"", ""domain"": ""Physical Sciences""}, ""keywords"": [""effective bounds"", ""class number"", ""regulators"", ""number fields"", ""Dirichlet L-functions""], ""abstract"": null, ""referenced_works"": [""W1030743702"", ""W1489132645"", ""W1498769399"", ""W2577105026"", ""W2907203254"", ""W2983337356"", ""W3007836994"", ""W3140985225""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 10}, {""year"": 2024, ""cited_by_count"": 6}, {""year"": 2023, ""cited_by_count"": 6}, {""year"": 2022, ""cited_by_count"": 6}, {""year"": 2021, ""cited_by_count"": 5}, {""year"": 2020, ""cited_by_count"": 8}, {""year"": 2019, ""cited_by_count"": 12}, {""year"": 2018, ""cited_by_count"": 10}, {""year"": 2017, ""cited_by_count"": 8}, {""year"": 2016, ""cited_by_count"": 6}, {""year"": 2015, ""cited_by_count"": 12}, {""year"": 2014, ""cited_by_count"": 11}, {""year"": 2013, ""cited_by_count"": 4}, {""year"": 2012, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R005,015,Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,Milne2020,unverified,unverified,match,supplied_notes_identity,"Supplied title-page notes support Milne, Algebraic Number Theory, v3.08, July 19, 2020. Not independently read in this session.","@misc{Milne2020,
  author = {Milne, James S.},
  title = {Algebraic Number Theory},
  year = {2020}, note = {Version 3.08},
  url = {https://www.jmilne.org/math/CourseNotes/ANT.pdf}
}",86,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib#L86,[],[],[],"[{""sample_id"": ""R005"", ""source_url"": ""https://www.jmilne.org/math/CourseNotes/ANT.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""24b83c789a89f25aebffb3cbe4ae5ca29edd0075acc072de093d140770630847"", ""locator"": ""PDF page 1"", ""source_version"": ""Author course notes v3.08"", ""paraphrased_observations"": ""The title page identifies J. S. Milne, Algebraic Number Theory, version 3.08, dated July 19, 2020."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R006,024,An-asymptotic-formula-for-the-number-of-totients-September-25-2026,Pomerance1986,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Pomerance1986,
  author = {Pomerance, Carl},
  title = {On the distribution of the values of {Euler}'s function},
  journal = {Acta Arithmetica},
  volume = {47},
  number = {1},
  year = {1986},
  pages = {63--70},
  doi = {10.4064/aa-47-1-63-70},
  note = {\url{https://math.dartmouth.edu/~carlp/PDF/57.pdf}}
}",129,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/references.bib#L129,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4064/aa-47-1-63-70"", ""title"": [""On the distribution of the values of Euler's function""], ""authors"": [{""given"": ""Carl"", ""family"": ""Pomerance"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Institute of Mathematics, Polish Academy of Sciences"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1986]]}, ""container_title"": [""Acta Arithmetica""], ""url"": ""https://doi.org/10.4064/aa-47-1-63-70"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4064%2Faa-47-1-63-70""}}]","[""https://api.crossref.org/works/10.4064%2Faa-47-1-63-70""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W922556180"", ""doi"": ""10.4064/aa-47-1-63-70"", ""pmid"": null, ""title"": ""On the distribution of the values of Euler's function"", ""publication_year"": 1986, ""publication_date"": ""1986-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5065324760"", ""name"": ""Carl B. Pomerance"", ""orcid"": ""https://orcid.org/0000-0002-0960-1692"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S141074886"", ""display_name"": ""Acta Arithmetica"", ""issn_l"": ""0065-1036"", ""type"": ""journal""}, ""biblio"": {""volume"": ""47"", ""issue"": ""1"", ""first_page"": ""63"", ""last_page"": ""70""}, ""cited_by_count"": 10, ""fwci"": 0, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.impan.pl/shop/publication/transaction/download/product/105027?download.pdf""}, ""best_oa_pdf_url"": ""https://www.impan.pl/shop/publication/transaction/download/product/105027?download.pdf"", ""primary_topic"": {""id"": ""T11166"", ""display_name"": ""Analytic Number Theory Research"", ""field"": ""Mathematics"", ""subfield"": ""Algebra and Number Theory"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Euler totient function"", ""multiplicative functions"", ""number theory""], ""abstract"": null, ""referenced_works"": [], ""counts_by_year"": [{""year"": 2014, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R007,024,An-asymptotic-formula-for-the-number-of-totients-September-25-2026,Erdos1958,unverified,unverified,match,supplied_notes_identity,"Supplied first-page notes support author/title identity despite imperfect OCR. Year, venue and pagination remain unverified; the filename does not establish publication date.","@article{Erdos1958,
  author = {Erd{\H{o}}s, Paul},
  title = {Some remarks on {Euler}'s {$\phi$} function},
  journal = {Acta Arithmetica},
  volume = {4},
  year = {1958},
  pages = {10--19},
  note = {\url{https://users.renyi.hu/~p_erdos/1958-18.pdf}}
}",94,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/references.bib#L94,[],[],[],"[{""sample_id"": ""R007"", ""source_url"": ""https://users.renyi.hu/~p_erdos/1958-18.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""57d6660b80ddc74dfb6e3cb7521f52249a3cb598a1380faebc6465ea73221981"", ""locator"": ""PDF page 1"", ""source_version"": null, ""paraphrased_observations"": ""The first page identifies P. Erdos and the Euler phi function title matching the cited work; OCR is imperfect. The filename is 1958-18. Identity evidence is stronger than the previous failed lookup; exact publication year, venue and page range still require a complete bibliographic record."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R008,024,An-asymptotic-formula-for-the-number-of-totients-September-25-2026,ErdosHall1976,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{ErdosHall1976,
  author = {Erd{\H{o}}s, Paul and Hall, R. R.},
  title = {Distinct values of {Euler}'s {$\phi$}-function},
  journal = {Mathematika},
  volume = {23},
  number = {1},
  year = {1976},
  pages = {1--3},
  doi = {10.1112/S0025579300006100},
  note = {\url{https://users.renyi.hu/~p_erdos/1976-11.pdf}}
}",116,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/references.bib#L116,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1112/s0025579300006100"", ""title"": [""Distinct values of Euler's ø‐function""], ""authors"": [{""given"": ""P."", ""family"": ""Erdős"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Department of MathematicsUniversity of YorkEngland""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""R. R."", ""family"": ""Hall"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Department of MathematicsUniversity of YorkEngland""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Wiley"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1976, 6]]}, ""container_title"": [""Mathematika""], ""url"": ""https://doi.org/10.1112/s0025579300006100"", ""license"": [{""start"": {""date-parts"": [[1976, 6, 1]], ""date-time"": ""1976-06-01T00:00:00Z"", ""timestamp"": 202435200000}, ""content-version"": ""vor"", ""delay-in-days"": 0, ""URL"": ""http://onlinelibrary.wiley.com/termsAndConditions#vor""}, {""start"": {""date-parts"": [[1976, 6, 1]], ""date-time"": ""1976-06-01T00:00:00Z"", ""timestamp"": 202435200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://doi.wiley.com/10.1002/tdm_license_1.1""}], ""source_url"": ""https://api.crossref.org/works/10.1112%2Fs0025579300006100""}}]","[""https://api.crossref.org/works/10.1112%2Fs0025579300006100""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2138353546"", ""doi"": ""10.1112/s0025579300006100"", ""pmid"": null, ""title"": ""Distinct values of Euler's ø‐function"", ""publication_year"": 1976, ""publication_date"": ""1976-06-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5077107126"", ""name"": ""Paul L. Erdos"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of York""]}, {""author_id"": ""A5109892042"", ""name"": ""R. R. Hall"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of York""]}], ""source"": {""source_id"": ""S87963603"", ""display_name"": ""Mathematika"", ""issn_l"": ""0025-5793"", ""type"": ""journal""}, ""biblio"": {""volume"": ""23"", ""issue"": ""1"", ""first_page"": ""1"", ""last_page"": ""3""}, ""cited_by_count"": 15, ""fwci"": 0.567, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11166"", ""display_name"": ""Analytic Number Theory Research"", ""field"": ""Mathematics"", ""subfield"": ""Algebra and Number Theory"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Euler totient function"", ""multiplicative functions"", ""number theory""], ""abstract"": null, ""referenced_works"": [], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R009,024,An-asymptotic-formula-for-the-number-of-totients-September-25-2026,Erdos1945,unverified,unverified,match,supplied_notes_identity,"Supplied first-page notes support author/title identity. February 9, 1945 is a received date, not proof of publication date or pagination.","@article{Erdos1945,
  author = {Erd{\H{o}}s, Paul},
  title = {Some remarks on {Euler}'s {$\phi$} function and some related problems},
  journal = {Bulletin of the American Mathematical Society},
  volume = {51},
  year = {1945},
  pages = {540--544},
  note = {\url{https://users.renyi.hu/~p_erdos/1945-08.pdf}}
}",105,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/references.bib#L105,[],[],[],"[{""sample_id"": ""R009"", ""source_url"": ""https://users.renyi.hu/~p_erdos/1945-08.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""a4d3978b3e81a493521702d62e04cb2a59785b26cc05f43d27edb3b2802a96aa"", ""locator"": ""PDF page 1"", ""source_version"": null, ""paraphrased_observations"": ""The first page identifies Paul Erdos and the title concerning Euler phi function and related problems. It includes a received date of February 9, 1945. This supports identity; a received date alone does not establish publication date or journal pagination."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R010,024,An-asymptotic-formula-for-the-number-of-totients-September-25-2026,Pillai1929,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Pillai1929,
  author = {Pillai, S. S.},
  title = {On some functions connected with {$\phi(n)$}},
  journal = {Bulletin of the American Mathematical Society},
  volume = {35},
  number = {6},
  year = {1929},
  pages = {832--836},
  note = {\url{https://www.ams.org/journals/bull/1929-35-06/S0002-9904-1929-04799-2/S0002-9904-1929-04799-2.pdf}}
}",70,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/references.bib#L70,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1090/s0002-9904-1929-04799-2"", ""title"": [""On some functions connected with 𝜙(𝑛)""], ""authors"": [{""given"": ""S. Sivasankaranarayana"", ""family"": ""Pillai"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""American Mathematical Society (AMS)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1929]]}, ""container_title"": [""Bulletin of the American Mathematical Society""], ""url"": ""https://doi.org/10.1090/s0002-9904-1929-04799-2"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1090%2Fs0002-9904-1929-04799-2""}}]","[""https://api.crossref.org/works/10.1090%2Fs0002-9904-1929-04799-2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R011,046,A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,KatzarkovRamachandran1998,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{KatzarkovRamachandran1998,
  author = {Katzarkov, Ludmil and Ramachandran, Mohan},
  title = {On the universal coverings of algebraic surfaces},
  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
  series = {4},
  volume = {31},
  number = {4},
  pages = {525--535},
  year = {1998},
  doi = {10.1016/S0012-9593(98)80105-5}
}",223,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib#L223,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/s0012-9593(98)80105-5"", ""title"": [""On the universal coverings of algebraic surfaces""], ""authors"": [{""given"": ""L"", ""family"": ""KATZARKOV"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""M"", ""family"": ""RAMACHANDRAN"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Societe Mathematique de France"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1998, 7]]}, ""container_title"": [""Annales Scientifiques de l’École Normale Supérieure""], ""url"": ""https://doi.org/10.1016/s0012-9593(98)80105-5"", ""license"": [{""start"": {""date-parts"": [[1998, 7, 1]], ""date-time"": ""1998-07-01T00:00:00Z"", ""timestamp"": 899251200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.elsevier.com/tdm/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fs0012-9593(98)80105-5""}}]","[""https://api.crossref.org/works/10.1016%2Fs0012-9593(98)80105-5""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2074896987"", ""doi"": ""10.1016/s0012-9593(98)80105-5"", ""pmid"": null, ""title"": ""On the universal coverings of algebraic surfaces"", ""publication_year"": 1998, ""publication_date"": ""1998-07-01"", ""type"": ""article"", ""language"": ""fr"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5067549542"", ""name"": ""Ludmil Vasilev Katzarkov"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of California, Irvine""]}, {""author_id"": ""A5037944917"", ""name"": ""Mohan Ramachandran"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": true, ""institutions"": [""University at Buffalo, State University of New York""]}], ""source"": {""source_id"": ""S182617021"", ""display_name"": ""Annales Scientifiques de l École Normale Supérieure"", ""issn_l"": ""0012-9593"", ""type"": ""journal""}, ""biblio"": {""volume"": ""31"", ""issue"": ""4"", ""first_page"": ""525"", ""last_page"": ""535""}, ""cited_by_count"": 27, ""fwci"": 1.7281, ""referenced_works_count"": 18, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10061"", ""display_name"": ""Algebraic Geometry and Number Theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""universal cover"", ""algebraic surfaces"", ""holomorphic convexity"", ""Shafarevich conjecture"", ""analytic functions"", ""fundamental group"", ""smooth projective varieties""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1016/s0012-9593(98)80105-5"", ""referenced_works"": [""W176958027"", ""W204424931"", ""W1571072992"", ""W1599815147"", ""W1603265994"", ""W1965342821"", ""W1978712663"", ""W1985502139"", ""W1987701666"", ""W1991129601"", ""W1991675553"", ""W1997379578"", ""W2004050596"", ""W2027430137"", ""W2057221822"", ""W2058578721"", ""W2062911690"", ""W2328808682""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 3}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 2}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R012,046,A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,MilnorSingular,minor_difference,minor_difference,minor_difference,direct_records,"Same Milnor book, author, publisher and AM-61 identity. Citation says 1968; publisher-deposited record for the discovered edition says 1969-12-31. Edition/date discrepancy unresolved, not evidence of a different book.","@book{MilnorSingular,
  author = {Milnor, John},
  title = {Singular Points of Complex Hypersurfaces},
  series = {Annals of Mathematics Studies},
  volume = {61},
  publisher = {Princeton University Press},
  address = {Princeton, NJ},
  year = {1968}
}",276,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib#L276,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1515/9781400881819"", ""title"": [""Singular Points of Complex Hypersurfaces. (AM-61)""], ""authors"": [{""given"": ""John"", ""family"": ""Milnor"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Princeton University Press"", ""type"": ""edited-book"", ""published"": {""date-parts"": [[1969, 12, 31]]}, ""container_title"": [], ""url"": ""https://doi.org/10.1515/9781400881819"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1515%2F9781400881819""}}]","[""https://api.crossref.org/works/10.1515%2F9781400881819""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R013,046,A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,BailyBorel1966,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{BailyBorel1966,
  author = {Baily, Jr., Walter L. and Borel, Armand},
  title = {Compactification of arithmetic quotients of bounded symmetric domains},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {84},
  number = {3},
  pages = {442--528},
  year = {1966},
  doi = {10.2307/1970457}
}",49,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib#L49,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.2307/1970457"", ""title"": [""Compactification of Arithmetic Quotients of Bounded Symmetric Domains""], ""authors"": [{""suffix"": ""join(' '"", ""given"": ""W. L."", ""family"": ""Baily"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""A."", ""family"": ""Borel"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""JSTOR"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1966, 11]]}, ""container_title"": [""The Annals of Mathematics""], ""url"": ""https://doi.org/10.2307/1970457"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.2307%2F1970457""}}]","[""https://api.crossref.org/works/10.2307%2F1970457""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2319666400"", ""doi"": ""10.2307/1970457"", ""pmid"": null, ""title"": ""Compactification of Arithmetic Quotients of Bounded Symmetric Domains"", ""publication_year"": 1966, ""publication_date"": ""1966-11-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5054810851"", ""name"": ""Walter L. Baily"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5109004255"", ""name"": ""Armand Borel"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": []}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""84"", ""issue"": ""3"", ""first_page"": ""442"", ""last_page"": ""442""}, ""cited_by_count"": 611, ""fwci"": 4.5893, ""referenced_works_count"": 10, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11680"", ""display_name"": ""Advanced Algebra and Geometry"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""compactification"", ""arithmetic quotients"", ""bounded symmetric domains"", ""Hermitian symmetric spaces"", ""arithmetic groups""], ""abstract"": null, ""referenced_works"": [""W44816256"", ""W182374552"", ""W1976036928"", ""W2030880005"", ""W2087640086"", ""W2097462658"", ""W2322023032"", ""W2327287139"", ""W2330668363"", ""W2335759464""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 4}, {""year"": 2025, ""cited_by_count"": 11}, {""year"": 2024, ""cited_by_count"": 12}, {""year"": 2023, ""cited_by_count"": 15}, {""year"": 2022, ""cited_by_count"": 20}, {""year"": 2021, ""cited_by_count"": 24}, {""year"": 2020, ""cited_by_count"": 28}, {""year"": 2019, ""cited_by_count"": 19}, {""year"": 2018, ""cited_by_count"": 25}, {""year"": 2017, ""cited_by_count"": 13}, {""year"": 2016, ""cited_by_count"": 27}, {""year"": 2015, ""cited_by_count"": 24}, {""year"": 2014, ""cited_by_count"": 13}, {""year"": 2013, ""cited_by_count"": 18}, {""year"": 2012, ""cited_by_count"": 15}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R014,046,A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,Gromov1987,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@incollection{Gromov1987,
  author = {Gromov, Mikhail},
  title = {Hyperbolic groups},
  booktitle = {Essays in Group Theory},
  editor = {Gersten, S. M.},
  series = {Mathematical Sciences Research Institute Publications},
  volume = {8},
  pages = {75--263},
  publisher = {Springer-Verlag},
  address = {New York},
  year = {1987},
  doi = {10.1007/978-1-4613-9586-7_3}
}",12,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib#L12,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/978-1-4613-9586-7_3"", ""title"": [""Hyperbolic Groups""], ""authors"": [{""given"": ""M."", ""family"": ""Gromov"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer New York"", ""type"": ""book-chapter"", ""published"": {""date-parts"": [[1987]]}, ""container_title"": [""Mathematical Sciences Research Institute Publications"", ""Essays in Group Theory""], ""url"": ""https://doi.org/10.1007/978-1-4613-9586-7_3"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1007%2F978-1-4613-9586-7_3""}}]","[""https://api.crossref.org/works/10.1007%2F978-1-4613-9586-7_3""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4235484205"", ""doi"": ""10.1007/978-1-4613-9586-7_3"", ""pmid"": null, ""title"": ""Hyperbolic Groups"", ""publication_year"": 1987, ""publication_date"": ""1987-01-01"", ""type"": ""book-chapter"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5109481429"", ""name"": ""Mikhail Leonidovich Gromov"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S4210187425"", ""display_name"": ""Mathematical Sciences Research Institute publications"", ""issn_l"": ""0940-4740"", ""type"": ""book series""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": ""75"", ""last_page"": ""263""}, ""cited_by_count"": 1550, ""fwci"": 16.2603, ""referenced_works_count"": 20, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10304"", ""display_name"": ""Geometric and Algebraic Topology"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""hyperbolic groups"", ""Gromov hyperbolicity"", ""geometric group theory"", ""Cayley graphs"", ""word metric"", ""hyperbolic geometry""], ""abstract"": null, ""referenced_works"": [""W652737649"", ""W1491682511"", ""W1561664481"", ""W1630072945"", ""W1967168980"", ""W1971554164"", ""W2028978432"", ""W2040034546"", ""W2052190887"", ""W2061130210"", ""W2063481322"", ""W2086321422"", ""W2087258949"", ""W2112861192"", ""W2315813414"", ""W2320193599"", ""W2327127028"", ""W4213195349"", ""W4244461330"", ""W4292101339""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 39}, {""year"": 2025, ""cited_by_count"": 106}, {""year"": 2024, ""cited_by_count"": 84}, {""year"": 2023, ""cited_by_count"": 83}, {""year"": 2022, ""cited_by_count"": 79}, {""year"": 2021, ""cited_by_count"": 81}, {""year"": 2020, ""cited_by_count"": 60}, {""year"": 2019, ""cited_by_count"": 56}, {""year"": 2018, ""cited_by_count"": 63}, {""year"": 2017, ""cited_by_count"": 67}, {""year"": 2016, ""cited_by_count"": 67}, {""year"": 2015, ""cited_by_count"": 60}, {""year"": 2014, ""cited_by_count"": 63}, {""year"": 2013, ""cited_by_count"": 57}, {""year"": 2012, ""cited_by_count"": 48}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R015,046,A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,Hartshorne,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@book{Hartshorne,
  author = {Hartshorne, Robin},
  title = {Algebraic Geometry},
  series = {Graduate Texts in Mathematics},
  volume = {52},
  publisher = {Springer-Verlag},
  address = {New York},
  year = {1977},
  doi = {10.1007/978-1-4757-3849-0}
}",286,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib#L286,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/978-1-4757-3849-0"", ""title"": [""Algebraic Geometry""], ""authors"": [{""given"": ""Robin"", ""family"": ""Hartshorne"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer New York"", ""type"": ""book"", ""published"": {""date-parts"": [[1977]]}, ""container_title"": [""Graduate Texts in Mathematics""], ""url"": ""https://doi.org/10.1007/978-1-4757-3849-0"", ""license"": [{""start"": {""date-parts"": [[1977, 1, 1]], ""date-time"": ""1977-01-01T00:00:00Z"", ""timestamp"": 220924800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2F978-1-4757-3849-0""}}]","[""https://api.crossref.org/works/10.1007%2F978-1-4757-3849-0""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4248269519"", ""doi"": ""10.1007/978-1-4757-3849-0"", ""pmid"": null, ""title"": ""Algebraic Geometry"", ""publication_year"": 1977, ""publication_date"": ""1977-01-01"", ""type"": ""book"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5055740712"", ""name"": ""Robin Hartshorne"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of California, Berkeley""]}], ""source"": {""source_id"": ""S4210201401"", ""display_name"": ""Graduate texts in mathematics"", ""issn_l"": ""0072-5285"", ""type"": ""book series""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 6136, ""fwci"": 9.203, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10061"", ""display_name"": ""Algebraic Geometry and Number Theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""algebraic geometry"", ""algebraic varieties"", ""commutative algebra"", ""algebraic curves"", ""algebraic surfaces""], ""abstract"": null, ""referenced_works"": [], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 132}, {""year"": 2025, ""cited_by_count"": 278}, {""year"": 2024, ""cited_by_count"": 298}, {""year"": 2023, ""cited_by_count"": 243}, {""year"": 2022, ""cited_by_count"": 299}, {""year"": 2021, ""cited_by_count"": 289}, {""year"": 2020, ""cited_by_count"": 288}, {""year"": 2019, ""cited_by_count"": 265}, {""year"": 2018, ""cited_by_count"": 264}, {""year"": 2017, ""cited_by_count"": 209}, {""year"": 2016, ""cited_by_count"": 253}, {""year"": 2015, ""cited_by_count"": 215}, {""year"": 2014, ""cited_by_count"": 217}, {""year"": 2013, ""cited_by_count"": 178}, {""year"": 2012, ""cited_by_count"": 169}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R016,065,Virasoro-Constraints-under-Projectivization-October-5-2026,Fan,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Fan}
H.~Fan,
\emph{Chern classes and Gromov--Witten theory of projective bundles},
Amer. J. Math. \textbf{143} (2021), no.~3, 811--832.
\href{https://doi.org/10.1353/ajm.2021.0020}{doi:10.1353/ajm.2021.0020}.",33,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/references.tex#L33,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1353/ajm.2021.0020"", ""title"": [""Chern classes and Gromov-Witten theory of projective bundles""], ""authors"": [{""given"": ""Honglu"", ""family"": ""Fan"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Project MUSE"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2021]]}, ""container_title"": [""American Journal of Mathematics""], ""url"": ""https://doi.org/10.1353/ajm.2021.0020"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1353%2Fajm.2021.0020""}}]","[""https://api.crossref.org/works/10.1353%2Fajm.2021.0020""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W3168070877"", ""doi"": ""10.1353/ajm.2021.0020"", ""pmid"": null, ""title"": ""Chern classes and Gromov-Witten theory of projective bundles"", ""publication_year"": 2021, ""publication_date"": ""2021-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5037318039"", ""name"": ""Honglu Fan"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S114406506"", ""display_name"": ""American Journal of Mathematics"", ""issn_l"": ""0002-9327"", ""type"": ""journal""}, ""biblio"": {""volume"": ""143"", ""issue"": ""3"", ""first_page"": ""811"", ""last_page"": ""832""}, ""cited_by_count"": 7, ""fwci"": 1.8309, ""referenced_works_count"": 13, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://muse.jhu.edu/article/794837/pdf""}, ""best_oa_pdf_url"": ""https://muse.jhu.edu/article/794837/pdf"", ""primary_topic"": {""id"": ""T10061"", ""display_name"": ""Algebraic Geometry and Number Theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Gromov-Witten invariants"", ""projective bundles"", ""Chern classes"", ""enumerative geometry"", ""moduli space of curves""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1353/ajm.2021.0020"", ""referenced_works"": [""W1506872116"", ""W1776945898"", ""W1910723461"", ""W1987408216"", ""W2054875997"", ""W2063405087"", ""W2121326231"", ""W2136817914"", ""W2468963720"", ""W2616070961"", ""W2950179003"", ""W2950264121"", ""W6600566992""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R017,065,Virasoro-Constraints-under-Projectivization-October-5-2026,Teleman,minor_difference,minor_difference,minor_difference,direct_records,"Same Teleman work and DOI. Citation uses 2012; Crossref published date is 2011-09-24, compatible with online-first versus print dating. Retrieved schema does not expose both date types.","\bibitem{Teleman}
C.~Teleman,
\emph{The structure of 2D semi-simple field theories},
Invent. Math. \textbf{188} (2012), no.~3, 525--588.
\href{https://doi.org/10.1007/s00222-011-0352-5}{doi:10.1007/s00222-011-0352-5}.",124,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/references.tex#L124,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s00222-011-0352-5"", ""title"": [""The structure of 2D semi-simple field theories""], ""authors"": [{""given"": ""Constantin"", ""family"": ""Teleman"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2011, 9, 24]]}, ""container_title"": [""Inventiones mathematicae""], ""url"": ""https://doi.org/10.1007/s00222-011-0352-5"", ""license"": [{""start"": {""date-parts"": [[2011, 9, 24]], ""date-time"": ""2011-09-24T00:00:00Z"", ""timestamp"": 1316822400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs00222-011-0352-5""}}]","[""https://api.crossref.org/works/10.1007%2Fs00222-011-0352-5""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2123872555"", ""doi"": ""10.1007/s00222-011-0352-5"", ""pmid"": null, ""title"": ""The structure of 2D semi-simple field theories"", ""publication_year"": 2011, ""publication_date"": ""2011-09-23"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5085784472"", ""name"": ""Constantin Teleman"", ""orcid"": ""https://orcid.org/0000-0001-6054-7962"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of California System"", ""University of California, Berkeley""]}], ""source"": {""source_id"": ""S98347115"", ""display_name"": ""Inventiones mathematicae"", ""issn_l"": ""0020-9910"", ""type"": ""journal""}, ""biblio"": {""volume"": ""188"", ""issue"": ""3"", ""first_page"": ""525"", ""last_page"": ""588""}, ""cited_by_count"": 219, ""fwci"": 12.573, ""referenced_works_count"": 38, ""open_access"": {""is_oa"": true, ""oa_status"": ""green"", ""oa_url"": ""https://arxiv.org/pdf/0712.0160""}, ""best_oa_pdf_url"": ""https://arxiv.org/pdf/0712.0160"", ""primary_topic"": {""id"": ""T10061"", ""display_name"": ""Algebraic Geometry and Number Theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""cohomological field theory"", ""Gromov-Witten invariants"", ""moduli space of curves""], ""abstract"": null, ""referenced_works"": [""W37769792"", ""W1525873992"", ""W1720786389"", ""W1959074930"", ""W1976363546"", ""W2002855434"", ""W2003561784"", ""W2022161026"", ""W2028752706"", ""W2042210864"", ""W2047753277"", ""W2058559695"", ""W2066050702"", ""W2081930158"", ""W2094152079"", ""W2097870565"", ""W2100363938"", ""W2115279314"", ""W2121326231"", ""W2123074243"", ""W2127469979"", ""W2130943078"", ""W2133309910"", ""W2139434720"", ""W2145264931"", ""W2150829394"", ""W2158823467"", ""W2164252329"", ""W2197794610"", ""W2227788553"", ""W2501718805"", ""W2949792811"", ""W2950938847"", ""W2951199322"", ""W2963265562"", ""W2965282030"", ""W3104092446"", ""W6737767207""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 5}, {""year"": 2025, ""cited_by_count"": 13}, {""year"": 2024, ""cited_by_count"": 14}, {""year"": 2023, ""cited_by_count"": 10}, {""year"": 2022, ""cited_by_count"": 11}, {""year"": 2021, ""cited_by_count"": 18}, {""year"": 2020, ""cited_by_count"": 11}, {""year"": 2019, ""cited_by_count"": 20}, {""year"": 2018, ""cited_by_count"": 23}, {""year"": 2017, ""cited_by_count"": 15}, {""year"": 2016, ""cited_by_count"": 15}, {""year"": 2015, ""cited_by_count"": 16}, {""year"": 2014, ""cited_by_count"": 13}, {""year"": 2013, ""cited_by_count"": 12}, {""year"": 2012, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R018,065,Virasoro-Constraints-under-Projectivization-October-5-2026,QRR,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{QRR}
T.~Coates and A.~Givental,
\emph{Quantum Riemann--Roch, Lefschetz and Serre},
Ann. of Math. (2) \textbf{165} (2007), no.~1, 15--53.
\href{https://doi.org/10.4007/annals.2007.165.15}{doi:10.4007/annals.2007.165.15}.",9,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/references.tex#L9,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2007.165.15"", ""title"": [""Quantum Riemann–Roch, Lefschetz and Serre""], ""authors"": [{""given"": ""Thomas"", ""family"": ""Coates"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Alexander"", ""family"": ""Givental"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2007, 1, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2007.165.15"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2007.165.15""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2007.165.15""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2121326231"", ""doi"": ""10.4007/annals.2007.165.15"", ""pmid"": null, ""title"": ""Quantum Riemann–Roch, Lefschetz and Serre"", ""publication_year"": 2007, ""publication_date"": ""2007-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5041367538"", ""name"": ""Thomas Coates"", ""orcid"": ""https://orcid.org/0000-0003-0779-9735"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Harvard University""]}, {""author_id"": ""A5041646126"", ""name"": ""Alexander B. Givental"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of California, Berkeley""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""165"", ""issue"": ""1"", ""first_page"": ""15"", ""last_page"": ""53""}, ""cited_by_count"": 255, ""fwci"": 18.308, ""referenced_works_count"": 45, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p02.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p02.pdf"", ""primary_topic"": {""id"": ""T10061"", ""display_name"": ""Algebraic Geometry and Number Theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""holomorphic vector bundles"", ""stable maps"", ""virtual fundamental class"", ""Grothendieck-Riemann-Roch theorem"", ""complete intersections"", ""supermanifolds"", ""genus zero invariants""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2007.165.15"", ""referenced_works"": [""W1514220978"", ""W1561332677"", ""W1602673883"", ""W1761439427"", ""W1839620406"", ""W1959074930"", ""W2007457483"", ""W2014053041"", ""W2014361573"", ""W2015077931"", ""W2023025917"", ""W2035601195"", ""W2042210864"", ""W2044687071"", ""W2051116379"", ""W2054875997"", ""W2059343253"", ""W2065761068"", ""W2084882460"", ""W2089521132"", ""W2094017794"", ""W2097850098"", ""W2116968704"", ""W2129221945"", ""W2135823380"", ""W2136480649"", ""W2139434720"", ""W2145264931"", ""W2146168872"", ""W2949130017"", ""W2949440181"", ""W2964020539"", ""W2995449042"", ""W3100979339"", ""W3101225833"", ""W3101376047"", ""W3102484428"", ""W3105736159"", ""W4251255418"", ""W4294189460"", ""W4389281511"", ""W6600185353"", ""W6680773454"", ""W6766281440"", ""W6850119119""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 6}, {""year"": 2024, ""cited_by_count"": 7}, {""year"": 2023, ""cited_by_count"": 7}, {""year"": 2022, ""cited_by_count"": 12}, {""year"": 2021, ""cited_by_count"": 20}, {""year"": 2020, ""cited_by_count"": 8}, {""year"": 2019, ""cited_by_count"": 23}, {""year"": 2018, ""cited_by_count"": 26}, {""year"": 2017, ""cited_by_count"": 14}, {""year"": 2016, ""cited_by_count"": 15}, {""year"": 2015, ""cited_by_count"": 19}, {""year"": 2014, ""cited_by_count"": 15}, {""year"": 2013, ""cited_by_count"": 14}, {""year"": 2012, ""cited_by_count"": 9}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R019,065,Virasoro-Constraints-under-Projectivization-October-5-2026,GetzlerVirasoro,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{GetzlerVirasoro}
E.~Getzler,
\emph{The Virasoro conjecture for Gromov--Witten invariants},
in \emph{Algebraic Geometry: Hirzebruch 70} (Warsaw, 1998),
Contemp. Math., vol.~241, Amer. Math. Soc., Providence, RI, 1999,
147--176.
\href{https://arxiv.org/abs/math/9812026}{arXiv:math/9812026}.",39,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/references.tex#L39,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""math/9812026"", ""version"": 4, ""id_versioned"": ""math/9812026v4"", ""title"": ""The Virasoro conjecture for Gromov-Witten invariants"", ""abstract"": ""The Virasoro conjecture is a conjectured sequence of relations among the descendent Gromov-Witten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and Calabi-Yau threefolds (S. Katz). We review the statement of the conjecture and its proof in genus 0, following Eguchi, Hori and Xiong. This version incorporates many changes and improvements compared with the first, and a small number of misprints from the second. It will appear in the AMS volume \""Hirzebruch 70\"", edited by P. Pragacz et al."", ""authors"": [""Ezra Getzler""], ""published"": ""1998-12-03T19:57:13Z"", ""updated"": ""1999-04-06T20:31:42Z"", ""primary_category"": ""math.AG"", ""categories"": [""math.AG"", ""hep-th""], ""doi"": null, ""journal_ref"": null, ""comment"": ""amslatex 1.2, 30 pages"", ""abs_url"": ""http://arxiv.org/abs/math/9812026v4"", ""pdf_url"": ""https://arxiv.org/pdf/math/9812026v4""}}]","[""http://arxiv.org/abs/math/9812026v4""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R020,065,Virasoro-Constraints-under-Projectivization-October-5-2026,IritaniKoto,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{IritaniKoto}
H.~Iritani and Y.~Koto,
\emph{Quantum cohomology of projective bundles},
preprint, 2023, revised January 31, 2026.
\href{https://arxiv.org/abs/2307.03696v4}{arXiv:2307.03696v4}.",86,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/references.tex#L86,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2307.03696"", ""version"": 4, ""id_versioned"": ""2307.03696v4"", ""title"": ""Quantum cohomology of projective bundles"", ""abstract"": ""We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of the quantum cohomology D-modules of the base space B. This has applications to the semisimplicity of big quantum cohomology."", ""authors"": [""Hiroshi Iritani"", ""Yuki Koto""], ""published"": ""2023-07-07T16:13:20Z"", ""updated"": ""2026-01-31T00:49:40Z"", ""primary_category"": ""math.AG"", ""categories"": [""math.AG"", ""math.SG""], ""doi"": null, ""journal_ref"": null, ""comment"": ""40 pages, 2 figures; v2: Figure 2 added, v3: Introducion improved, v4: proof of Theorem 1.1 streamlined, log q term in the change of coordinates for Theorem 1.7 removed, error in Theorem 5.1(5) corrected, a reconstruction algorithm added in Section 5.8"", ""abs_url"": ""http://arxiv.org/abs/2307.03696v4"", ""pdf_url"": ""https://arxiv.org/pdf/2307.03696v4""}}]","[""http://arxiv.org/abs/2307.03696v4""]",[],"[{""sample_id"": ""R020"", ""source_url"": ""https://arxiv.org/pdf/2307.03696v4"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""5139f8e0c9d46f8ccb8cb415396a0fb1fb357719b7dcfbca46234a9735b57624"", ""locator"": ""PDF pages 2 and 4; Theorem 1.1, Remarks 1.2/1.4/1.9, Corollary 1.8"", ""source_version"": ""arXiv:2307.03696v4"", ""paraphrased_observations"": ""Theorem 1.1 covers a rank at least two vector bundle over a smooth projective base with globally generated dual. Remark 1.2 removes that restriction by a line-bundle twist preserving the projectivization. Remark 1.4 concerns genus-zero invariants. Corollary 1.8 gives generic semisimplicity for the projective bundle if and only if for its base. Remark 1.9 specifies formal semisimplicity and does not establish convergence. These inspected passages do not independently establish the nearby downstream Virasoro consequence."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R021,082,Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,OberlinSeegerTaoThieleWright2012,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{OberlinSeegerTaoThieleWright2012,
  author = {Oberlin, Richard and Seeger, Andreas and Tao, Terence and Thiele, Christoph and Wright, James},
  title = {A variation norm {Carleson} theorem},
  journal = {Journal of the European Mathematical Society},
  volume = {14},
  number = {2},
  pages = {421--464},
  year = {2012},
  doi = {10.4171/JEMS/307},
  url = {https://arxiv.org/abs/0910.1555}
}",150,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/references.bib#L150,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4171/jems/307"", ""title"": [""A variation norm Carleson theorem""], ""authors"": [{""given"": ""Richard"", ""family"": ""Oberlin"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Louisiana State University, Baton Rouge, USA""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Andreas"", ""family"": ""Seeger"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""University of Wisconsin, Madison, USA""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""ORCID"": ""https://orcid.org/0000-0002-0140-7641"", ""authenticated-orcid"": false, ""given"": ""Terence"", ""family"": ""Tao"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""University of California Los Angeles, United States""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Christoph"", ""family"": ""Thiele"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Universität Bonn, Germany""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""James"", ""family"": ""Wright"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""University of Edinburgh, UK""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""European Mathematical Society - EMS - Publishing House GmbH"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2012, 2, 4]]}, ""container_title"": [""Journal of the European Mathematical Society""], ""url"": ""https://doi.org/10.4171/jems/307"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4171%2Fjems%2F307""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""0910.1555"", ""version"": 2, ""id_versioned"": ""0910.1555v2"", ""title"": ""A variation norm Carleson theorem"", ""abstract"": ""We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\\max\\{r',2\\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory."", ""authors"": [""Richard Oberlin"", ""Andreas Seeger"", ""Terence Tao"", ""Christoph Thiele"", ""James Wright""], ""published"": ""2009-10-08T17:11:17Z"", ""updated"": ""2010-08-25T15:03:37Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA""], ""doi"": null, ""journal_ref"": null, ""comment"": ""41 pages"", ""abs_url"": ""http://arxiv.org/abs/0910.1555v2"", ""pdf_url"": ""https://arxiv.org/pdf/0910.1555v2""}}]","[""https://api.crossref.org/works/10.4171%2Fjems%2F307"", ""http://arxiv.org/abs/0910.1555v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1984895635"", ""doi"": ""10.4171/jems/307"", ""pmid"": null, ""title"": ""A variation norm Carleson theorem"", ""publication_year"": 2012, ""publication_date"": ""2012-02-04"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5085375797"", ""name"": ""Richard Oberlin"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Louisiana State University"", ""University of Wisconsin–Madison"", ""University of California, Los Angeles""]}, {""author_id"": ""A5001159999"", ""name"": ""Andreas Seeger"", ""orcid"": ""https://orcid.org/0000-0001-6234-8852"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Louisiana State University"", ""University of Wisconsin–Madison"", ""University of California, Los Angeles""]}, {""author_id"": ""A5066728260"", ""name"": ""Terence Tao"", ""orcid"": ""https://orcid.org/0000-0002-0140-7641"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Louisiana State University"", ""University of Wisconsin–Madison"", ""University of California, Los Angeles""]}, {""author_id"": ""A5011998711"", ""name"": ""Christoph M. Thiele"", ""orcid"": ""https://orcid.org/0000-0001-6158-7667"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Louisiana State University"", ""University of Bonn"", ""University of Wisconsin–Madison"", ""University of California, Los Angeles""]}, {""author_id"": ""A5017084784"", ""name"": ""James Wright"", ""orcid"": ""https://orcid.org/0009-0008-3205-8916"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Louisiana State University"", ""University of Wisconsin–Madison"", ""University of California, Los Angeles"", ""University of Edinburgh""]}], ""source"": {""source_id"": ""S88235189"", ""display_name"": ""Journal of the European Mathematical Society"", ""issn_l"": ""1435-9855"", ""type"": ""journal""}, ""biblio"": {""volume"": ""14"", ""issue"": ""2"", ""first_page"": ""421"", ""last_page"": ""464""}, ""cited_by_count"": 128, ""fwci"": 16.8525, ""referenced_works_count"": 31, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://ems.press/content/serial-article-files/4274""}, ""best_oa_pdf_url"": ""https://ems.press/content/serial-article-files/4274"", ""primary_topic"": {""id"": ""T11049"", ""display_name"": ""Advanced Harmonic Analysis Research"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Fourier series"", ""Fourier integrals"", ""Lp estimates"", ""transference"", ""nonlinear Fourier transform"", ""ergodic theory""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4171/jems/307"", ""referenced_works"": [""W592834996"", ""W777981297"", ""W1483973558"", ""W1559907478"", ""W1589506572"", ""W1871058307"", ""W1970883504"", ""W1979283834"", ""W1979685827"", ""W1984734725"", ""W1986689940"", ""W1988619203"", ""W2004095444"", ""W2018784102"", ""W2019659573"", ""W2020943029"", ""W2025584671"", ""W2027376140"", ""W2053652533"", ""W2054777142"", ""W2059363534"", ""W2070059687"", ""W2078012491"", ""W2086510139"", ""W2087883984"", ""W2094006795"", ""W2109976819"", ""W2186959210"", ""W2324292457"", ""W2333624702"", ""W2607438301""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 3}, {""year"": 2025, ""cited_by_count"": 8}, {""year"": 2024, ""cited_by_count"": 6}, {""year"": 2023, ""cited_by_count"": 5}, {""year"": 2022, ""cited_by_count"": 5}, {""year"": 2021, ""cited_by_count"": 11}, {""year"": 2020, ""cited_by_count"": 7}, {""year"": 2019, ""cited_by_count"": 9}, {""year"": 2018, ""cited_by_count"": 7}, {""year"": 2017, ""cited_by_count"": 16}, {""year"": 2016, ""cited_by_count"": 14}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 10}, {""year"": 2013, ""cited_by_count"": 11}, {""year"": 2012, ""cited_by_count"": 8}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R022,082,Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,Lepingle1976,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Lepingle1976,
  author = {L\'epingle, Dominique},
  title = {La variation d'ordre {$p$} des semi-martingales},
  journal = {Zeitschrift f\""ur Wahrscheinlichkeitstheorie und Verwandte Gebiete},
  volume = {36},
  number = {4},
  pages = {295--316},
  year = {1976},
  doi = {10.1007/BF00532696}
}",127,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/references.bib#L127,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf00532696"", ""title"": [""La variation d'ordre p des semi-martingales""], ""authors"": [{""given"": ""D."", ""family"": ""Lepingle"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1976]]}, ""container_title"": [""Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete""], ""url"": ""https://doi.org/10.1007/bf00532696"", ""license"": [{""start"": {""date-parts"": [[1976, 1, 1]], ""date-time"": ""1976-01-01T00:00:00Z"", ""timestamp"": 189302400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf00532696""}}]","[""https://api.crossref.org/works/10.1007%2Fbf00532696""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2078012491"", ""doi"": ""10.1007/bf00532696"", ""pmid"": null, ""title"": ""La variation d'ordre p des semi-martingales"", ""publication_year"": 1976, ""publication_date"": ""1976-01-01"", ""type"": ""article"", ""language"": ""de"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5081523351"", ""name"": ""Dominique Lépingle"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Université d'Orléans""]}], ""source"": {""source_id"": ""S105693234"", ""display_name"": ""Probability Theory and Related Fields"", ""issn_l"": ""0178-8051"", ""type"": ""journal""}, ""biblio"": {""volume"": ""36"", ""issue"": ""4"", ""first_page"": ""295"", ""last_page"": ""316""}, ""cited_by_count"": 236, ""fwci"": 2.0285, ""referenced_works_count"": 17, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://link.springer.com/content/pdf/10.1007/BF00532696.pdf""}, ""best_oa_pdf_url"": ""https://link.springer.com/content/pdf/10.1007/BF00532696.pdf"", ""primary_topic"": {""id"": ""T10067"", ""display_name"": ""Stochastic processes and financial applications"", ""field"": ""Economics, Econometrics and Finance"", ""subfield"": ""Finance"", ""domain"": ""Social Sciences""}, ""keywords"": [""semimartingales"", ""p-variation"", ""convergence in probability"", ""almost sure convergence"", ""stationary independent increments""], ""abstract"": null, ""referenced_works"": [""W974865560"", ""W1561844546"", ""W1563505767"", ""W1828451219"", ""W1964702073"", ""W1966085307"", ""W2004294736"", ""W2023426016"", ""W2035984893"", ""W2040169677"", ""W2044110302"", ""W2047230870"", ""W2092039627"", ""W2130763187"", ""W2164944564"", ""W2177292643"", ""W2185883865""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 6}, {""year"": 2025, ""cited_by_count"": 14}, {""year"": 2024, ""cited_by_count"": 10}, {""year"": 2023, ""cited_by_count"": 17}, {""year"": 2022, ""cited_by_count"": 13}, {""year"": 2021, ""cited_by_count"": 13}, {""year"": 2020, ""cited_by_count"": 12}, {""year"": 2019, ""cited_by_count"": 12}, {""year"": 2018, ""cited_by_count"": 8}, {""year"": 2017, ""cited_by_count"": 9}, {""year"": 2016, ""cited_by_count"": 11}, {""year"": 2015, ""cited_by_count"": 6}, {""year"": 2014, ""cited_by_count"": 7}, {""year"": 2013, ""cited_by_count"": 9}, {""year"": 2012, ""cited_by_count"": 11}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R023,082,Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,GrafakosEtAl2017,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"@misc{GrafakosEtAl2017,
  author = {Grafakos, Loukas and Oliveira e Silva, Diogo and Pramanik, Malabika and Seeger, Andreas and Stovall, Betsy},
  title = {Some problems in harmonic analysis},
  year = {2017},
  howpublished = {arXiv:1701.06637},
  url = {https://arxiv.org/abs/1701.06637},
  note = {Section 9, contributed by Christoph Thiele, Problem 13}
}",1,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/references.bib#L1,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1701.06637"", ""version"": 1, ""id_versioned"": ""1701.06637v1"", ""title"": ""Some Problems in Harmonic Analysis"", ""abstract"": ""In May 2016, we organized a conference in harmonic analysis in honor of Professor Michael Christ, on the campus of the University of Wisconsin in Madison. We are happy to present sixteen open problems, almost all of which were contributed by participants of a problem session held in the afternoon of May 19, 2016."", ""authors"": [""Loukas Grafakos"", ""Diogo Oliveira e Silva"", ""Malabika Pramanik"", ""Andreas Seeger"", ""Betsy Stovall""], ""published"": ""2017-01-23T21:19:09Z"", ""updated"": ""2017-01-23T21:19:09Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA"", ""math.AP""], ""doi"": null, ""journal_ref"": null, ""comment"": ""19 pages"", ""abs_url"": ""http://arxiv.org/abs/1701.06637v1"", ""pdf_url"": ""https://arxiv.org/pdf/1701.06637v1""}}]","[""http://arxiv.org/abs/1701.06637v1""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R024,082,Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,LinSlavikova2026,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"@misc{LinSlavikova2026,
  author = {Lin, Fred Yu-Hsiang and Slav\'ikov\'a, Lenka},
  title = {Rough averages of triangular {Hilbert} transforms},
  year = {2026},
  howpublished = {arXiv:2607.20206},
  url = {https://arxiv.org/abs/2607.20206},
  note = {July 22, 2026}
}",227,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/references.bib#L227,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2607.20206"", ""version"": 1, ""id_versioned"": ""2607.20206v1"", ""title"": ""Rough averages of triangular Hilbert transforms"", ""abstract"": ""We study a bilinear singular integral operator obtained by taking rough averages of certain directional variants of the triangular Hilbert transform. This operator can be interpreted as the twisted paraproduct with a rough homogeneous kernel. Under a balance condition on the $L^q$ integrability of the kernel on the unit sphere, we establish a range of $L^{p_1} \\times L^{p_2} \\rightarrow L^p$ bounds for this operator. Our results are optimal within the known boundedness range for the twisted paraproduct with a smooth kernel."", ""authors"": [""Fred Yu-Hsiang Lin"", ""Lenka Slavíková""], ""published"": ""2026-07-22T14:26:36Z"", ""updated"": ""2026-07-22T14:26:36Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA""], ""doi"": null, ""journal_ref"": null, ""comment"": null, ""abs_url"": ""http://arxiv.org/abs/2607.20206v1"", ""pdf_url"": ""https://arxiv.org/pdf/2607.20206v1""}}]","[""http://arxiv.org/abs/2607.20206v1""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R025,082,Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,ZorinKranich2017,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{ZorinKranich2017,
  author = {Zorin-Kranich, Pavel},
  title = {Cancellation for the simplex {Hilbert} transform},
  journal = {Mathematical Research Letters},
  volume = {24},
  number = {2},
  pages = {581--592},
  year = {2017},
  doi = {10.4310/MRL.2017.v24.n2.a16},
  url = {https://arxiv.org/abs/1507.02436}
}",45,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/references.bib#L45,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4310/mrl.2017.v24.n2.a16"", ""title"": [""Cancellation for the simplex Hilbert transform""], ""authors"": [{""given"": ""Pavel"", ""family"": ""Zorin-Kranich"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Mathematisches Institut, Universität Bonn, Germany""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""International Press of Boston"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2017]]}, ""container_title"": [""Mathematical Research Letters""], ""url"": ""https://doi.org/10.4310/mrl.2017.v24.n2.a16"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4310%2Fmrl.2017.v24.n2.a16""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1507.02436"", ""version"": 1, ""id_versioned"": ""1507.02436v1"", ""title"": ""Cancellation for the simplex Hilbert transform"", ""abstract"": ""We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the multilinear Hilbert transform. Our main tool is the Hilbert space regularity lemma due to Gowers, which enables a very short proof."", ""authors"": [""Pavel Zorin-Kranich""], ""published"": ""2015-07-09T09:53:17Z"", ""updated"": ""2015-07-09T09:53:17Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA""], ""doi"": ""10.4310/MRL.2017.v24.n2.a16"", ""journal_ref"": ""Math. Res. Lett. 24.2 (2017), pp. 581-592"", ""comment"": ""8 pages"", ""abs_url"": ""http://arxiv.org/abs/1507.02436v1"", ""pdf_url"": ""https://arxiv.org/pdf/1507.02436v1""}}]","[""https://api.crossref.org/works/10.4310%2Fmrl.2017.v24.n2.a16"", ""http://arxiv.org/abs/1507.02436v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1676190354"", ""doi"": ""10.4310/mrl.2017.v24.n2.a16"", ""pmid"": null, ""title"": ""Cancellation for the simplex Hilbert transform"", ""publication_year"": 2017, ""publication_date"": ""2017-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5029215074"", ""name"": ""Pavel Zorin‐Kranich"", ""orcid"": ""https://orcid.org/0000-0002-2153-7689"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Bonn""]}], ""source"": {""source_id"": ""S126920787"", ""display_name"": ""Mathematical Research Letters"", ""issn_l"": ""1073-2780"", ""type"": ""journal""}, ""biblio"": {""volume"": ""24"", ""issue"": ""2"", ""first_page"": ""581"", ""last_page"": ""592""}, ""cited_by_count"": 13, ""fwci"": 2.7795, ""referenced_works_count"": 8, ""open_access"": {""is_oa"": true, ""oa_status"": ""green"", ""oa_url"": ""https://arxiv.org/pdf/1507.02436""}, ""best_oa_pdf_url"": ""https://arxiv.org/pdf/1507.02436"", ""primary_topic"": {""id"": ""T11049"", ""display_name"": ""Advanced Harmonic Analysis Research"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""multilinear harmonic analysis""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4310/mrl.2017.v24.n2.a16"", ""referenced_works"": [""W609604325"", ""W1584610719"", ""W1718341735"", ""W2157116313"", ""W2242479782"", ""W2949149512"", ""W3102814824"", ""W3106120639""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 3}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R026,113,A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,EfronStein1981,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem[Efron and Stein(1981)]{EfronStein1981}
Bradley Efron and Charles Stein.
\newblock The jackknife estimate of variance.
\newblock \emph{The Annals of Statistics}, 9\penalty0
  (3):\penalty0 586--596, 1981.
\newblock \doi{10.1214/aos/1176345462}.",3354,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex#L3354,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1214/aos/1176345462"", ""title"": [""The Jackknife Estimate of Variance""], ""authors"": [{""given"": ""B."", ""family"": ""Efron"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""C."", ""family"": ""Stein"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Institute of Mathematical Statistics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1981, 5, 1]]}, ""container_title"": [""The Annals of Statistics""], ""url"": ""https://doi.org/10.1214/aos/1176345462"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1214%2Faos%2F1176345462""}}]","[""https://api.crossref.org/works/10.1214%2Faos%2F1176345462""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1992068214"", ""doi"": ""10.1214/aos/1176345462"", ""pmid"": null, ""title"": ""The Jackknife Estimate of Variance"", ""publication_year"": 1981, ""publication_date"": ""1981-05-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5108567924"", ""name"": ""Bradley Efron"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5102712577"", ""name"": ""C. Stein"", ""orcid"": ""https://orcid.org/0009-0008-8107-578X"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": []}], ""source"": {""source_id"": ""S119757635"", ""display_name"": ""The Annals of Statistics"", ""issn_l"": ""0090-5364"", ""type"": ""journal""}, ""biblio"": {""volume"": ""9"", ""issue"": ""3"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 1477, ""fwci"": 10.8939, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11871"", ""display_name"": ""Advanced Statistical Methods and Models"", ""field"": ""Mathematics"", ""subfield"": ""Statistics and Probability"", ""domain"": ""Physical Sciences""}, ""keywords"": [""jackknife variance estimation"", ""ANOVA decomposition"", ""independent identically distributed random variables""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1214/aos/1176345462"", ""referenced_works"": [], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 39}, {""year"": 2025, ""cited_by_count"": 46}, {""year"": 2024, ""cited_by_count"": 63}, {""year"": 2023, ""cited_by_count"": 67}, {""year"": 2022, ""cited_by_count"": 72}, {""year"": 2021, ""cited_by_count"": 115}, {""year"": 2020, ""cited_by_count"": 97}, {""year"": 2019, ""cited_by_count"": 102}, {""year"": 2018, ""cited_by_count"": 100}, {""year"": 2017, ""cited_by_count"": 72}, {""year"": 2016, ""cited_by_count"": 69}, {""year"": 2015, ""cited_by_count"": 84}, {""year"": 2014, ""cited_by_count"": 60}, {""year"": 2013, ""cited_by_count"": 53}, {""year"": 2012, ""cited_by_count"": 49}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R027,113,A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,CaiLiu2020,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem[Cai and Liu(2020)]{CaiLiu2020}
Jin-Yi Cai and Tianyu Liu.
\newblock Counting perfect matchings and the eight-vertex model.
\newblock In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors,
  \emph{47th International Colloquium on Automata, Languages, and Programming
  (ICALP 2020)}, volume 168 of \emph{Leibniz International Proceedings in
  Informatics}, pages 23:1--23:18. Schloss Dagstuhl--Leibniz-Zentrum f{\""u}r
  Informatik, 2020.
\newblock \doi{10.4230/LIPIcs.ICALP.2020.23}.
\newblock URL
  \url{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.23}.",3323,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex#L3323,"[{""provider"": ""DataCite original deposit"", ""record"": {""record"": {""doi"": ""10.4230/lipics.icalp.2020.23"", ""titles"": [{""title"": ""Counting Perfect Matchings and the Eight-Vertex Model"", ""lang"": ""en""}], ""creators"": [{""name"": ""Cai, Jin-Yi"", ""nameType"": ""Personal"", ""givenName"": ""Jin-Yi"", ""familyName"": ""Cai"", ""affiliation"": [""University of Wisconsin-Madison, Madison, WI, USA""], ""nameIdentifiers"": []}, {""name"": ""Liu, Tianyu"", ""nameType"": ""Personal"", ""givenName"": ""Tianyu"", ""familyName"": ""Liu"", ""affiliation"": [""University of Wisconsin-Madison, Madison, WI, USA""], ""nameIdentifiers"": []}], ""publisher"": ""Schloss Dagstuhl – Leibniz-Zentrum für Informatik"", ""publication_year"": 2020, ""resource_type"": {""resourceTypeGeneral"": ""ConferencePaper"", ""ris"": ""CPAPER"", ""bibtex"": ""inproceedings"", ""citeproc"": """", ""schemaOrg"": ""Article"", ""resourceType"": ""Track A: Algorithms, Complexity and Games""}, ""url"": ""https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.23"", ""rights"": [{""lang"": ""en"", ""rights"": ""Creative Commons Attribution 3.0 Unported license"", ""rightsUri"": ""https://creativecommons.org/licenses/by/3.0/legalcode"", ""rightsIdentifier"": ""cc by 3.0""}, {""rights"": ""info:eu-repo/semantics/openAccess""}], ""related_identifiers"": [{""relationType"": ""IsVersionOf"", ""relatedIdentifier"": ""arXiv:1904.10493"", ""resourceTypeGeneral"": ""Text"", ""relatedIdentifierType"": ""arXiv""}, {""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""10.4230/LIPIcs.ICALP.2020"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""978-3-95977-138-2"", ""relatedIdentifierType"": ""ISBN""}, {""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""1868-8969"", ""relatedIdentifierType"": ""ISSN""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1103/PhysRevLett.26.832"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1016/0003-4916(72)90335-1"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1016/j.tcs.2017.05.001"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1017/9781107477063"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""arXiv:1702.07938"", ""relatedIdentifierType"": ""arXiv""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1137/1.9781611975482.136"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""arXiv:1811.03126"", ""relatedIdentifierType"": ""arXiv""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1103/PhysRevB.2.723"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1145/2785964"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1016/j.ic.2008.04.003"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1007/s00453-008-9246-3"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1002/rsa.20560"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1109/SFCS.1983.35"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1016/0031-8914(61)90063-5"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.4230/LIPIcs.APPROX-RANDOM.2018.52"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""arXiv:1301.2880"", ""relatedIdentifierType"": ""arXiv""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1063/1.1665111"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1080/14786436108243366"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1137/S0097539700377025"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""Cites"", ""relatedIdentifier"": ""10.1137/070682575"", ""relatedIdentifierType"": ""DOI""}], ""version"": null}, ""source_url"": ""https://api.datacite.org/dois/10.4230%2Flipics.icalp.2020.23""}}]","[""https://api.datacite.org/dois/10.4230%2Flipics.icalp.2020.23""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R028,113,A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,Hastings1970,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem[Hastings(1970)]{Hastings1970}
W.~K. Hastings.
\newblock Monte Carlo sampling methods using Markov chains and their applications.
\newblock \emph{Biometrika}, 57\penalty0 (1):\penalty0 97--109, 1970.
\newblock \doi{10.1093/biomet/57.1.97}.",3367,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex#L3367,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1093/biomet/57.1.97"", ""title"": [""Monte Carlo sampling methods using Markov chains and their applications""], ""authors"": [{""given"": ""W. K."", ""family"": ""Hastings"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""University of Toronto""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Oxford University Press (OUP)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1970, 4, 1]]}, ""container_title"": [""Biometrika""], ""url"": ""https://doi.org/10.1093/biomet/57.1.97"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1093%2Fbiomet%2F57.1.97""}}]","[""https://api.crossref.org/works/10.1093%2Fbiomet%2F57.1.97""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2138309709"", ""doi"": ""10.1093/biomet/57.1.97"", ""pmid"": null, ""title"": ""Monte Carlo sampling methods using Markov chains and their applications"", ""publication_year"": 1970, ""publication_date"": ""1970-04-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5031680938"", ""name"": ""W. Keith Hastings"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Toronto""]}], ""source"": {""source_id"": ""S172180718"", ""display_name"": ""Biometrika"", ""issn_l"": ""0006-3444"", ""type"": ""journal""}, ""biblio"": {""volume"": ""57"", ""issue"": ""1"", ""first_page"": ""97"", ""last_page"": ""109""}, ""cited_by_count"": 15503, ""fwci"": 1.4603, ""referenced_works_count"": 16, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T12056"", ""display_name"": ""Markov Chains and Monte Carlo Methods"", ""field"": ""Mathematics"", ""subfield"": ""Statistics and Probability"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Markov chain Monte Carlo"", ""Metropolis algorithm"", ""Monte Carlo sampling"", ""error estimation"", ""random orthogonal matrices""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1093/biomet/57.1.97"", ""referenced_works"": [""W209994976"", ""W1541225004"", ""W1967046553"", ""W1974637548"", ""W1991918076"", ""W2014208555"", ""W2017395506"", ""W2034183584"", ""W2060605975"", ""W2077722317"", ""W2116414161"", ""W2147555604"", ""W2324992469"", ""W2535728692"", ""W2905049930"", ""W4248173958""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 460}, {""year"": 2025, ""cited_by_count"": 715}, {""year"": 2024, ""cited_by_count"": 730}, {""year"": 2023, ""cited_by_count"": 773}, {""year"": 2022, ""cited_by_count"": 832}, {""year"": 2021, ""cited_by_count"": 937}, {""year"": 2020, ""cited_by_count"": 919}, {""year"": 2019, ""cited_by_count"": 887}, {""year"": 2018, ""cited_by_count"": 742}, {""year"": 2017, ""cited_by_count"": 752}, {""year"": 2016, ""cited_by_count"": 752}, {""year"": 2015, ""cited_by_count"": 711}, {""year"": 2014, ""cited_by_count"": 679}, {""year"": 2013, ""cited_by_count"": 649}, {""year"": 2012, ""cited_by_count"": 662}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R029,113,A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,MetropolisEtAl1953,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem[Metropolis et~al.(1953)Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller]{MetropolisEtAl1953}
Nicholas Metropolis, Arianna~W. Rosenbluth, Marshall~N. Rosenbluth,
Augusta~H. Teller, and Edward Teller.
\newblock Equation of state calculations by fast computing machines.
\newblock \emph{The Journal of Chemical Physics}, 21\penalty0
  (6):\penalty0 1087--1092, 1953.
\newblock \doi{10.1063/1.1699114}.",3409,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex#L3409,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1063/1.1699114"", ""title"": [""Equation of State Calculations by Fast Computing Machines""], ""authors"": [{""given"": ""Nicholas"", ""family"": ""Metropolis"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Los Alamos Scientific Laboratory, Los Alamos, New Mexico""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Arianna W."", ""family"": ""Rosenbluth"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Los Alamos Scientific Laboratory, Los Alamos, New Mexico""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Marshall N."", ""family"": ""Rosenbluth"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Los Alamos Scientific Laboratory, Los Alamos, New Mexico""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Augusta H."", ""family"": ""Teller"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Los Alamos Scientific Laboratory, Los Alamos, New Mexico""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Edward"", ""family"": ""Teller"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Department of Physics, University of Chicago, Chicago, Illinois""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""AIP Publishing"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1953, 6, 1]]}, ""container_title"": [""The Journal of Chemical Physics""], ""url"": ""https://doi.org/10.1063/1.1699114"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1063%2F1.1699114""}}]","[""https://api.crossref.org/works/10.1063%2F1.1699114""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2056760934"", ""doi"": ""10.1063/1.1699114"", ""pmid"": null, ""title"": ""Equation of State Calculations by Fast Computing Machines"", ""publication_year"": 1953, ""publication_date"": ""1953-06-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5007322687"", ""name"": ""N. Metropolis"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Los Alamos National Laboratory""]}, {""author_id"": ""A5009563598"", ""name"": ""Arianna W. Rosenbluth"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Los Alamos National Laboratory""]}, {""author_id"": ""A5110000776"", ""name"": ""M. N. Rosenbluth"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Los Alamos National Laboratory""]}, {""author_id"": ""A5080021425"", ""name"": ""Augusta H. Teller"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Los Alamos National Laboratory""]}, {""author_id"": ""A5034295195"", ""name"": ""Edward Teller"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Chicago""]}], ""source"": {""source_id"": ""S77047749"", ""display_name"": ""The Journal of Chemical Physics"", ""issn_l"": ""0021-9606"", ""type"": ""journal""}, ""biblio"": {""volume"": ""21"", ""issue"": ""6"", ""first_page"": ""1087"", ""last_page"": ""1092""}, ""cited_by_count"": 37615, ""fwci"": 5.2318, ""referenced_works_count"": 1, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10402"", ""display_name"": ""Phase Equilibria and Thermodynamics"", ""field"": ""Engineering"", ""subfield"": ""Biomedical Engineering"", ""domain"": ""Physical Sciences""}, ""keywords"": [""equation of state"", ""interacting molecules"", ""Monte Carlo integration"", ""configuration space"", ""hard-sphere model"", ""two-dimensional fluids""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1063/1.1699114"", ""referenced_works"": [""W2052864234""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 829}, {""year"": 2025, ""cited_by_count"": 1207}, {""year"": 2024, ""cited_by_count"": 1310}, {""year"": 2023, ""cited_by_count"": 1322}, {""year"": 2022, ""cited_by_count"": 1379}, {""year"": 2021, ""cited_by_count"": 1441}, {""year"": 2020, ""cited_by_count"": 1454}, {""year"": 2019, ""cited_by_count"": 1377}, {""year"": 2018, ""cited_by_count"": 1311}, {""year"": 2017, ""cited_by_count"": 1324}, {""year"": 2016, ""cited_by_count"": 1313}, {""year"": 2015, ""cited_by_count"": 1352}, {""year"": 2014, ""cited_by_count"": 1349}, {""year"": 2013, ""cited_by_count"": 1353}, {""year"": 2012, ""cited_by_count"": 1417}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R030,113,A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,Kasteleyn1963,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem[Kasteleyn(1963)]{Kasteleyn1963}
P.~W. Kasteleyn.
\newblock Dimer statistics and phase transitions.
\newblock \emph{Journal of Mathematical Physics}, 4\penalty0 (2):\penalty0
  287--293, 1963.
\newblock \doi{10.1063/1.1703953}.",3402,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex#L3402,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1063/1.1703953"", ""title"": [""Dimer Statistics and Phase Transitions""], ""authors"": [{""given"": ""P. W."", ""family"": ""Kasteleyn"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Koninklijke/Shell-Laboratorium, Amsterdam, Netherlands, (Shell Internationale Research Maatschappij N.V.)""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""AIP Publishing"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1963, 2, 1]]}, ""container_title"": [""Journal of Mathematical Physics""], ""url"": ""https://doi.org/10.1063/1.1703953"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1063%2F1.1703953""}}]","[""https://api.crossref.org/works/10.1063%2F1.1703953""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1982250981"", ""doi"": ""10.1063/1.1703953"", ""pmid"": null, ""title"": ""Dimer Statistics and Phase Transitions"", ""publication_year"": 1963, ""publication_date"": ""1963-02-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5091285972"", ""name"": ""Piet Kasteleyn"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Shell (Netherlands)""]}], ""source"": {""source_id"": ""S158241587"", ""display_name"": ""Journal of Mathematical Physics"", ""issn_l"": ""0022-2488"", ""type"": ""journal""}, ""biblio"": {""volume"": ""4"", ""issue"": ""2"", ""first_page"": ""287"", ""last_page"": ""293""}, ""cited_by_count"": 968, ""fwci"": 3.1709, ""referenced_works_count"": 17, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10591"", ""display_name"": ""Theoretical and Computational Physics"", ""field"": ""Physics and Astronomy"", ""subfield"": ""Condensed Matter Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""phase transitions"", ""Ising model"", ""Pfaffian"", ""configurational partition function"", ""planar lattices"", ""quadratic lattices"", ""hexagonal lattice"", ""Onsager solution"", ""lattice graphs"", ""combinatorial enumeration""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1063/1.1703953"", ""referenced_works"": [""W1484543240"", ""W1653035750"", ""W1980106694"", ""W1985061810"", ""W1990811265"", ""W2010957693"", ""W2026549070"", ""W2029747337"", ""W2034612530"", ""W2056525688"", ""W2065813152"", ""W2074670256"", ""W2084472170"", ""W2087054195"", ""W2094655176"", ""W2097568349"", ""W4236672535""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 17}, {""year"": 2025, ""cited_by_count"": 31}, {""year"": 2024, ""cited_by_count"": 24}, {""year"": 2023, ""cited_by_count"": 34}, {""year"": 2022, ""cited_by_count"": 26}, {""year"": 2021, ""cited_by_count"": 37}, {""year"": 2020, ""cited_by_count"": 30}, {""year"": 2019, ""cited_by_count"": 37}, {""year"": 2018, ""cited_by_count"": 35}, {""year"": 2017, ""cited_by_count"": 34}, {""year"": 2016, ""cited_by_count"": 29}, {""year"": 2015, ""cited_by_count"": 28}, {""year"": 2014, ""cited_by_count"": 18}, {""year"": 2013, ""cited_by_count"": 25}, {""year"": 2012, ""cited_by_count"": 20}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R031,115,Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026,CDGJM2006,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{CDGJM2006,
  author = {Cryan, Mary and Dyer, Martin and Goldberg, Leslie Ann
            and Jerrum, Mark and Martin, Russell},
  title = {Rapidly Mixing {Markov} Chains for Sampling Contingency Tables
           with a Constant Number of Rows},
  journal = {SIAM Journal on Computing},
  volume = {36},
  number = {1},
  pages = {247--278},
  year = {2006},
  doi = {10.1137/S0097539703434243},
  note = {\href{https://doi.org/10.1137/S0097539703434243}{doi:10.1137/S0097539703434243}}
}",173,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/refs/sources.bib#L173,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1137/s0097539703434243"", ""title"": [""Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows""], ""authors"": [{""given"": ""Mary"", ""family"": ""Cryan"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Martin"", ""family"": ""Dyer"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Leslie Ann"", ""family"": ""Goldberg"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Mark"", ""family"": ""Jerrum"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Russell"", ""family"": ""Martin"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Society for Industrial & Applied Mathematics (SIAM)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2006, 1]]}, ""container_title"": [""SIAM Journal on Computing""], ""url"": ""https://doi.org/10.1137/s0097539703434243"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1137%2Fs0097539703434243""}}]","[""https://api.crossref.org/works/10.1137%2Fs0097539703434243""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2162508244"", ""doi"": ""10.1137/s0097539703434243"", ""pmid"": null, ""title"": ""Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows"", ""publication_year"": 2006, ""publication_date"": ""2006-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5057611636"", ""name"": ""Mary Cryan"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5019297501"", ""name"": ""Martin Dyer"", ""orcid"": ""https://orcid.org/0000-0002-2018-0374"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5029293569"", ""name"": ""Leslie Ann Goldberg"", ""orcid"": ""https://orcid.org/0000-0003-1879-6089"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5090122841"", ""name"": ""Mark Jerrum"", ""orcid"": ""https://orcid.org/0000-0003-0863-7279"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5106500064"", ""name"": ""Russell Martin"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": []}], ""source"": {""source_id"": ""S153560523"", ""display_name"": ""SIAM Journal on Computing"", ""issn_l"": ""0097-5397"", ""type"": ""journal""}, ""biblio"": {""volume"": ""36"", ""issue"": ""1"", ""first_page"": ""247"", ""last_page"": ""278""}, ""cited_by_count"": 33, ""fwci"": 1.5083, ""referenced_works_count"": 24, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T12056"", ""display_name"": ""Markov Chains and Monte Carlo Methods"", ""field"": ""Mathematics"", ""subfield"": ""Statistics and Probability"", ""domain"": ""Physical Sciences""}, ""keywords"": [""contingency tables"", ""FPRAS"", ""comparison method"", ""convex sets"", ""multidimensional knapsack problem"", ""random walks""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1137/s0097539703434243"", ""referenced_works"": [""W1527978610"", ""W1759989895"", ""W1970853627"", ""W1976418263"", ""W1976445742"", ""W2000597435"", ""W2003554015"", ""W2005107874"", ""W2014108043"", ""W2033281046"", ""W2039689894"", ""W2043694962"", ""W2062142815"", ""W2079008867"", ""W2088515005"", ""W2092636674"", ""W2129740858"", ""W2140808818"", ""W2330405256"", ""W2611804663"", ""W3111890340"", ""W4230573295"", ""W4253228418"", ""W4302328504""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 3}, {""year"": 2017, ""cited_by_count"": 5}, {""year"": 2016, ""cited_by_count"": 2}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R032,115,Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026,AGW2021,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"@misc{AGW2021,
  author        = {Arman, Andrii and Gao, Pu and Wormald, Nicholas},
  title         = {Linear-time uniform generation of random sparse
                   contingency tables with specified marginals},
  howpublished  = {\href{https://arxiv.org/abs/2104.09413v2}{arXiv:2104.09413v2}},
  year          = {2021},
  eprint        = {2104.09413},
  archivePrefix = {arXiv},
  primaryClass  = {math.CO},
  note          = {Version 2, 15 June 2021},
  url           = {https://arxiv.org/abs/2104.09413v2}
}",1,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/refs/sources.bib#L1,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2104.09413"", ""version"": 2, ""id_versioned"": ""2104.09413v2"", ""title"": ""Linear-time uniform generation of random sparse contingency tables with specified marginals"", ""abstract"": ""We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When $Δ^4< M/5$, where $Δ$ is the maximum of the row and column sums and $M$ is the sum of all entries of the matrix, our algorithm runs in time linear in $M$ in expectation. Most previously published algorithms for this problem are approximate samplers based on Markov chain Monte Carlo, whose provable bounds on the mixing time are typically polynomials with rather large degrees."", ""authors"": [""Andrii Arman"", ""Pu Gao"", ""Nicholas Wormald""], ""published"": ""2021-04-19T16:03:24Z"", ""updated"": ""2021-06-15T20:02:34Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO"", ""cs.DS"", ""math.PR""], ""doi"": null, ""journal_ref"": null, ""comment"": ""27 pages, 1 figure"", ""abs_url"": ""http://arxiv.org/abs/2104.09413v2"", ""pdf_url"": ""https://arxiv.org/pdf/2104.09413v2""}}]","[""http://arxiv.org/abs/2104.09413v2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R033,115,Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026,DZ2016,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"@misc{DZ2016,
  author = {DeSalvo, Stephen and Zhao, James Y.},
  title = {Random Sampling of Contingency Tables via Probabilistic Divide-and-Conquer},
  year = {2016},
  howpublished = {\href{https://arxiv.org/abs/1507.00070v4}{arXiv:1507.00070v4}},
  eprint = {1507.00070},
  archivePrefix = {arXiv},
  primaryClass = {math.ST},
  note = {Version 4, 29 February 2016},
  url = {https://arxiv.org/abs/1507.00070v4}
}",136,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/refs/sources.bib#L136,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1507.00070"", ""version"": 4, ""id_versioned"": ""1507.00070v4"", ""title"": ""Random Sampling of Contingency Tables via Probabilistic Divide-and-Conquer"", ""abstract"": ""We present a new approach for random sampling of contingency tables of any size and constraints based on a recently introduced $\\textit{probabilistic divide-and-conquer}$ technique. A simple exact sampling algorithm is presented for $2\\times n$ tables, as well as a generalization where each entry of the table has a specified marginal distribution."", ""authors"": [""Stephen DeSalvo"", ""James Y. Zhao""], ""published"": ""2015-07-01T00:06:21Z"", ""updated"": ""2016-02-29T19:14:46Z"", ""primary_category"": ""math.ST"", ""categories"": [""math.ST"", ""math.PR""], ""doi"": null, ""journal_ref"": null, ""comment"": ""32 Pages, rewritten"", ""abs_url"": ""http://arxiv.org/abs/1507.00070v4"", ""pdf_url"": ""https://arxiv.org/pdf/1507.00070v4""}}]","[""http://arxiv.org/abs/1507.00070v4""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R034,115,Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026,DKM1997,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{DKM1997,
  author = {Dyer, Martin and Kannan, Ravi and Mount, John},
  title = {Sampling contingency tables},
  journal = {Random Structures \& Algorithms},
  volume = {10},
  number = {4},
  pages = {487--506},
  year = {1997},
  note = {\href{https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/contingency.pdf}{Published article}},
  url = {https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/contingency.pdf}
}",55,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/refs/sources.bib#L55,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1002/(sici)1098-2418(199707)10:4<487::aid-rsa4>3.0.co;2-q"", ""title"": [""Sampling contingency tables""], ""authors"": [{""given"": ""Martin"", ""family"": ""Dyer"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Ravi"", ""family"": ""Kannan"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""John"", ""family"": ""Mount"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Wiley"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1997, 7]]}, ""container_title"": [""Random Structures and Algorithms""], ""url"": ""https://doi.org/10.1002/(sici)1098-2418(199707)10:4<487::aid-rsa4>3.0.co;2-q"", ""license"": [{""start"": {""date-parts"": [[2015, 9, 1]], ""date-time"": ""2015-09-01T00:00:00Z"", ""timestamp"": 1441065600000}, ""content-version"": ""tdm"", ""delay-in-days"": 6636, ""URL"": ""http://doi.wiley.com/10.1002/tdm_license_1.1""}], ""source_url"": ""https://api.crossref.org/works/10.1002%2F(sici)1098-2418(199707)10%3A4%3C487%3A%3Aaid-rsa4%3E3.0.co%3B2-q""}}]","[""https://api.crossref.org/works/10.1002%2F(sici)1098-2418(199707)10%3A4%3C487%3A%3Aaid-rsa4%3E3.0.co%3B2-q""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R035,115,Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026,Dyer2003,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@inproceedings{Dyer2003,
  author = {Dyer, Martin},
  title = {Approximate counting by dynamic programming},
  booktitle = {Proceedings of the Thirty-Fifth Annual ACM Symposium on
               Theory of Computing},
  pages = {693--699},
  publisher = {ACM},
  year = {2003},
  note = {\href{https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/knapsack.pdf}{Published article}},
  url = {https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/knapsack.pdf}
}",161,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/refs/sources.bib#L161,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1145/780542.780643"", ""title"": [""Approximate counting by dynamic programming""], ""authors"": [{""given"": ""Martin"", ""family"": ""Dyer"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""University of Leeds, Leeds, UK""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""ACM"", ""type"": ""proceedings-article"", ""published"": {""date-parts"": [[2003, 6, 9]]}, ""container_title"": [""Proceedings of the thirty-fifth annual ACM symposium on Theory of computing""], ""url"": ""https://doi.org/10.1145/780542.780643"", ""license"": [{""start"": {""date-parts"": [[2003, 6, 9]], ""date-time"": ""2003-06-09T00:00:00Z"", ""timestamp"": 1055116800000}, ""content-version"": ""vor"", ""delay-in-days"": 0, ""URL"": ""https://www.acm.org/publications/policies/copyright_policy#Background""}], ""source_url"": ""https://api.crossref.org/works/10.1145%2F780542.780643""}}]","[""https://api.crossref.org/works/10.1145%2F780542.780643""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R036,127,Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,Xu2004,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Xu2004}
Yuan Xu,
\emph{On discrete orthogonal polynomials of several variables},
Advances in Applied Mathematics \textbf{33} (2004), 615--632.
\href{https://doi.org/10.1016/j.aam.2004.03.002}{\nolinkurl{doi:10.1016/j.aam.2004.03.002}}.
Preprint: \href{https://arxiv.org/abs/math/0401418v2}{arXiv:math/0401418v2}.",79,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/references.tex#L79,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.aam.2004.03.002"", ""title"": [""On discrete orthogonal polynomials of several variables""], ""authors"": [{""given"": ""Yuan"", ""family"": ""Xu"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2004, 10]]}, ""container_title"": [""Advances in Applied Mathematics""], ""url"": ""https://doi.org/10.1016/j.aam.2004.03.002"", ""license"": [{""start"": {""date-parts"": [[2004, 10, 1]], ""date-time"": ""2004-10-01T00:00:00Z"", ""timestamp"": 1096588800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2004, 10, 1]], ""date-time"": ""2004-10-01T00:00:00Z"", ""timestamp"": 1096588800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/legal/tdmrep-license""}, {""start"": {""date-parts"": [[2013, 7, 17]], ""date-time"": ""2013-07-17T00:00:00Z"", ""timestamp"": 1374019200000}, ""content-version"": ""vor"", ""delay-in-days"": 3211, ""URL"": ""http://www.elsevier.com/open-access/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.aam.2004.03.002""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""math/0401418"", ""version"": 2, ""id_versioned"": ""math/0401418v2"", ""title"": ""On discrete orthogonal polynomials of several variables"", ""abstract"": ""Let $V$ be a set of isolated points in $\\RR^d$. Define a linear functional $\\CL$ on the space of real polynomials restricted on $V$, $\\CL f = \\sum_{x \\in V} f(x)ρ(x)$, where $ρ$ is a nonzero function on $V$. Polynomial subspaces that contain discrete orthogonal polynomials with respect to the bilinear form $<f,g> = \\CL(f g)$ are identified. One result shows that the discrete orthogonal polynomials still satisfy a three-term relation and Favard's theorem holds in this general setting."", ""authors"": [""Yuan Xu""], ""published"": ""2004-01-29T17:38:30Z"", ""updated"": ""2004-04-09T18:30:33Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA""], ""doi"": null, ""journal_ref"": null, ""comment"": ""15 pages, 2 figures"", ""abs_url"": ""http://arxiv.org/abs/math/0401418v2"", ""pdf_url"": ""https://arxiv.org/pdf/math/0401418v2""}}]","[""https://api.crossref.org/works/10.1016%2Fj.aam.2004.03.002"", ""http://arxiv.org/abs/math/0401418v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2019091792"", ""doi"": ""10.1016/j.aam.2004.03.002"", ""pmid"": null, ""title"": ""On discrete orthogonal polynomials of several variables"", ""publication_year"": 2004, ""publication_date"": ""2004-06-25"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5101667089"", ""name"": ""Yuan Xu"", ""orcid"": ""https://orcid.org/0000-0001-9228-3818"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Oregon""]}], ""source"": {""source_id"": ""S164906323"", ""display_name"": ""Advances in Applied Mathematics"", ""issn_l"": ""0196-8858"", ""type"": ""journal""}, ""biblio"": {""volume"": ""33"", ""issue"": ""3"", ""first_page"": ""615"", ""last_page"": ""632""}, ""cited_by_count"": 49, ""fwci"": 3.565, ""referenced_works_count"": 15, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1016/j.aam.2004.03.002""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11191"", ""display_name"": ""Mathematical functions and polynomials"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""discrete orthogonal polynomials"", ""several variables"", ""multivariate orthogonal polynomials"", ""discrete point sets"", ""three-term recurrence relation"", ""bilinear forms""], ""abstract"": null, ""referenced_works"": [""W154636239"", ""W1528387448"", ""W1654411218"", ""W1823994934"", ""W1971245105"", ""W1982576917"", ""W1999656260"", ""W2016995179"", ""W2079883141"", ""W2130010995"", ""W2319466955"", ""W2483071985"", ""W4214631368"", ""W4230199773"", ""W6606345541""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 7}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 5}, {""year"": 2016, ""cited_by_count"": 3}, {""year"": 2015, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 7}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R037,127,Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,Kane2014,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Kane2014}
Daniel M. Kane,
\emph{The correct exponent for the Gotsman--Linial conjecture},
Computational Complexity \textbf{23} (2014), no.~2, 151--175.
\href{https://doi.org/10.1007/s00037-014-0086-z}{\nolinkurl{doi:10.1007/s00037-014-0086-z}}.
Preprint: \href{https://arxiv.org/abs/1210.1283v1}{arXiv:1210.1283v1}.",53,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/references.tex#L53,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s00037-014-0086-z"", ""title"": [""The correct exponent for the Gotsman–Linial Conjecture""], ""authors"": [{""given"": ""Daniel M."", ""family"": ""Kane"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2014, 5, 13]]}, ""container_title"": [""computational complexity""], ""url"": ""https://doi.org/10.1007/s00037-014-0086-z"", ""license"": [{""start"": {""date-parts"": [[2014, 5, 13]], ""date-time"": ""2014-05-13T00:00:00Z"", ""timestamp"": 1399939200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs00037-014-0086-z""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1210.1283"", ""version"": 1, ""id_versioned"": ""1210.1283v1"", ""title"": ""The Correct Exponent for the Gotsman-Linial Conjecture"", ""abstract"": ""We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree $d$ in at most $n$ variables has average sensitivity at most $\\sqrt{n}(\\log(n))^{O(d\\log(d))}2^{O(d^2\\log(d)}$. For fixed $d$ the exponent in terms of $n$ in this bound is known to be optimal. This bound makes significant progress towards the Gotsman-Linial Conjecture which would put the correct bound at $Θ(d\\sqrt{n})$."", ""authors"": [""Daniel M. Kane""], ""published"": ""2012-10-04T03:08:19Z"", ""updated"": ""2012-10-04T03:08:19Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO"", ""cs.CC"", ""math.PR""], ""doi"": null, ""journal_ref"": null, ""comment"": null, ""abs_url"": ""http://arxiv.org/abs/1210.1283v1"", ""pdf_url"": ""https://arxiv.org/pdf/1210.1283v1""}}]","[""https://api.crossref.org/works/10.1007%2Fs00037-014-0086-z"", ""http://arxiv.org/abs/1210.1283v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2071833075"", ""doi"": ""10.1007/s00037-014-0086-z"", ""pmid"": null, ""title"": ""The correct exponent for the Gotsman–Linial Conjecture"", ""publication_year"": 2014, ""publication_date"": ""2014-05-12"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5035962853"", ""name"": ""Daniel M. Kane"", ""orcid"": ""https://orcid.org/0000-0002-0220-9962"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Stanford University""]}], ""source"": {""source_id"": ""S193995496"", ""display_name"": ""Computational Complexity"", ""issn_l"": ""1016-3328"", ""type"": ""journal""}, ""biblio"": {""volume"": ""23"", ""issue"": ""2"", ""first_page"": ""151"", ""last_page"": ""175""}, ""cited_by_count"": 26, ""fwci"": 4.0571, ""referenced_works_count"": 17, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10720"", ""display_name"": ""Complexity and Algorithms in Graphs"", ""field"": ""Computer Science"", ""subfield"": ""Computational Theory and Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""average sensitivity"", ""polynomial threshold functions"", ""Boolean functions"", ""polynomial degree""], ""abstract"": null, ""referenced_works"": [""W1483560004"", ""W1486564402"", ""W1497671620"", ""W1973898585"", ""W1997142544"", ""W2005888001"", ""W2043702877"", ""W2060775054"", ""W2078134499"", ""W2078929687"", ""W2106552140"", ""W2117294010"", ""W2133263321"", ""W2144919274"", ""W2163332813"", ""W2334237942"", ""W3100065552""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 3}, {""year"": 2018, ""cited_by_count"": 5}, {""year"": 2017, ""cited_by_count"": 3}, {""year"": 2016, ""cited_by_count"": 4}, {""year"": 2014, ""cited_by_count"": 5}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R038,127,Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,TsengVolberg2026,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{TsengVolberg2026}
Chun-Kai Tseng and Alexander Volberg,
\emph{Small moments of the sensitivity of polynomial threshold functions},
preprint, 2026.
\href{https://arxiv.org/abs/2606.16004v2}{arXiv:2606.16004v2}.",73,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/references.tex#L73,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2606.16004"", ""version"": 2, ""id_versioned"": ""2606.16004v2"", ""title"": ""Small moments of the sensitivity of polynomial threshold functions"", ""abstract"": ""In the first version of Chang, Slote, Volberg, and Zhang's paper \\cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \\cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \\cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \\sqrt{\\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \\cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\\mathbb E[s(x)^η]$ where $η< 1/2$. It is interesting to note the phase transition that happens at $η=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that."", ""authors"": [""Chun-Kai Tseng"", ""Alexander Volberg""], ""published"": ""2026-06-14T20:18:11Z"", ""updated"": ""2026-06-21T17:31:54Z"", ""primary_category"": ""math.PR"", ""categories"": [""math.PR"", ""math.AP"", ""math.CO""], ""doi"": null, ""journal_ref"": null, ""comment"": ""15 pages"", ""abs_url"": ""http://arxiv.org/abs/2606.16004v2"", ""pdf_url"": ""https://arxiv.org/pdf/2606.16004v2""}}]","[""http://arxiv.org/abs/2606.16004v2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R039,127,Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,ChangSloteVolbergZhang2026,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{ChangSloteVolbergZhang2026}
Fan Chang, Joseph Slote, Alexander Volberg, and Haonan Zhang,
\emph{The Boolean surface area of polynomial threshold functions},
preprint, 2026.
\href{https://arxiv.org/abs/2604.08095v2}{arXiv:2604.08095v2}.",3,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/references.tex#L3,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2604.08095"", ""version"": 2, ""id_versioned"": ""2604.08095v2"", ""title"": ""The Boolean surface area of polynomial threshold functions"", ""abstract"": ""Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \\[ \\mathbf{BSA}[f]=\\mathbb{E}|\\nabla f|=\\mathbb{E}\\sqrt{s_{f}(x)}, \\] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \\[ \\mathbf{BSA}[f]\\le C_d(\\log(en))^{C_d}. \\] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \\cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \\cite{DK} on average sensitivity, which independently yields the weaker bound \\[ \\mathbf{BSA}[f]\\le \\exp(C_d\\sqrt{\\log n}). \\]"", ""authors"": [""Fan Chang"", ""Joseph Slote"", ""Alexander Volberg"", ""Haonan Zhang""], ""published"": ""2026-04-09T11:11:06Z"", ""updated"": ""2026-04-27T06:28:32Z"", ""primary_category"": ""cs.CC"", ""categories"": [""cs.CC"", ""math.AP"", ""math.CA"", ""math.PR""], ""doi"": null, ""journal_ref"": null, ""comment"": ""18 pages, 1 figure"", ""abs_url"": ""http://arxiv.org/abs/2604.08095v2"", ""pdf_url"": ""https://arxiv.org/pdf/2604.08095v2""}}]","[""http://arxiv.org/abs/2604.08095v2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R040,127,Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,Chapman2018,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Chapman2018}
Brynmor Chapman,
\emph{The Gotsman--Linial conjecture is false},
in Proceedings of the Twenty-Ninth Annual ACM--SIAM Symposium on Discrete
Algorithms (SODA 2018), Society for Industrial and Applied Mathematics,
2018, 692--699.
\href{https://doi.org/10.1137/1.9781611975031.45}{\nolinkurl{doi:10.1137/1.9781611975031.45}}.
Preprint: \href{https://arxiv.org/abs/2108.02288v1}{arXiv:2108.02288v1}.",9,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/references.tex#L9,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1137/1.9781611975031.45"", ""title"": [""The Gotsman-Linial Conjecture is False""], ""authors"": [{""given"": ""Brynmor"", ""family"": ""Chapman"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Society for Industrial and Applied Mathematics"", ""type"": ""book-chapter"", ""published"": {""date-parts"": [[2018, 1]]}, ""container_title"": [""Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms""], ""url"": ""https://doi.org/10.1137/1.9781611975031.45"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1137%2F1.9781611975031.45""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2108.02288"", ""version"": 1, ""id_versioned"": ""2108.02288v1"", ""title"": ""The Gotsman-Linial Conjecture is False"", ""abstract"": ""In 1991, Craig Gotsman and Nathan Linial conjectured that for all $n$ and $d$, the average sensitivity of a degree-$d$ polynomial threshold function on $n$ variables is maximized by the degree-$d$ symmetric polynomial which computes the parity function on the $d$ layers of the hypercube with Hamming weight closest to $n/2$. We refute the conjecture for almost all $d$ and for almost all $n$, and we confirm the conjecture in many of the remaining cases."", ""authors"": [""Brynmor Chapman""], ""published"": ""2021-08-04T21:19:31Z"", ""updated"": ""2021-08-04T21:19:31Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO"", ""cs.CC""], ""doi"": ""10.1137/1.9781611975031.45"", ""journal_ref"": ""Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 692-699, 2018"", ""comment"": ""Appeared in SODA 2018"", ""abs_url"": ""http://arxiv.org/abs/2108.02288v1"", ""pdf_url"": ""https://arxiv.org/pdf/2108.02288v1""}}]","[""https://api.crossref.org/works/10.1137%2F1.9781611975031.45"", ""http://arxiv.org/abs/2108.02288v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2779077750"", ""doi"": ""10.1137/1.9781611975031.45"", ""pmid"": null, ""title"": ""The Gotsman-Linial Conjecture is False"", ""publication_year"": 2018, ""publication_date"": ""2018-01-01"", ""type"": ""conference-paper"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5063179780"", ""name"": ""Brynmor Chapman"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Massachusetts Institute of Technology""]}], ""source"": {""source_id"": ""S4306463922"", ""display_name"": ""Society for Industrial and Applied Mathematics eBooks"", ""issn_l"": null, ""type"": ""ebook platform""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": ""692"", ""last_page"": ""699""}, ""cited_by_count"": 3, ""fwci"": 0.2746, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": true, ""oa_status"": ""green"", ""oa_url"": ""https://arxiv.org/pdf/2108.02288""}, ""best_oa_pdf_url"": ""https://arxiv.org/pdf/2108.02288"", ""primary_topic"": {""id"": ""T10720"", ""display_name"": ""Complexity and Algorithms in Graphs"", ""field"": ""Computer Science"", ""subfield"": ""Computational Theory and Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""average sensitivity"", ""polynomial threshold functions"", ""parity function"", ""symmetric polynomials"", ""hypercube"", ""Hamming weight"", ""Boolean functions"", ""counterexample""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1137/1.9781611975031.45"", ""referenced_works"": [], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R041,148,The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,RapaportVarju2024,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{RapaportVarju2024,
  author = {Rapaport, Ariel and Varj\'{u}, P\'{e}ter P.},
  title = {Self-similar measures associated to a homogeneous system of three maps},
  journal = {Duke Mathematical Journal},
  volume = {173},
  number = {3},
  year = {2024},
  pages = {513--602},
  doi = {10.1215/00127094-2023-0019},
  eprint = {2010.01022v3},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2010.01022v3},
  note = {\href{https://doi.org/10.1215/00127094-2023-0019}{doi:10.1215/00127094-2023-0019}}
}",78,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/references.bib#L78,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1215/00127094-2023-0019"", ""title"": [""Self-similar measures associated to a homogeneous system of three maps""], ""authors"": [{""given"": ""Ariel"", ""family"": ""Rapaport"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Department of Mathematics, Technion, Haifa, Israel""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Péter P."", ""family"": ""Varjú"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK""}], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Duke University Press"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2024, 2, 15]]}, ""container_title"": [""Duke Mathematical Journal""], ""url"": ""https://doi.org/10.1215/00127094-2023-0019"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1215%2F00127094-2023-0019""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2010.01022"", ""version"": 3, ""id_versioned"": ""2010.01022v3"", ""title"": ""Self-similar measures associated to a homogeneous system of three maps"", ""abstract"": ""We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and Bárány, Käenmäki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup."", ""authors"": [""Ariel Rapaport"", ""Péter P. Varjú""], ""published"": ""2020-10-02T14:19:33Z"", ""updated"": ""2023-03-08T10:11:00Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS""], ""doi"": null, ""journal_ref"": null, ""comment"": ""85 pages, minor revision based on referee's report, final accepted version to appear in Duke Math. J"", ""abs_url"": ""http://arxiv.org/abs/2010.01022v3"", ""pdf_url"": ""https://arxiv.org/pdf/2010.01022v3""}}]","[""https://api.crossref.org/works/10.1215%2F00127094-2023-0019"", ""http://arxiv.org/abs/2010.01022v3""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W3091280398"", ""doi"": ""10.1215/00127094-2023-0019"", ""pmid"": null, ""title"": ""Self-similar measures associated to a homogeneous system of three maps"", ""publication_year"": 2024, ""publication_date"": ""2024-02-15"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5045452900"", ""name"": ""Ariel Rapaport"", ""orcid"": ""https://orcid.org/0000-0001-7621-9314"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Technion – Israel Institute of Technology""]}, {""author_id"": ""A5090834349"", ""name"": ""Péter P. Varjú"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Cambridge""]}], ""source"": {""source_id"": ""S171170845"", ""display_name"": ""Duke Mathematical Journal"", ""issn_l"": ""0012-7094"", ""type"": ""journal""}, ""biblio"": {""volume"": ""173"", ""issue"": ""3"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 12, ""fwci"": 4.8541, ""referenced_works_count"": 36, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""self-similar measures"", ""iterated function systems"", ""Hausdorff dimension"", ""Bernoulli convolutions"", ""function fields""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1215/00127094-2023-0019"", ""referenced_works"": [""W1496620437"", ""W1559103132"", ""W1601621518"", ""W1602594669"", ""W1971967210"", ""W2018665155"", ""W2044440663"", ""W2051374214"", ""W2113119296"", ""W2140035157"", ""W2148932839"", ""W2269341939"", ""W2325571074"", ""W2327970496"", ""W2482071716"", ""W2529844022"", ""W2544904738"", ""W2906962869"", ""W2962852664"", ""W2963166899"", ""W2963562570"", ""W2972725323"", ""W2972959157"", ""W2979343135"", ""W2997523743"", ""W3020808117"", ""W3099200545"", ""W3104213773"", ""W3119097474"", ""W3119103025"", ""W3158017144"", ""W3189566504"", ""W4213358012"", ""W4243950761"", ""W4249775755"", ""W4318815121""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 6}, {""year"": 2021, ""cited_by_count"": 4}, {""year"": 2020, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R042,148,The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,Baker2021,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Baker2021,
  author = {Baker, Simon},
  title = {Iterated function systems with super-exponentially close cylinders},
  journal = {Advances in Mathematics},
  volume = {379},
  year = {2021},
  pages = {107548},
  doi = {10.1016/j.aim.2020.107548},
  eprint = {1909.04343v2},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1909.04343v2},
  note = {\href{https://doi.org/10.1016/j.aim.2020.107548}{doi:10.1016/j.aim.2020.107548}}
}",251,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/references.bib#L251,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.aim.2020.107548"", ""title"": [""Iterated function systems with super-exponentially close cylinders""], ""authors"": [{""given"": ""Simon"", ""family"": ""Baker"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2021, 3]]}, ""container_title"": [""Advances in Mathematics""], ""url"": ""https://doi.org/10.1016/j.aim.2020.107548"", ""license"": [{""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/legal/tdmrep-license""}, {""start"": {""date-parts"": [[2025, 3, 5]], ""date-time"": ""2025-03-05T00:00:00Z"", ""timestamp"": 1741132800000}, ""content-version"": ""vor"", ""delay-in-days"": 1465, ""URL"": ""http://www.elsevier.com/open-access/userlicense/1.0/""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-017""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-037""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-012""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-029""}, {""start"": {""date-parts"": [[2021, 3, 1]], ""date-time"": ""2021-03-01T00:00:00Z"", ""timestamp"": 1614556800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-004""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.aim.2020.107548""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1909.04343"", ""version"": 2, ""id_versioned"": ""1909.04343v2"", ""title"": ""Iterated function systems with super-exponentially close cylinders"", ""abstract"": ""Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in $\\mathbb{R}$ does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the $L^q$ dimension of self-similar measures in $\\mathbb{R}$. With these statements in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence $(ε_n)_{n=1}^{\\infty}$ of positive real numbers, there exists an iterated function system $\\{φ_i\\}_{i\\in \\mathcal{I}}$ that does not contain exact overlaps and $$\\min\\left\\{|φ_{\\mathbf{a}}(0)-φ_{\\mathbf{b}}(0)|: \\mathbf{a},\\mathbf{b}\\in \\mathcal{I}^n,\\, \\mathbf{a}\\neq \\mathbf{b},\\, r_{\\mathbf{a}}=r_{\\mathbf{b}}\\right\\}\\leq ε_n$$ for all $n\\in \\mathbb{N}.$"", ""authors"": [""Simon Baker""], ""published"": ""2019-09-10T08:08:58Z"", ""updated"": ""2019-09-12T12:05:04Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS"", ""math.CA"", ""math.MG""], ""doi"": null, ""journal_ref"": null, ""comment"": null, ""abs_url"": ""http://arxiv.org/abs/1909.04343v2"", ""pdf_url"": ""https://arxiv.org/pdf/1909.04343v2""}}]","[""https://api.crossref.org/works/10.1016%2Fj.aim.2020.107548"", ""http://arxiv.org/abs/1909.04343v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W3119097474"", ""doi"": ""10.1016/j.aim.2020.107548"", ""pmid"": null, ""title"": ""Iterated function systems with super-exponentially close cylinders"", ""publication_year"": 2021, ""publication_date"": ""2021-01-10"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5061943510"", ""name"": ""Simon J Baker"", ""orcid"": ""https://orcid.org/0000-0002-0716-6236"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Birmingham""]}], ""source"": {""source_id"": ""S122441808"", ""display_name"": ""Advances in Mathematics"", ""issn_l"": ""0001-8708"", ""type"": ""journal""}, ""biblio"": {""volume"": ""379"", ""issue"": null, ""first_page"": ""107548"", ""last_page"": ""107548""}, ""cited_by_count"": 11, ""fwci"": 2.4396, ""referenced_works_count"": 18, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1016/j.aim.2020.107548""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""iterated function systems"", ""self-similar measures"", ""Hausdorff dimension""], ""abstract"": null, ""referenced_works"": [""W1569256320"", ""W2007343403"", ""W2113119296"", ""W2491962810"", ""W2525870670"", ""W2956468610"", ""W2962852664"", ""W2963562570"", ""W2964089240"", ""W2969292053"", ""W2972959157"", ""W3099200545"", ""W3196884212"", ""W4249775755"", ""W6633988135"", ""W6759362407"", ""W6768045204"", ""W6992018761""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 4}, {""year"": 2020, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R043,148,The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,Hutchinson1981,unverified,unverified,unverified,direct_records,"DOI exists and registry author/year agree, but both Crossref and OpenAlex return an empty title. Work identity is only partially corroborated; title and full pagination remain unverified.","@article{Hutchinson1981,
  author = {Hutchinson, John E.},
  title = {Fractals and self-similarity},
  journal = {Indiana University Mathematics Journal},
  volume = {30},
  number = {5},
  year = {1981},
  pages = {713--747},
  doi = {10.1512/iumj.1981.30.30055},
  url = {https://doi.org/10.1512/iumj.1981.30.30055},
  note = {\href{https://doi.org/10.1512/iumj.1981.30.30055}{doi:10.1512/iumj.1981.30.30055}}
}",158,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/references.bib#L158,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1512/iumj.1981.30.30055"", ""title"": [""""], ""authors"": [{""given"": ""John"", ""family"": ""Hutchinson"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Indiana University Mathematics Journal"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1981]]}, ""container_title"": [""Indiana University Mathematics Journal""], ""url"": ""https://doi.org/10.1512/iumj.1981.30.30055"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1512%2Fiumj.1981.30.30055""}}]","[""https://api.crossref.org/works/10.1512%2Fiumj.1981.30.30055""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4249775755"", ""doi"": ""10.1512/iumj.1981.30.30055"", ""pmid"": null, ""title"": """", ""publication_year"": 1981, ""publication_date"": ""1981-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5087436752"", ""name"": ""John Hutchinson"", ""orcid"": ""https://orcid.org/0000-0002-5927-6058"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S196210972"", ""display_name"": ""Indiana University Mathematics Journal"", ""issn_l"": ""0022-2518"", ""type"": ""journal""}, ""biblio"": {""volume"": ""30"", ""issue"": ""5"", ""first_page"": ""713"", ""last_page"": ""713""}, ""cited_by_count"": 3103, ""fwci"": null, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30055/pdf""}, ""best_oa_pdf_url"": ""http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1981/30/30055/pdf"", ""primary_topic"": null, ""keywords"": [], ""abstract"": null, ""referenced_works"": [], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 108}, {""year"": 2025, ""cited_by_count"": 196}, {""year"": 2024, ""cited_by_count"": 161}, {""year"": 2023, ""cited_by_count"": 162}, {""year"": 2022, ""cited_by_count"": 163}, {""year"": 2021, ""cited_by_count"": 137}, {""year"": 2020, ""cited_by_count"": 144}, {""year"": 2019, ""cited_by_count"": 128}, {""year"": 2018, ""cited_by_count"": 141}, {""year"": 2017, ""cited_by_count"": 112}, {""year"": 2016, ""cited_by_count"": 94}, {""year"": 2015, ""cited_by_count"": 99}, {""year"": 2014, ""cited_by_count"": 104}, {""year"": 2013, ""cited_by_count"": 94}, {""year"": 2012, ""cited_by_count"": 97}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R044,148,The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,FengHu2009,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{FengHu2009,
  author = {Feng, De-Jun and Hu, Huyi},
  title = {Dimension theory of iterated function systems},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {62},
  number = {11},
  year = {2009},
  pages = {1435--1500},
  doi = {10.1002/cpa.20276},
  eprint = {1002.2036v1},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1002.2036v1},
  note = {\href{https://doi.org/10.1002/cpa.20276}{doi:10.1002/cpa.20276}}
}",1,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/references.bib#L1,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1002/cpa.20276"", ""title"": [""Dimension theory of iterated function systems""], ""authors"": [{""given"": ""De‐Jun"", ""family"": ""Feng"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Huyi"", ""family"": ""Hu"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Wiley"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2009, 3, 3]]}, ""container_title"": [""Communications on Pure and Applied Mathematics""], ""url"": ""https://doi.org/10.1002/cpa.20276"", ""license"": [{""start"": {""date-parts"": [[2009, 3, 3]], ""date-time"": ""2009-03-03T00:00:00Z"", ""timestamp"": 1236038400000}, ""content-version"": ""vor"", ""delay-in-days"": 0, ""URL"": ""http://onlinelibrary.wiley.com/termsAndConditions#vor""}], ""source_url"": ""https://api.crossref.org/works/10.1002%2Fcpa.20276""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1002.2036"", ""version"": 1, ""id_versioned"": ""1002.2036v1"", ""title"": ""Dimension theory of iterated function systems"", ""abstract"": ""Let $\\{S_i\\}_{i=1}^\\ell$ be an iterated function system (IFS) on $\\R^d$ with attractor $K$. Let $(Σ,σ)$ denote the one-sided full shift over the alphabet $\\{1,..., \\ell\\}$. We define the projection entropy function $h_π$ on the space of invariant measures on $Σ$ associated with the coding map $π: Σ\\to K$, and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on $K$. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under $π$ is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures."", ""authors"": [""De-Jun Feng"", ""Huyi Hu""], ""published"": ""2010-02-10T07:56:12Z"", ""updated"": ""2010-02-10T07:56:12Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS"", ""math.CA""], ""doi"": null, ""journal_ref"": ""Comm. Pure Appl. Math., 62 (2009), no. 11, 1435--1500"", ""comment"": ""60 pages"", ""abs_url"": ""http://arxiv.org/abs/1002.2036v1"", ""pdf_url"": ""https://arxiv.org/pdf/1002.2036v1""}}]","[""https://api.crossref.org/works/10.1002%2Fcpa.20276"", ""http://arxiv.org/abs/1002.2036v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2962852664"", ""doi"": ""10.1002/cpa.20276"", ""pmid"": null, ""title"": ""Dimension theory of iterated function systems"", ""publication_year"": 2009, ""publication_date"": ""2009-03-03"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5112614265"", ""name"": ""De‐Jun Feng"", ""orcid"": ""https://orcid.org/0000-0003-4375-4139"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Chinese University of Hong Kong""]}, {""author_id"": ""A5051406466"", ""name"": ""Huyi Hu"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": true, ""institutions"": [""Michigan State University""]}], ""source"": {""source_id"": ""S134818455"", ""display_name"": ""Communications on Pure and Applied Mathematics"", ""issn_l"": ""0010-3640"", ""type"": ""journal""}, ""biblio"": {""volume"": ""62"", ""issue"": ""11"", ""first_page"": ""1435"", ""last_page"": ""1500""}, ""cited_by_count"": 166, ""fwci"": 5.5741, ""referenced_works_count"": 69, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""iterated function systems"", ""Hausdorff dimension"", ""invariant measures"", ""conformal iterated function systems"", ""affine iterated function systems"", ""Lyapunov exponents"", ""variational principle"", ""ergodic measures"", ""attractors"", ""full shift""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1002/cpa.20276"", ""referenced_works"": [""W20854365"", ""W39337161"", ""W212966973"", ""W572526246"", ""W997267355"", ""W1493938220"", ""W1498335665"", ""W1515015816"", ""W1521685998"", ""W1587745056"", ""W1589522393"", ""W1592457332"", ""W1594563152"", ""W1600502364"", ""W1604601822"", ""W1971471503"", ""W1979242098"", ""W1979468839"", ""W1987395549"", ""W1997037277"", ""W1998727969"", ""W2013241441"", ""W2014573070"", ""W2017926248"", ""W2024738451"", ""W2032121576"", ""W2050414795"", ""W2051333261"", ""W2054460150"", ""W2056099894"", ""W2058059510"", ""W2073232781"", ""W2074326293"", ""W2075508378"", ""W2081277635"", ""W2084842067"", ""W2087049094"", ""W2087201972"", ""W2087751256"", ""W2089571721"", ""W2091029068"", ""W2100145758"", ""W2106318693"", ""W2110671120"", ""W2111963011"", ""W2112136864"", ""W2116255401"", ""W2120496796"", ""W2128554376"", ""W2129003903"", ""W2152337671"", ""W2159767801"", ""W2162696617"", ""W2166136302"", ""W2167712533"", ""W2332374329"", ""W2339378878"", ""W2490674248"", ""W2801434754"", ""W2963483447"", ""W2993508114"", ""W3152478022"", ""W4206197472"", ""W4231009334"", ""W4240810995"", ""W4244297378"", ""W4247786464"", ""W4249775755"", ""W4285719527""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 4}, {""year"": 2025, ""cited_by_count"": 12}, {""year"": 2024, ""cited_by_count"": 15}, {""year"": 2023, ""cited_by_count"": 9}, {""year"": 2022, ""cited_by_count"": 9}, {""year"": 2021, ""cited_by_count"": 24}, {""year"": 2020, ""cited_by_count"": 15}, {""year"": 2019, ""cited_by_count"": 16}, {""year"": 2018, ""cited_by_count"": 10}, {""year"": 2017, ""cited_by_count"": 13}, {""year"": 2016, ""cited_by_count"": 8}, {""year"": 2015, ""cited_by_count"": 5}, {""year"": 2014, ""cited_by_count"": 8}, {""year"": 2013, ""cited_by_count"": 4}, {""year"": 2012, ""cited_by_count"": 8}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R045,148,The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,BaranyVerma2026,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"@misc{BaranyVerma2026,
  author = {B\'{a}r\'{a}ny, Bal\'{a}zs and Verma, Manuj},
  title = {Hausdorff dimension of self-similar measures and sets with common fixed point structure},
  year = {2026},
  eprint = {2507.05835v2},
  note = {\href{https://arxiv.org/abs/2507.05835v2}{arXiv:2507.05835v2}},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  url = {https://arxiv.org/abs/2507.05835v2}
}",119,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/references.bib#L119,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2507.05835"", ""version"": 2, ""id_versioned"": ""2507.05835v2"", ""title"": ""Hausdorff dimension of self-similar measures and sets with common fixed point structure"", ""abstract"": ""In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of a Hausdorff co-dimension one exceptional set of natural parameters, such systems satisfy a weak exponential separation. This significantly strengthens the previous result of the first author and Szvák. As an application, we give the Hausdorff dimension of self-affine measures supported on the generalised 4-corner set."", ""authors"": [""Balázs Bárány"", ""Manuj Verma""], ""published"": ""2025-07-08T09:57:46Z"", ""updated"": ""2026-07-14T09:06:38Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS""], ""doi"": null, ""journal_ref"": null, ""comment"": null, ""abs_url"": ""http://arxiv.org/abs/2507.05835v2"", ""pdf_url"": ""https://arxiv.org/pdf/2507.05835v2""}}]","[""http://arxiv.org/abs/2507.05835v2""]",[],"[{""sample_id"": ""R045"", ""source_url"": ""https://arxiv.org/pdf/2507.05835v2"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""b01fade7cbd6fb36052d0836df5a6dca93c584a2c8a6bb349e2d2037d77b03e8"", ""locator"": ""PDF pages 8, 10–11; discussion before Section 3 and Theorem 3.5"", ""source_version"": ""arXiv:2507.05835v2"", ""paraphrased_observations"": ""Page 8 explains that the weak exponential separation condition permits exact overlaps. Theorem 3.5 gives Hausdorff dimension of a self-similar measure under WESC as the minimum of one and random-walk entropy divided by the Lyapunov exponent. The surrounding text distinguishes this general contraction-ratio version from the earlier homogeneous case. This is a statement/attribution check, not verification of the proof."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R046,154,Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,Jamneshan2023,minor_difference,minor_difference,minor_difference,direct_records,"Same Jamneshan work and DOI. Citation uses 2023; Crossref published date is 2022-06-21. Revised arXiv v4 identity also matches. Likely online-first/issue-year difference, not a wrong-work citation.","@article{Jamneshan2023,
  author = {Jamneshan, A.},
  title = {An uncountable {Furstenberg--Zimmer} structure theory},
  journal = {Ergodic Theory Dynam. Systems},
  volume = {43},
  number = {7},
  year = {2023},
  pages = {2404--2436},
  doi = {10.1017/etds.2022.43},
  note = {Corrected version: arXiv:2103.17167v4}
}",129,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib#L129,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1017/etds.2022.43"", ""title"": [""An uncountable Furstenberg–Zimmer structure theory""], ""authors"": [{""given"": ""ASGAR"", ""family"": ""JAMNESHAN"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Cambridge University Press (CUP)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2022, 6, 21]]}, ""container_title"": [""Ergodic Theory and Dynamical Systems""], ""url"": ""https://doi.org/10.1017/etds.2022.43"", ""license"": [{""start"": {""date-parts"": [[2022, 6, 21]], ""date-time"": ""2022-06-21T00:00:00Z"", ""timestamp"": 1655769600000}, ""content-version"": ""unspecified"", ""delay-in-days"": 0, ""URL"": ""https://creativecommons.org/licenses/by/4.0""}], ""source_url"": ""https://api.crossref.org/works/10.1017%2Fetds.2022.43""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""2103.17167"", ""version"": 4, ""id_versioned"": ""2103.17167v4"", ""title"": ""An uncountable Furstenberg--Zimmer structure theory"", ""abstract"": ""Furstenberg--Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogues of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg--Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi."", ""authors"": [""Asgar Jamneshan""], ""published"": ""2021-03-31T15:40:57Z"", ""updated"": ""2025-09-27T14:06:05Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS"", ""math.LO""], ""doi"": ""10.1017/etds.2022.43"", ""journal_ref"": null, ""comment"": ""38 pages, 4 figures, 1 table; [v4] After publication the author found two issues; we refer to the erratum, appeared online at https://doi.org/10.1017/etds.2025.10214 . The main results remain unchanged, and this version contains all the corrections outlined in the erratum"", ""abs_url"": ""http://arxiv.org/abs/2103.17167v4"", ""pdf_url"": ""https://arxiv.org/pdf/2103.17167v4""}}]","[""https://api.crossref.org/works/10.1017%2Fetds.2022.43"", ""http://arxiv.org/abs/2103.17167v4""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W3150594803"", ""doi"": ""10.1017/etds.2022.43"", ""pmid"": null, ""title"": ""An uncountable Furstenberg–Zimmer structure theory"", ""publication_year"": 2022, ""publication_date"": ""2022-06-21"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5073749739"", ""name"": ""Asgar Jamneshan"", ""orcid"": ""https://orcid.org/0000-0002-1450-6569"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Koç University""]}], ""source"": {""source_id"": ""S14709335"", ""display_name"": ""Ergodic Theory and Dynamical Systems"", ""issn_l"": ""0143-3857"", ""type"": ""journal""}, ""biblio"": {""volume"": ""43"", ""issue"": ""7"", ""first_page"": ""2404"", ""last_page"": ""2436""}, ""cited_by_count"": 10, ""fwci"": 3.3958, ""referenced_works_count"": 55, ""open_access"": {""is_oa"": true, ""oa_status"": ""hybrid"", ""oa_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1141F58ED704C5F30DB76741CD3E0362/S0143385722000438a.pdf/div-class-title-an-uncountable-furstenberg-zimmer-structure-theory-div.pdf""}, ""best_oa_pdf_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1141F58ED704C5F30DB76741CD3E0362/S0143385722000438a.pdf/div-class-title-an-uncountable-furstenberg-zimmer-structure-theory-div.pdf"", ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""measure-preserving dynamical systems"", ""internal logic"", ""uncountable groups""], ""abstract"": ""Abstract Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi."", ""abstract_license"": ""cc-by"", ""referenced_works"": [""W402698555"", ""W1022649828"", ""W1481868958"", ""W1483888214"", ""W1489644082"", ""W1534018603"", ""W1549382427"", ""W1554682571"", ""W1581696870"", ""W1793232940"", ""W1901037400"", ""W1921448755"", ""W1950693991"", ""W1970366772"", ""W1974560332"", ""W1976245876"", ""W1985280810"", ""W2003286481"", ""W2017170340"", ""W2039661070"", ""W2053369303"", ""W2070464931"", ""W2115508955"", ""W2136575564"", ""W2141304880"", ""W2319590377"", ""W2324147299"", ""W2324802422"", ""W2326323046"", ""W2328246511"", ""W2332886906"", ""W2333586672"", ""W2334218032"", ""W2335400162"", ""W2342433841"", ""W2574237409"", ""W2575907820"", ""W2597642898"", ""W2777506625"", ""W2962958747"", ""W2963316687"", ""W2990392967"", ""W3089621865"", ""W3091648749"", ""W3102258829"", ""W3103596274"", ""W3109381375"", ""W3153182186"", ""W3169820778"", ""W4226288723"", ""W4235653545"", ""W4292027056"", ""W4308482081"", ""W4312887919"", ""W6750114732""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 3}]}}]","[{""sample_id"": ""R046"", ""source_url"": ""https://arxiv.org/pdf/2103.17167v4"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""b3ef73623931c3bcc11b1918441cbb84610a38148a1c9577710ba47bf947e248"", ""locator"": ""PDF page 2, Theorem 1.1"", ""source_version"": ""arXiv:2103.17167v4"", ""paraphrased_observations"": ""The uncountable Furstenberg–Zimmer theorem is stated for any group and probability algebra dynamical system, with a transfinite relatively compact tower and a weakly mixing extension at the top. Ergodicity is not assumed in the statement. This supports the nonergodic structure-theory attribution, not every adjacent assertion about pointwise averages or joinings."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R047,154,Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,HuangShaoYe2019,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{HuangShaoYe2019,
  author = {Huang, W. and Shao, S. and Ye, X.},
  title = {Pointwise convergence of multiple ergodic averages and strictly ergodic models},
  journal = {J. Anal. Math.},
  volume = {139},
  number = {1},
  year = {2019},
  pages = {265--305},
  doi = {10.1007/s11854-019-0061-3}
}",97,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib#L97,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s11854-019-0061-3"", ""title"": [""Pointwise convergence of multiple ergodic averages and strictly ergodic models""], ""authors"": [{""given"": ""Wen"", ""family"": ""Huang"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Song"", ""family"": ""Shao"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Xiangdong"", ""family"": ""Ye"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2019, 10]]}, ""container_title"": [""Journal d'Analyse Mathématique""], ""url"": ""https://doi.org/10.1007/s11854-019-0061-3"", ""license"": [{""start"": {""date-parts"": [[2019, 10, 1]], ""date-time"": ""2019-10-01T00:00:00Z"", ""timestamp"": 1569888000000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}, {""start"": {""date-parts"": [[2019, 10, 1]], ""date-time"": ""2019-10-01T00:00:00Z"", ""timestamp"": 1569888000000}, ""content-version"": ""vor"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs11854-019-0061-3""}}]","[""https://api.crossref.org/works/10.1007%2Fs11854-019-0061-3""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2979633430"", ""doi"": ""10.1007/s11854-019-0061-3"", ""pmid"": null, ""title"": ""Pointwise convergence of multiple ergodic averages and strictly ergodic models"", ""publication_year"": 2019, ""publication_date"": ""2019-10-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5100514605"", ""name"": ""Wen Huang"", ""orcid"": ""https://orcid.org/0009-0000-7920-6914"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Science and Technology of China"", ""Chinese Academy of Sciences""]}, {""author_id"": ""A5101887175"", ""name"": ""Song Shao"", ""orcid"": ""https://orcid.org/0000-0001-9653-4557"", ""position"": ""middle"", ""is_corresponding"": true, ""institutions"": [""University of Science and Technology of China"", ""Chinese Academy of Sciences""]}, {""author_id"": ""A5101650805"", ""name"": ""Xiangdong Ye"", ""orcid"": ""https://orcid.org/0000-0002-9527-544X"", ""position"": ""last"", ""is_corresponding"": true, ""institutions"": [""University of Science and Technology of China"", ""Chinese Academy of Sciences""]}], ""source"": {""source_id"": ""S67718071"", ""display_name"": ""Journal d Analyse Mathématique"", ""issn_l"": ""0021-7670"", ""type"": ""journal""}, ""biblio"": {""volume"": ""139"", ""issue"": ""1"", ""first_page"": ""265"", ""last_page"": ""305""}, ""cited_by_count"": 33, ""fwci"": 5.3385, ""referenced_works_count"": 55, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""multiple ergodic averages"", ""pointwise convergence"", ""ergodic systems"", ""multiple recurrence"", ""ergodic theory""], ""abstract"": null, ""referenced_works"": [""W55131882"", ""W566109589"", ""W1483336251"", ""W1483888214"", ""W1496765929"", ""W1532303937"", ""W1549382427"", ""W1672337913"", ""W1730937491"", ""W1950693991"", ""W1964679191"", ""W1970366772"", ""W1970890666"", ""W1985360540"", ""W1997120704"", ""W2004473150"", ""W2018350044"", ""W2018662377"", ""W2019707963"", ""W2035125948"", ""W2039015864"", ""W2052232008"", ""W2060171891"", ""W2065650425"", ""W2081138872"", ""W2101888865"", ""W2103560726"", ""W2115508955"", ""W2127881962"", ""W2141304880"", ""W2149558765"", ""W2157116313"", ""W2159405134"", ""W2163059972"", ""W2277173515"", ""W2323433666"", ""W2324080278"", ""W2488970184"", ""W2521449021"", ""W2525793566"", ""W2596437707"", ""W2962721974"", ""W2962941654"", ""W2963267448"", ""W2963548102"", ""W2963867777"", ""W2964113089"", ""W3042328837"", ""W3099104772"", ""W3102258829"", ""W3102843666"", ""W3103963141"", ""W4230830673"", ""W4244307463"", ""W4247766333""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 3}, {""year"": 2025, ""cited_by_count"": 4}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 3}, {""year"": 2022, ""cited_by_count"": 5}, {""year"": 2021, ""cited_by_count"": 5}, {""year"": 2020, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 4}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R048,154,Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,Austin2015,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Austin2015,
  author = {Austin, T.},
  title = {Pleasant extensions retaining algebraic structure, {I}},
  journal = {J. Anal. Math.},
  volume = {125},
  year = {2015},
  pages = {1--36},
  doi = {10.1007/s11854-015-0001-9}
}",108,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib#L108,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s11854-015-0001-9"", ""title"": [""Pleasant extensions retaining algebraic structure, I""], ""authors"": [{""given"": ""Tim"", ""family"": ""Austin"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2015, 1]]}, ""container_title"": [""Journal d'Analyse Mathématique""], ""url"": ""https://doi.org/10.1007/s11854-015-0001-9"", ""license"": [{""start"": {""date-parts"": [[2015, 1, 1]], ""date-time"": ""2015-01-01T00:00:00Z"", ""timestamp"": 1420070400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs11854-015-0001-9""}}]","[""https://api.crossref.org/works/10.1007%2Fs11854-015-0001-9""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4244930114"", ""doi"": ""10.1007/s11854-015-0001-9"", ""pmid"": null, ""title"": ""Pleasant extensions retaining algebraic structure, I"", ""publication_year"": 2015, ""publication_date"": ""2015-01-01"", ""type"": ""article"", ""language"": ""fr"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5069962326"", ""name"": ""Tim Austin"", ""orcid"": ""https://orcid.org/0009-0004-5850-7071"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Courant Institute of Mathematical Sciences"", ""New York University""]}], ""source"": {""source_id"": ""S67718071"", ""display_name"": ""Journal d Analyse Mathématique"", ""issn_l"": ""0021-7670"", ""type"": ""journal""}, ""biblio"": {""volume"": ""125"", ""issue"": ""1"", ""first_page"": ""1"", ""last_page"": ""36""}, ""cited_by_count"": 10, ""fwci"": 2.3613, ""referenced_works_count"": 34, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""asymptotic behavior""], ""abstract"": null, ""referenced_works"": [""W807981581"", ""W1483888214"", ""W1490170438"", ""W1566472539"", ""W1607458528"", ""W1950693991"", ""W1970366772"", ""W1970890666"", ""W1978545700"", ""W1999483518"", ""W2072038186"", ""W2081138872"", ""W2083567821"", ""W2088212677"", ""W2103825873"", ""W2111080364"", ""W2119691520"", ""W2127353638"", ""W2141304880"", ""W2149558765"", ""W2157116313"", ""W2159405134"", ""W2504324508"", ""W2583149881"", ""W2962721974"", ""W2962941654"", ""W2962958747"", ""W2963267448"", ""W2963548102"", ""W2964138827"", ""W3102258829"", ""W3102474281"", ""W3102843666"", ""W4292027056""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 2}, {""year"": 2015, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R049,154,Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,Tropp2018,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Tropp2018,
  author = {Tropp, J. A.},
  title = {Second-order matrix concentration inequalities},
  journal = {Appl. Comput. Harmon. Anal.},
  volume = {44},
  number = {3},
  year = {2018},
  pages = {700--736},
  doi = {10.1016/j.acha.2016.07.005}
}",174,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib#L174,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.acha.2016.07.005"", ""title"": [""Second-order matrix concentration inequalities""], ""authors"": [{""given"": ""Joel A."", ""family"": ""Tropp"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2018, 5]]}, ""container_title"": [""Applied and Computational Harmonic Analysis""], ""url"": ""https://doi.org/10.1016/j.acha.2016.07.005"", ""license"": [{""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/legal/tdmrep-license""}, {""start"": {""date-parts"": [[2022, 5, 1]], ""date-time"": ""2022-05-01T00:00:00Z"", ""timestamp"": 1651363200000}, ""content-version"": ""vor"", ""delay-in-days"": 1461, ""URL"": ""http://www.elsevier.com/open-access/userlicense/1.0/""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-017""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-037""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-012""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-029""}, {""start"": {""date-parts"": [[2018, 5, 1]], ""date-time"": ""2018-05-01T00:00:00Z"", ""timestamp"": 1525132800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-004""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.acha.2016.07.005""}}]","[""https://api.crossref.org/works/10.1016%2Fj.acha.2016.07.005""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1922270005"", ""doi"": ""10.1016/j.acha.2016.07.005"", ""pmid"": null, ""title"": ""Second-order matrix concentration inequalities"", ""publication_year"": 2016, ""publication_date"": ""2016-07-29"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5000879912"", ""name"": ""Joel A. Tropp"", ""orcid"": ""https://orcid.org/0000-0003-1024-1791"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S71199174"", ""display_name"": ""Applied and Computational Harmonic Analysis"", ""issn_l"": ""1063-5203"", ""type"": ""journal""}, ""biblio"": {""volume"": ""44"", ""issue"": ""3"", ""first_page"": ""700"", ""last_page"": ""736""}, ""cited_by_count"": 27, ""fwci"": 1.2903, ""referenced_works_count"": 23, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.sciencedirect.com/science/article/am/pii/S1063520316300434?via%3Dihub""}, ""best_oa_pdf_url"": ""https://www.sciencedirect.com/science/article/am/pii/S1063520316300434?via%3Dihub"", ""primary_topic"": {""id"": ""T11716"", ""display_name"": ""Random Matrices and Applications"", ""field"": ""Mathematics"", ""subfield"": ""Statistics and Probability"", ""domain"": ""Physical Sciences""}, ""keywords"": [""matrix concentration inequalities"", ""random matrices"", ""dimension dependence""], ""abstract"": null, ""referenced_works"": [""W591099569"", ""W1574576267"", ""W1586554030"", ""W1938444229"", ""W2025202270"", ""W2047492021"", ""W2077649571"", ""W2091281360"", ""W2107411554"", ""W2119335498"", ""W2131172946"", ""W2144164089"", ""W2955719082"", ""W2964178386"", ""W4206039841"", ""W6601968686"", ""W6617577314"", ""W6618378251"", ""W6621647036"", ""W6633361761"", ""W6637365148"", ""W6679168169"", ""W6750895352""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 4}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 5}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 3}, {""year"": 2020, ""cited_by_count"": 6}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R050,154,Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,DerrienLesigne1996,unverified,unverified,unverified,direct_records,Derrien and Lesigne 1996: bounded discovery searches did not retrieve an original publication record; no matching original was inspected.,"@article{DerrienLesigne1996,
  author = {Derrien, J.-M. and Lesigne, E.},
  title = {Un th{\'e}or{\`e}me ergodique polyn{\^o}mial ponctuel pour les endomorphismes exacts et les {$K$}-syst{\`e}mes},
  journal = {Ann. Inst. H. Poincar{\'e} Probab. Statist.},
  volume = {32},
  number = {6},
  year = {1996},
  pages = {765--778}
}",66,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib#L66,[],[],[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R051,164,Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,ZorinKranich,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{ZorinKranich}
P.~Zorin-Kranich,
\emph{A nilpotent IP polynomial multiple recurrence theorem},
J. Anal. Math. \textbf{123} (2014), 183--225.
\url{https://doi.org/10.1007/s11854-014-0018-5}.
Revised version: arXiv:1206.0287v4, 4 June 2018,
\url{https://arxiv.org/abs/1206.0287v4}.",229,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/references.tex#L229,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s11854-014-0018-5"", ""title"": [""A nilpotent IP polynomial multiple recurrence theorem""], ""authors"": [{""given"": ""Pavel"", ""family"": ""Zorin-Kranich"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2014, 7]]}, ""container_title"": [""Journal d'Analyse Mathématique""], ""url"": ""https://doi.org/10.1007/s11854-014-0018-5"", ""license"": [{""start"": {""date-parts"": [[2014, 7, 1]], ""date-time"": ""2014-07-01T00:00:00Z"", ""timestamp"": 1404172800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs11854-014-0018-5""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1206.0287"", ""version"": 4, ""id_versioned"": ""1206.0287v4"", ""title"": ""A nilpotent IP polynomial multiple recurrence theorem"", ""abstract"": ""We generalize the IP-polynomial Szemerédi theorem due to Bergelson and McCutcheon and the nilpotent Szemerédi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman."", ""authors"": [""Pavel Zorin-Kranich""], ""published"": ""2012-06-01T19:35:30Z"", ""updated"": ""2018-06-04T20:30:42Z"", ""primary_category"": ""math.DS"", ""categories"": [""math.DS"", ""math.CO""], ""doi"": ""10.1007/s11854-014-0018-5"", ""journal_ref"": ""Journal d'Analyse Mathématique 123 (2014) 183-225"", ""comment"": ""v4: switch to TeXlive 2016 and biblatex"", ""abs_url"": ""http://arxiv.org/abs/1206.0287v4"", ""pdf_url"": ""https://arxiv.org/pdf/1206.0287v4""}}]","[""https://api.crossref.org/works/10.1007%2Fs11854-014-0018-5"", ""http://arxiv.org/abs/1206.0287v4""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2163779747"", ""doi"": ""10.1007/s11854-014-0018-5"", ""pmid"": null, ""title"": ""A nilpotent IP polynomial multiple recurrence theorem"", ""publication_year"": 2014, ""publication_date"": ""2014-07-01"", ""type"": ""article"", ""language"": ""fr"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5029215074"", ""name"": ""Pavel Zorin‐Kranich"", ""orcid"": ""https://orcid.org/0000-0002-2153-7689"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Amsterdam""]}], ""source"": {""source_id"": ""S67718071"", ""display_name"": ""Journal d Analyse Mathématique"", ""issn_l"": ""0021-7670"", ""type"": ""journal""}, ""biblio"": {""volume"": ""123"", ""issue"": ""1"", ""first_page"": ""183"", ""last_page"": ""225""}, ""cited_by_count"": 11, ""fwci"": 4.0624, ""referenced_works_count"": 30, ""open_access"": {""is_oa"": true, ""oa_status"": ""green"", ""oa_url"": ""https://arxiv.org/pdf/1206.0287""}, ""best_oa_pdf_url"": ""https://arxiv.org/pdf/1206.0287"", ""primary_topic"": {""id"": ""T11329"", ""display_name"": ""Limits and Structures in Graph Theory"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""nilpotent groups"", ""multiple recurrence"", ""Hales-Jewett theorem"", ""polynomial maps"", ""ergodic theory""], ""abstract"": null, ""referenced_works"": [""W1022649828"", ""W1490170438"", ""W1510024482"", ""W1541824250"", ""W1549382427"", ""W1554682571"", ""W1966794383"", ""W1970366772"", ""W1983242711"", ""W1985404726"", ""W1997681421"", ""W1999853872"", ""W2017788193"", ""W2026998787"", ""W2028782705"", ""W2045286421"", ""W2047891343"", ""W2056043123"", ""W2060449236"", ""W2067546659"", ""W2102115128"", ""W2110056376"", ""W2149883925"", ""W2159405134"", ""W2170315318"", ""W2574237409"", ""W2777506625"", ""W2963575640"", ""W3146830351"", ""W4230191592""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 5}, {""year"": 2013, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R052,164,Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,GreenTaoRegularity,minor_difference,minor_difference,minor_difference,direct_records,Same Green–Tao work/authors and explicitly cited arXiv v3. Citation title includes “an associated counting lemma”; arXiv title omits “an”. Journal/book chapter metadata were not independently verified. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{GreenTaoRegularity}
B.~Green and T.~Tao,
\emph{An arithmetic regularity lemma, an associated counting lemma, and applications},
in \emph{An Irregular Mind: Szemer\'edi is 70},
Bolyai Society Mathematical Studies, Springer, 2010, pp.~261--334.
Corrected version: arXiv:1002.2028v3, 24 November 2020,
\url{https://arxiv.org/abs/1002.2028v3}.",98,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/references.tex#L98,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1002.2028"", ""version"": 3, ""id_versioned"": ""1002.2028v3"", ""title"": ""An arithmetic regularity lemma, associated counting lemma, and applications"", ""abstract"": ""Szemeredi's regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece (a partition into cells with edge densities), a small error (corresponding to irregular cells), and a uniform piece (the pseudorandom deviations from the edge densities). We establish an arithmetic regularity lemma that similarly decomposes bounded functions f : [N] -> C, into a (well-equidistributed, virtual) -step nilsequence, an error which is small in L^2 and a further error which is miniscule in the Gowers U^{s+1}-norm, where s is a positive integer. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f. We provide a number of applications of these lemmas: a proof of Szemeredi's theorem on arithmetic progressions, a proof of a conjecture of Bergelson, Host and Kra, and a generalisation of certain results of Gowers and Wolf. Our result is dependent on the inverse conjecture for the Gowers U^{s+1} norm, recently established for general s by the authors and T. Ziegler. REVISED November 2020: This paper has been revised so that only systems of linear forms satisfying a condition called the flag property are covered by the counting lemma. Translation-invariant systems, as well as systems of Cauchy-Schwarz complexity 1 and some other naturally-occurring systems have this property. We thank Daniel Altman for drawing our attention to what appeared to be a minor technical issue in one of our proofs but which ultimately led us to realise that the the counting lemma fails quite badly (with rather simple examples, which we shall describe) without some assumption of this type. Additionally, we supply a short appendix showing that the Gowers-Wolf complexity of a system is bounded by its Cauchy-Schwarz complexity."", ""authors"": [""Ben Green"", ""Terence Tao""], ""published"": ""2010-02-10T17:45:01Z"", ""updated"": ""2020-11-24T22:44:32Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO"", ""math.DS""], ""doi"": null, ""journal_ref"": null, ""comment"": ""61 pages, corrected version of article which appeared in An Irregular Mind, Szemeredi is 70"", ""abs_url"": ""http://arxiv.org/abs/1002.2028v3"", ""pdf_url"": ""https://arxiv.org/pdf/1002.2028v3""}}]","[""http://arxiv.org/abs/1002.2028v3""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R053,164,Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,HostKra,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{HostKra}
B.~Host and B.~Kra,
\emph{Nonconventional ergodic averages and nilmanifolds},
Ann. of Math. (2) \textbf{161} (2005), no.~1, 397--488.
\url{https://doi.org/10.4007/annals.2005.161.397}.",150,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/references.tex#L150,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2005.161.397"", ""title"": [""Nonconventional ergodic averages and nilmanifolds""], ""authors"": [{""given"": ""Bernard"", ""family"": ""Host"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Bryna"", ""family"": ""Kra"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 1, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2005.161.397"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2005.161.397""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2005.161.397""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2141304880"", ""doi"": ""10.4007/annals.2005.161.397"", ""pmid"": null, ""title"": ""Nonconventional ergodic averages and nilmanifolds"", ""publication_year"": 2005, ""publication_date"": ""2005-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5079711840"", ""name"": ""Bernard Host"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Université Paris-Est Marne-la-Vallée""]}, {""author_id"": ""A5014788190"", ""name"": ""Bryna Kra"", ""orcid"": ""https://orcid.org/0000-0002-5301-3839"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Northwestern University""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""161"", ""issue"": ""1"", ""first_page"": ""397"", ""last_page"": ""488""}, ""cited_by_count"": 475, ""fwci"": 29.3262, ""referenced_works_count"": 36, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p08.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p08.pdf"", ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""ergodic averages"", ""mean square convergence"", ""nilmanifolds"", ""characteristic factors"", ""Szemerédi's theorem"", ""upper density""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2005.161.397"", ""referenced_works"": [""W1489644082"", ""W1507992226"", ""W1549382427"", ""W1607458528"", ""W1970366772"", ""W1974782890"", ""W1999254158"", ""W2011036058"", ""W2012588304"", ""W2039548965"", ""W2045286421"", ""W2053652533"", ""W2076884693"", ""W2083309269"", ""W2083567821"", ""W2090467845"", ""W2119691520"", ""W2141416360"", ""W2149558765"", ""W2159405134"", ""W2324080278"", ""W2327101409"", ""W2334405238"", ""W2335400162"", ""W2488970184"", ""W2521449021"", ""W2574237409"", ""W2578107452"", ""W2583149881"", ""W2613316504"", ""W3010677823"", ""W4205356207"", ""W4285719527"", ""W6602750635"", ""W6731585797"", ""W6826470420""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 12}, {""year"": 2025, ""cited_by_count"": 19}, {""year"": 2024, ""cited_by_count"": 16}, {""year"": 2023, ""cited_by_count"": 29}, {""year"": 2022, ""cited_by_count"": 20}, {""year"": 2021, ""cited_by_count"": 18}, {""year"": 2020, ""cited_by_count"": 23}, {""year"": 2019, ""cited_by_count"": 18}, {""year"": 2018, ""cited_by_count"": 25}, {""year"": 2017, ""cited_by_count"": 12}, {""year"": 2016, ""cited_by_count"": 30}, {""year"": 2015, ""cited_by_count"": 24}, {""year"": 2014, ""cited_by_count"": 22}, {""year"": 2013, ""cited_by_count"": 25}, {""year"": 2012, ""cited_by_count"": 24}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R054,164,Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,GowersDense,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{GowersDense}
W.~T.~Gowers,
\emph{Decompositions, approximate structure, transference, and the Hahn--Banach theorem},
Bull. Lond. Math. Soc. \textbf{42} (2010), no.~4, 573--606.
\url{https://doi.org/10.1112/blms/bdq018}.",60,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/references.tex#L60,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1112/blms/bdq018"", ""title"": [""Decompositions, approximate structure, transference, and the Hahn-Banach theorem""], ""authors"": [{""given"": ""W. T."", ""family"": ""Gowers"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Wiley"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2010, 6, 27]]}, ""container_title"": [""Bulletin of the London Mathematical Society""], ""url"": ""https://doi.org/10.1112/blms/bdq018"", ""license"": [{""start"": {""date-parts"": [[2010, 6, 27]], ""date-time"": ""2010-06-27T00:00:00Z"", ""timestamp"": 1277596800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://doi.wiley.com/10.1002/tdm_license_1.1""}, {""start"": {""date-parts"": [[2010, 6, 27]], ""date-time"": ""2010-06-27T00:00:00Z"", ""timestamp"": 1277596800000}, ""content-version"": ""vor"", ""delay-in-days"": 0, ""URL"": ""http://onlinelibrary.wiley.com/termsAndConditions#vor""}], ""source_url"": ""https://api.crossref.org/works/10.1112%2Fblms%2Fbdq018""}}]","[""https://api.crossref.org/works/10.1112%2Fblms%2Fbdq018""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2150482562"", ""doi"": ""10.1112/blms/bdq018"", ""pmid"": null, ""title"": ""Decompositions, approximate structure, transference, and the Hahn-Banach theorem"", ""publication_year"": 2010, ""publication_date"": ""2010-06-27"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""name"": ""W. T. Gowers"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Cambridge""]}], ""source"": {""source_id"": ""S39770307"", ""display_name"": ""Bulletin of the London Mathematical Society"", ""issn_l"": ""0024-6093"", ""type"": ""journal""}, ""biblio"": {""volume"": ""42"", ""issue"": ""4"", ""first_page"": ""573"", ""last_page"": ""606""}, ""cited_by_count"": 49, ""fwci"": 4.0054, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": true, ""oa_status"": ""green"", ""oa_url"": ""https://arxiv.org/pdf/0811.3103""}, ""best_oa_pdf_url"": ""https://arxiv.org/pdf/0811.3103"", ""primary_topic"": {""id"": ""T11329"", ""display_name"": ""Limits and Structures in Graph Theory"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""decomposition theorem"", ""transference principle"", ""Hahn-Banach theorem"", ""additive combinatorics"", ""extremal combinatorics"", ""Green-Tao theorem""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1112/blms/bdq018"", ""referenced_works"": [], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 4}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 5}, {""year"": 2018, ""cited_by_count"": 4}, {""year"": 2017, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 8}, {""year"": 2015, ""cited_by_count"": 4}, {""year"": 2014, ""cited_by_count"": 4}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R055,164,Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026,GTZ,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{GTZ}
B.~Green, T.~Tao and T.~Ziegler,
\emph{An inverse theorem for the Gowers $U^{s+1}[N]$-norm},
Ann. of Math. (2) \textbf{176} (2012), no.~2, 1231--1372.
\url{https://doi.org/10.4007/annals.2012.176.2.11}.
Further revision: arXiv:1009.3998v5, 23 April 2026,
\url{https://arxiv.org/abs/1009.3998v5}.",106,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/references.tex#L106,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2012.176.2.11"", ""title"": [""An inverse theorem for the Gowers U^(s+1)[N]-norm""], ""authors"": [{""given"": ""Ben"", ""family"": ""Green"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Terence"", ""family"": ""Tao"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Tamar"", ""family"": ""Ziegler"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2012, 9, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2012.176.2.11"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2012.176.2.11""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1009.3998"", ""version"": 5, ""id_versioned"": ""1009.3998v5"", ""title"": ""An inverse theorem for the Gowers U^{s+1}[N]-norm"", ""abstract"": ""We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} > δthen there is a bounded-complexity s-step nilsequence F(g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. A 6-page erratum to the original paper was provided in April 2024 and is available as a separate PDF on the webpages of the first and second authors."", ""authors"": [""Ben Green"", ""Terence Tao"", ""Tamar Ziegler""], ""published"": ""2010-09-21T05:41:11Z"", ""updated"": ""2026-04-23T12:20:55Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO"", ""math.DS""], ""doi"": null, ""journal_ref"": null, ""comment"": ""Appeared as Annals of Math. 176 (2012), 1231--1372. Preprint is 116 pages. This version corrects a mistake in the proof of Lemma 13.2 (the wrong filtration was specified) and two small typos on page 28"", ""abs_url"": ""http://arxiv.org/abs/1009.3998v5"", ""pdf_url"": ""https://arxiv.org/pdf/1009.3998v5""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2012.176.2.11"", ""http://arxiv.org/abs/1009.3998v5""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2777506625"", ""doi"": ""10.4007/annals.2012.176.2.11"", ""pmid"": null, ""title"": ""An inverse theorem for the Gowers U^(s+1)[N]-norm"", ""publication_year"": 2012, ""publication_date"": ""2012-07-03"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5100610718"", ""name"": ""Ben Green"", ""orcid"": ""https://orcid.org/0000-0002-2224-1193"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of Cambridge""]}, {""author_id"": ""A5066728260"", ""name"": ""Terence Tao"", ""orcid"": ""https://orcid.org/0000-0002-0140-7641"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""University of California, Los Angeles""]}, {""author_id"": ""A5091670796"", ""name"": ""Tamar Ziegler"", ""orcid"": ""https://orcid.org/0000-0001-6423-7072"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Technion – Israel Institute of Technology""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""176"", ""issue"": ""2"", ""first_page"": ""1231"", ""last_page"": ""1372""}, ""cited_by_count"": 153, ""fwci"": 15.539, ""referenced_works_count"": 56, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf"", ""primary_topic"": {""id"": ""T11329"", ""display_name"": ""Limits and Structures in Graph Theory"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""nilsequences""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2012.176.2.11"", ""referenced_works"": [""W647913949"", ""W984484859"", ""W1510024482"", ""W1554682571"", ""W1582583909"", ""W1585475679"", ""W1603206290"", ""W1617056419"", ""W1663694896"", ""W1675214544"", ""W1837752427"", ""W1950693991"", ""W1970366772"", ""W1970446299"", ""W1985404726"", ""W1985761146"", ""W1996922551"", ""W2011552750"", ""W2017788193"", ""W2020229390"", ""W2039015864"", ""W2042383482"", ""W2044872526"", ""W2064334110"", ""W2064469747"", ""W2072504509"", ""W2074891711"", ""W2091329824"", ""W2097673562"", ""W2115342189"", ""W2119691520"", ""W2127881962"", ""W2140492814"", ""W2141304880"", ""W2149558765"", ""W2158593242"", ""W2159063988"", ""W2165859273"", ""W2176761779"", ""W2335400162"", ""W2342433841"", ""W2484570010"", ""W2504324508"", ""W2912209775"", ""W2950588683"", ""W2963404131"", ""W2963575640"", ""W3102258829"", ""W3102618960"", ""W3106502785"", ""W3124591719"", ""W4248158774"", ""W4255834797"", ""W4388432573"", ""W4394665965"", ""W4394667327""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 7}, {""year"": 2025, ""cited_by_count"": 10}, {""year"": 2024, ""cited_by_count"": 7}, {""year"": 2023, ""cited_by_count"": 14}, {""year"": 2022, ""cited_by_count"": 9}, {""year"": 2021, ""cited_by_count"": 9}, {""year"": 2020, ""cited_by_count"": 10}, {""year"": 2019, ""cited_by_count"": 13}, {""year"": 2018, ""cited_by_count"": 13}, {""year"": 2017, ""cited_by_count"": 11}, {""year"": 2016, ""cited_by_count"": 15}, {""year"": 2015, ""cited_by_count"": 13}, {""year"": 2014, ""cited_by_count"": 10}, {""year"": 2013, ""cited_by_count"": 6}, {""year"": 2012, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R056,192,Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,ODonnellServedio,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{ODonnellServedio}
R. O'Donnell and R. A. Servedio,
Learning monotone decision trees in polynomial time,
\emph{SIAM Journal on Computing} \textbf{37} (2007), 827--844.
\url{https://doi.org/10.1137/060669309}.",45,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/bibliography.tex#L45,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1137/060669309"", ""title"": [""Learning Monotone Decision Trees in Polynomial Time""], ""authors"": [{""given"": ""Ryan"", ""family"": ""O'Donnell"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Rocco A."", ""family"": ""Servedio"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Society for Industrial & Applied Mathematics (SIAM)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2007, 1]]}, ""container_title"": [""SIAM Journal on Computing""], ""url"": ""https://doi.org/10.1137/060669309"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1137%2F060669309""}}]","[""https://api.crossref.org/works/10.1137%2F060669309""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2029370139"", ""doi"": ""10.1137/060669309"", ""pmid"": null, ""title"": ""Learning Monotone Decision Trees in Polynomial Time"", ""publication_year"": 2007, ""publication_date"": ""2007-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5051167298"", ""name"": ""Ryan O’Donnell"", ""orcid"": ""https://orcid.org/0009-0005-5609-5795"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5014866889"", ""name"": ""Rocco A. Servedio"", ""orcid"": ""https://orcid.org/0000-0003-2407-543X"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": []}], ""source"": {""source_id"": ""S153560523"", ""display_name"": ""SIAM Journal on Computing"", ""issn_l"": ""0097-5397"", ""type"": ""journal""}, ""biblio"": {""volume"": ""37"", ""issue"": ""3"", ""first_page"": ""827"", ""last_page"": ""844""}, ""cited_by_count"": 99, ""fwci"": 6.805, ""referenced_works_count"": 20, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10720"", ""display_name"": ""Complexity and Algorithms in Graphs"", ""field"": ""Computer Science"", ""subfield"": ""Computational Theory and Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""monotone Boolean functions"", ""decision tree induction"", ""average sensitivity"", ""real functions"", ""computational learning theory"", ""polynomial-time algorithms""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1137/060669309"", ""referenced_works"": [""W10927739"", ""W1519375940"", ""W1526215707"", ""W1529556207"", ""W1969880832"", ""W1986296546"", ""W1989370476"", ""W2008470071"", ""W2011717937"", ""W2029286977"", ""W2042194938"", ""W2043842763"", ""W2070902649"", ""W2088390444"", ""W2100654536"", ""W2117362057"", ""W2168115971"", ""W2619938522"", ""W3148810677"", ""W4238893454""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 3}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2023, ""cited_by_count"": 3}, {""year"": 2022, ""cited_by_count"": 4}, {""year"": 2021, ""cited_by_count"": 13}, {""year"": 2020, ""cited_by_count"": 7}, {""year"": 2019, ""cited_by_count"": 4}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 3}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 9}, {""year"": 2013, ""cited_by_count"": 10}, {""year"": 2012, ""cited_by_count"": 4}]}}]","[{""sample_id"": ""R056"", ""source_url"": ""https://www.cs.columbia.edu/~rocco/Public/os-ccc.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""5c303739310f8463d47603e2e9f87e4d0cab1e3e46ce6c304e5523ade70c8e21"", ""locator"": ""PDF pages 6–7 (printed pages 5–6), Lemma 3 and Theorem 3"", ""source_version"": ""Author draft 2005-12-03"", ""paraphrased_observations"": ""Lemma 3 expresses the signed sum of singleton Fourier coefficients as an expectation over a subcube partition. Theorem 3 and its proof bound this signed sum by the square root of the expected number of fixed coordinates, and then by the square root of log of partition size. The signed-sum inequality applies to Boolean functions; interpreting it as total influence requires monotonicity. The inspected source is an author draft dated December 3, 2005, not the final 2007 journal PDF."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R057,192,Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,Wang,unverified,unverified,unverified,direct_records,Wang ePrint 2020/002: original ePrint site access denied; discovery results did not establish this report.,"\bibitem{Wang}
Q. Wang, \emph{On a Conjecture of O'Donnell},
Cryptology ePrint Archive, Report 2020/002 (2020).
\url{https://eprint.iacr.org/2020/002}.",51,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/bibliography.tex#L51,[],[],[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R058,192,Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,Durrett,unverified,unverified,match,supplied_notes_identity,"Supplied notes identify Durrett, Probability: Theory and Examples, author manuscript Version 5, January 11, 2019. Commercial fifth-edition metadata remains unverified.","\bibitem{Durrett}
R. Durrett, \emph{Probability: Theory and Examples}, fifth edition,
Cambridge University Press, 2019.
Author's Version~5, January~11, 2019,
\url{https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf}.",8,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/bibliography.tex#L8,[],[],[],"[{""sample_id"": ""R058"", ""source_url"": ""https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""aeac36cbf5e44c53d69fa60a2d29a393e2d0e8c955ee103bd845d925fd910886"", ""locator"": ""PDF page 1"", ""source_version"": ""Author manuscript Version 5"", ""paraphrased_observations"": ""The title page identifies Rick Durrett, Probability: Theory and Examples, Version 5, January 11, 2019. It is an author-hosted manuscript and is not by itself evidence for all details of a commercially published fifth edition."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R059,192,Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,KudinPasalic,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{KudinPasalic}
S. Kudin and E. Pa\v{s}ali\'c,
Proving the conjecture of O'Donnell in certain cases and disproving
its general validity,
\emph{Discrete Applied Mathematics} \textbf{289} (2021), 345--353.
\url{https://doi.org/10.1016/j.dam.2020.11.005}.",27,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/bibliography.tex#L27,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.dam.2020.11.005"", ""title"": [""Proving the conjecture of O’Donnell in certain cases and disproving its general validity""], ""authors"": [{""given"": ""S."", ""family"": ""Kudin"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""E."", ""family"": ""Pasalic"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2021, 1]]}, ""container_title"": [""Discrete Applied Mathematics""], ""url"": ""https://doi.org/10.1016/j.dam.2020.11.005"", ""license"": [{""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/legal/tdmrep-license""}, {""start"": {""date-parts"": [[2025, 1, 31]], ""date-time"": ""2025-01-31T00:00:00Z"", ""timestamp"": 1738281600000}, ""content-version"": ""vor"", ""delay-in-days"": 1491, ""URL"": ""http://www.elsevier.com/open-access/userlicense/1.0/""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-017""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-037""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-012""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-029""}, {""start"": {""date-parts"": [[2021, 1, 1]], ""date-time"": ""2021-01-01T00:00:00Z"", ""timestamp"": 1609459200000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-004""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.dam.2020.11.005""}}]","[""https://api.crossref.org/works/10.1016%2Fj.dam.2020.11.005""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W3100495479"", ""doi"": ""10.1016/j.dam.2020.11.005"", ""pmid"": null, ""title"": ""Proving the conjecture of O’Donnell in certain cases and disproving its general validity"", ""publication_year"": 2020, ""publication_date"": ""2020-11-19"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5075799044"", ""name"": ""Sadmir Kudin"", ""orcid"": ""https://orcid.org/0009-0000-4516-7160"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of Primorska""]}, {""author_id"": ""A5039271966"", ""name"": ""Enes Pašalić"", ""orcid"": ""https://orcid.org/0000-0001-6343-8796"", ""position"": ""last"", ""is_corresponding"": true, ""institutions"": [""University of Primorska""]}], ""source"": {""source_id"": ""S171741597"", ""display_name"": ""Discrete Applied Mathematics"", ""issn_l"": ""0166-218X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""289"", ""issue"": null, ""first_page"": ""345"", ""last_page"": ""353""}, ""cited_by_count"": 0, ""fwci"": 0, ""referenced_works_count"": 24, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1016/j.dam.2020.11.005""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11130"", ""display_name"": ""Coding theory and cryptography"", ""field"": ""Computer Science"", ""subfield"": ""Artificial Intelligence"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Boolean functions"", ""social choice"", ""counterexample"", ""Boolean hypercube"", ""Walsh-Fourier transform""], ""abstract"": null, ""referenced_works"": [""W1512363553"", ""W1565496777"", ""W1571317543"", ""W1589266564"", ""W1838666760"", ""W1891181203"", ""W1945290888"", ""W1980279697"", ""W2010723297"", ""W2026474140"", ""W2095027936"", ""W2504274129"", ""W2565555098"", ""W2776245944"", ""W2779214179"", ""W2798612817"", ""W2903744383"", ""W2911925211"", ""W3004304484"", ""W4212960282"", ""W4293362586"", ""W6635302384"", ""W6724579519"", ""W6750921750""], ""counts_by_year"": []}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R060,192,Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,Jha,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{Jha}
S. K. Jha, \emph{On the Sum of Linear Coefficients of a Boolean Valued
Function}, arXiv:1611.01029v2 (2016).
\url{https://arxiv.org/abs/1611.01029v2}.",22,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/bibliography.tex#L22,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1611.01029"", ""version"": 2, ""id_versioned"": ""1611.01029v2"", ""title"": ""On the Sum of Linear Coefficients of a Boolean Valued Function"", ""abstract"": ""Let $f:\\{-1,1\\}^{n}\\rightarrow \\{-1,1\\}$ be a Boolean valued function having total degree $d$. Then a conjecture due to Servedio and Gopalan asserts that $\\sum_{i=1}^{n}\\widehat{f}(i)\\leq \\sum_{j=1}^{d}\\widehat{\\text{Maj}}_{d}(j)$ where $\\text{Maj}_{d}$ is the majority function on $d$ bits. Here we give some alternative formalisms of this conjecture involving the discrete derivative operators on $f$."", ""authors"": [""Sumit Kumar Jha""], ""published"": ""2016-11-03T14:10:17Z"", ""updated"": ""2016-11-06T09:06:47Z"", ""primary_category"": ""cs.CC"", ""categories"": [""cs.CC""], ""doi"": null, ""journal_ref"": null, ""comment"": null, ""abs_url"": ""http://arxiv.org/abs/1611.01029v2"", ""pdf_url"": ""https://arxiv.org/pdf/1611.01029v2""}}]","[""http://arxiv.org/abs/1611.01029v2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R061,210,Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,McKay2008,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{McKay2008,
  author = {McKay, Tom},
  title = {On plethysm conjectures of {Stanley} and {Foulkes}},
  journal = {Journal of Algebra},
  volume = {319},
  number = {5},
  year = {2008},
  pages = {2050--2071},
  note = {\href{https://doi.org/10.1016/j.jalgebra.2007.12.003}{doi: 10.1016/j.jalgebra.2007.12.003}},
  doi = {10.1016/j.jalgebra.2007.12.003}
}",72,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/references.bib#L72,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.jalgebra.2007.12.003"", ""title"": [""On Plethysm conjectures of Stanley and Foulkes""], ""authors"": [{""given"": ""Tom"", ""family"": ""McKay"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2008, 3]]}, ""container_title"": [""Journal of Algebra""], ""url"": ""https://doi.org/10.1016/j.jalgebra.2007.12.003"", ""license"": [{""start"": {""date-parts"": [[2008, 3, 1]], ""date-time"": ""2008-03-01T00:00:00Z"", ""timestamp"": 1204329600000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2013, 7, 17]], ""date-time"": ""2013-07-17T00:00:00Z"", ""timestamp"": 1374019200000}, ""content-version"": ""vor"", ""delay-in-days"": 1964, ""URL"": ""https://www.elsevier.com/open-access/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.jalgebra.2007.12.003""}}]","[""https://api.crossref.org/works/10.1016%2Fj.jalgebra.2007.12.003""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1985585355"", ""doi"": ""10.1016/j.jalgebra.2007.12.003"", ""pmid"": null, ""title"": ""On Plethysm conjectures of Stanley and Foulkes"", ""publication_year"": 2008, ""publication_date"": ""2008-01-30"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""name"": ""Tom McKay"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of East Anglia""]}], ""source"": {""source_id"": ""S184147796"", ""display_name"": ""Journal of Algebra"", ""issn_l"": ""0021-8693"", ""type"": ""journal""}, ""biblio"": {""volume"": ""319"", ""issue"": ""5"", ""first_page"": ""2050"", ""last_page"": ""2071""}, ""cited_by_count"": 20, ""fwci"": 0.5855, ""referenced_works_count"": 24, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1016/j.jalgebra.2007.12.003""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10948"", ""display_name"": ""Advanced Combinatorial Mathematics"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""plethysm"", ""symmetric group representations"", ""Specht modules"", ""Young tableaux"", ""partitions"", ""dominance order"", ""column addition""], ""abstract"": null, ""referenced_works"": [""W140315353"", ""W1503154552"", ""W1509187629"", ""W1526458291"", ""W1964737112"", ""W1966056518"", ""W1996834903"", ""W1997431816"", ""W2012855802"", ""W2047023672"", ""W2052855066"", ""W2057979593"", ""W2074693285"", ""W2107061370"", ""W2111604338"", ""W2134984950"", ""W2488278335"", ""W2503354009"", ""W4214684581"", ""W4300990447"", ""W6679742276"", ""W6724813748"", ""W6821281073"", ""W7042800306""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 3}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 2}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R062,210,Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,DentSiemons2000,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{DentSiemons2000,
  author = {Dent, Suzie C. and Siemons, Johannes},
  title = {On a conjecture of {Foulkes}},
  journal = {Journal of Algebra},
  volume = {226},
  number = {1},
  year = {2000},
  pages = {236--249},
  note = {\href{https://doi.org/10.1006/jabr.1999.8169}{doi: 10.1006/jabr.1999.8169}},
  doi = {10.1006/jabr.1999.8169}
}",48,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/references.bib#L48,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1006/jabr.1999.8169"", ""title"": [""On a Conjecture of Foulkes""], ""authors"": [{""given"": ""Suzie C."", ""family"": ""Dent"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Johannes"", ""family"": ""Siemons"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2000, 4]]}, ""container_title"": [""Journal of Algebra""], ""url"": ""https://doi.org/10.1006/jabr.1999.8169"", ""license"": [{""start"": {""date-parts"": [[2000, 4, 1]], ""date-time"": ""2000-04-01T00:00:00Z"", ""timestamp"": 954547200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2013, 7, 17]], ""date-time"": ""2013-07-17T00:00:00Z"", ""timestamp"": 1374019200000}, ""content-version"": ""vor"", ""delay-in-days"": 4855, ""URL"": ""https://www.elsevier.com/open-access/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1006%2Fjabr.1999.8169""}}]","[""https://api.crossref.org/works/10.1006%2Fjabr.1999.8169""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2057979593"", ""doi"": ""10.1006/jabr.1999.8169"", ""pmid"": null, ""title"": ""On a Conjecture of Foulkes"", ""publication_year"": 2000, ""publication_date"": ""2000-04-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5042921951"", ""name"": ""Suzie C. Dent"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of East Anglia""]}, {""author_id"": ""A5025187027"", ""name"": ""Johannes Siemons"", ""orcid"": ""https://orcid.org/0000-0003-3986-0545"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of East Anglia""]}], ""source"": {""source_id"": ""S184147796"", ""display_name"": ""Journal of Algebra"", ""issn_l"": ""0021-8693"", ""type"": ""journal""}, ""biblio"": {""volume"": ""226"", ""issue"": ""1"", ""first_page"": ""236"", ""last_page"": ""249""}, ""cited_by_count"": 25, ""fwci"": 0, ""referenced_works_count"": 22, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1006/jabr.1999.8169""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10849"", ""display_name"": ""Finite Group Theory Research"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""permutation modules"", ""symmetric group representations"", ""module decomposition""], ""abstract"": null, ""referenced_works"": [""W165378800"", ""W850723756"", ""W1503154552"", ""W1515761639"", ""W1600137463"", ""W1964737112"", ""W1994354349"", ""W2006844964"", ""W2012855802"", ""W2033944624"", ""W2066268301"", ""W2074693285"", ""W2312858861"", ""W2322515038"", ""W2330548796"", ""W2488278335"", ""W2503354009"", ""W3143671738"", ""W4206811917"", ""W4214684581"", ""W4253466435"", ""W4301015504""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 2}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 2}, {""year"": 2015, ""cited_by_count"": 2}, {""year"": 2014, ""cited_by_count"": 2}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R063,210,Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,MatolcsiRuzsa2010,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@incollection{MatolcsiRuzsa2010,
  author = {Matolcsi, M{\'a}t{\'e} and Ruzsa, Imre Z.},
  title = {Sumsets and the convex hull},
  booktitle = {Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson},
  editor = {Chudnovsky, David and Chudnovsky, Gregory},
  publisher = {Springer},
  year = {2010},
  pages = {221--227},
  note = {\href{https://arxiv.org/abs/0810.1485v1}{arXiv:0810.1485v1}. \href{https://doi.org/10.1007/978-0-387-68361-4_16}{doi: 10.1007/978-0-387-68361-4\_16}},
  doi = {10.1007/978-0-387-68361-4_16},
  eprint = {0810.1485v1},
  archivePrefix = {arXiv},
  primaryClass = {math.CO},
  url = {https://arxiv.org/abs/0810.1485v1}
}",216,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/references.bib#L216,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/978-0-387-68361-4_16"", ""title"": [""Sumsets and the Convex Hull""], ""authors"": [{""given"": ""Máté"", ""family"": ""Matolcsi"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Imre Z."", ""family"": ""Ruzsa"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Springer New York"", ""type"": ""book-chapter"", ""published"": {""date-parts"": [[2010]]}, ""container_title"": [""Additive Number Theory""], ""url"": ""https://doi.org/10.1007/978-0-387-68361-4_16"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1007%2F978-0-387-68361-4_16""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""0810.1485"", ""version"": 1, ""id_versioned"": ""0810.1485v1"", ""title"": ""Sumsets and the convex hull"", ""abstract"": ""We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times."", ""authors"": [""Mate Matolcsi"", ""Imre Z. Ruzsa""], ""published"": ""2008-10-08T17:09:42Z"", ""updated"": ""2008-10-08T17:09:42Z"", ""primary_category"": ""math.CO"", ""categories"": [""math.CO""], ""doi"": null, ""journal_ref"": null, ""comment"": ""6 pages"", ""abs_url"": ""http://arxiv.org/abs/0810.1485v1"", ""pdf_url"": ""https://arxiv.org/pdf/0810.1485v1""}}]","[""https://api.crossref.org/works/10.1007%2F978-0-387-68361-4_16"", ""http://arxiv.org/abs/0810.1485v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1723055616"", ""doi"": ""10.1007/978-0-387-68361-4_16"", ""pmid"": null, ""title"": ""Sumsets and the Convex Hull"", ""publication_year"": 2010, ""publication_date"": ""2010-01-01"", ""type"": ""book-chapter"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5016481249"", ""name"": ""Máté Matolcsi"", ""orcid"": ""https://orcid.org/0000-0003-4889-697X"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""HUN-REN Alfréd Rényi Institute of Mathematics"", ""Budapest University of Technology and Economics""]}, {""author_id"": ""A5079708499"", ""name"": ""Imre Z. Ruzsa"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""HUN-REN Alfréd Rényi Institute of Mathematics""]}], ""source"": {""source_id"": ""S4306463937"", ""display_name"": ""Springer eBooks"", ""issn_l"": null, ""type"": ""ebook platform""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": ""221"", ""last_page"": ""227""}, ""cited_by_count"": 7, ""fwci"": 2.3671, ""referenced_works_count"": 2, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10948"", ""display_name"": ""Advanced Combinatorial Mathematics"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""sumsets"", ""convex hull"", ""additive combinatorics"", ""set inclusion""], ""abstract"": null, ""referenced_works"": [""W1555768633"", ""W1999339208""], ""counts_by_year"": [{""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2014, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R064,210,Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,MuellerNeunhoeffer2005,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{MuellerNeunhoeffer2005,
  author = {M{\""u}ller, J{\""u}rgen and Neunh{\""o}ffer, Max},
  title = {Some computations regarding {Foulkes}' conjecture},
  journal = {Experimental Mathematics},
  volume = {14},
  number = {3},
  year = {2005},
  pages = {277--283},
  note = {\href{https://doi.org/10.1080/10586458.2005.10128928}{doi: 10.1080/10586458.2005.10128928}},
  doi = {10.1080/10586458.2005.10128928}
}",60,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/references.bib#L60,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1080/10586458.2005.10128928"", ""title"": [""Some Computations Regarding Foulkes' Conjecture""], ""authors"": [{""given"": ""Jurgen"", ""family"": ""Müller"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Max"", ""family"": ""Neunhöffer"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Informa UK Limited"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 1]]}, ""container_title"": [""Experimental Mathematics""], ""url"": ""https://doi.org/10.1080/10586458.2005.10128928"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1080%2F10586458.2005.10128928""}}]","[""https://api.crossref.org/works/10.1080%2F10586458.2005.10128928""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1997431816"", ""doi"": ""10.1080/10586458.2005.10128928"", ""pmid"": null, ""title"": ""Some Computations Regarding Foulkes' Conjecture"", ""publication_year"": 2005, ""publication_date"": ""2005-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5101717295"", ""name"": ""Jürgen Müller"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""RWTH Aachen University""]}, {""author_id"": ""A5001762342"", ""name"": ""Max Neunhöffer"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""RWTH Aachen University""]}], ""source"": {""source_id"": ""S119636414"", ""display_name"": ""Experimental Mathematics"", ""issn_l"": ""1058-6458"", ""type"": ""journal""}, ""biblio"": {""volume"": ""14"", ""issue"": ""3"", ""first_page"": ""277"", ""last_page"": ""283""}, ""cited_by_count"": 27, ""fwci"": 1.058, ""referenced_works_count"": 9, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10849"", ""display_name"": ""Finite Group Theory Research"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""symmetric group"", ""permutation actions"", ""distributed computing"", ""workstation clusters"", ""computational group theory""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1080/10586458.2005.10128928"", ""referenced_works"": [""W140315353"", ""W165378800"", ""W1503154552"", ""W1527800887"", ""W1964737112"", ""W2015855792"", ""W2074693285"", ""W2152815562"", ""W2574262255""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 2}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 3}, {""year"": 2013, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R065,210,Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,BurgisserChristandlIkenmeyer2011,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{BurgisserChristandlIkenmeyer2011,
  author = {B{\""u}rgisser, Peter and Christandl, Matthias and Ikenmeyer, Christian},
  title = {Even partitions in plethysms},
  journal = {Journal of Algebra},
  volume = {328},
  number = {1},
  year = {2011},
  pages = {322--329},
  note = {\href{https://arxiv.org/abs/1003.4474v2}{arXiv:1003.4474v2}. \href{https://doi.org/10.1016/j.jalgebra.2010.10.031}{doi: 10.1016/\allowbreak j.jalgebra.2010.10.031}},
  doi = {10.1016/j.jalgebra.2010.10.031},
  eprint = {1003.4474v2},
  archivePrefix = {arXiv},
  primaryClass = {math.GR}
}",262,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/references.bib#L262,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.jalgebra.2010.10.031"", ""title"": [""Even partitions in plethysms""], ""authors"": [{""given"": ""Peter"", ""family"": ""Bürgisser"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Matthias"", ""family"": ""Christandl"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Christian"", ""family"": ""Ikenmeyer"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2011, 2]]}, ""container_title"": [""Journal of Algebra""], ""url"": ""https://doi.org/10.1016/j.jalgebra.2010.10.031"", ""license"": [{""start"": {""date-parts"": [[2011, 2, 1]], ""date-time"": ""2011-02-01T00:00:00Z"", ""timestamp"": 1296518400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2015, 2, 15]], ""date-time"": ""2015-02-15T00:00:00Z"", ""timestamp"": 1423958400000}, ""content-version"": ""vor"", ""delay-in-days"": 1475, ""URL"": ""https://www.elsevier.com/open-access/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.jalgebra.2010.10.031""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1003.4474"", ""version"": 2, ""id_versioned"": ""1003.4474v2"", ""title"": ""Even Partitions in Plethysms"", ""abstract"": ""We prove that for all natural numbers k,n,d with k <= d and every partition lambda of size kn with at most k parts there exists an irreducible GL(d, C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n) (C^d)). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory."", ""authors"": [""Peter Bürgisser"", ""Matthias Christandl"", ""Christian Ikenmeyer""], ""published"": ""2010-03-23T17:24:49Z"", ""updated"": ""2010-12-15T10:55:48Z"", ""primary_category"": ""math.GR"", ""categories"": [""math.GR"", ""quant-ph""], ""doi"": ""10.1016/j.jalgebra.2010.10.031"", ""journal_ref"": ""Journal of Algebra 328, 322-329 (2011)"", ""comment"": ""9 pages"", ""abs_url"": ""http://arxiv.org/abs/1003.4474v2"", ""pdf_url"": ""https://arxiv.org/pdf/1003.4474v2""}}]","[""https://api.crossref.org/works/10.1016%2Fj.jalgebra.2010.10.031"", ""http://arxiv.org/abs/1003.4474v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2094971045"", ""doi"": ""10.1016/j.jalgebra.2010.10.031"", ""pmid"": null, ""title"": ""Even partitions in plethysms"", ""publication_year"": 2010, ""publication_date"": ""2010-11-30"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5029244957"", ""name"": ""Peter Bürgisser"", ""orcid"": ""https://orcid.org/0000-0001-8169-0514"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Paderborn University""]}, {""author_id"": ""A5006329724"", ""name"": ""Matthias Christandl"", ""orcid"": ""https://orcid.org/0000-0003-2281-3355"", ""position"": ""middle"", ""is_corresponding"": true, ""institutions"": [""ETH Zurich"", ""Institute for Theoretical Physics""]}, {""author_id"": ""A5000361549"", ""name"": ""Christian Ikenmeyer"", ""orcid"": ""https://orcid.org/0000-0003-4654-177X"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Paderborn University""]}], ""source"": {""source_id"": ""S184147796"", ""display_name"": ""Journal of Algebra"", ""issn_l"": ""0021-8693"", ""type"": ""journal""}, ""biblio"": {""volume"": ""328"", ""issue"": ""1"", ""first_page"": ""322"", ""last_page"": ""329""}, ""cited_by_count"": 31, ""fwci"": 3.0041, ""referenced_works_count"": 20, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1016/j.jalgebra.2010.10.031""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10948"", ""display_name"": ""Advanced Combinatorial Mathematics"", ""field"": ""Mathematics"", ""subfield"": ""Discrete Mathematics and Combinatorics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""plethysm"", ""irreducible representations"", ""highest weights"", ""symmetric powers"", ""geometric complexity theory"", ""quantum information"", ""Young tableaux""], ""abstract"": null, ""referenced_works"": [""W1502291261"", ""W1534413569"", ""W1561337879"", ""W1617404992"", ""W1625204656"", ""W1995509652"", ""W2007051555"", ""W2017282821"", ""W2052237616"", ""W2058207656"", ""W2074207216"", ""W2087312739"", ""W2148706005"", ""W2170831059"", ""W3102121785"", ""W3105095896"", ""W6629829254"", ""W6631923086"", ""W6685152964"", ""W6821281073""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 2}, {""year"": 2020, ""cited_by_count"": 2}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 5}, {""year"": 2015, ""cited_by_count"": 2}, {""year"": 2014, ""cited_by_count"": 4}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R066,215,The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,GawedzkiKupiainen,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{GawedzkiKupiainen,
  author = {Gaw{\k e}dzki, Krzysztof and Kupiainen, Antti},
  title = {Continuum limit of the hierarchical {$O(N)$} non-linear {$\sigma$}-model},
  journal = {Communications in Mathematical Physics}, volume = {106}, pages = {533--550}, year = {1986},
  doi = {10.1007/BF01463394}
}",79,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/references.bib#L79,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01463394"", ""title"": [""Continuum limit of the hierarchicalO(N) non-linear ?-model""], ""authors"": [{""given"": ""K."", ""family"": ""Gawedzki"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""A."", ""family"": ""Kupiainen"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1986, 12]]}, ""container_title"": [""Communications In Mathematical Physics""], ""url"": ""https://doi.org/10.1007/bf01463394"", ""license"": [{""start"": {""date-parts"": [[1986, 12, 1]], ""date-time"": ""1986-12-01T00:00:00Z"", ""timestamp"": 533779200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01463394""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01463394""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1497066249"", ""doi"": ""10.1007/bf01463394"", ""pmid"": null, ""title"": ""Continuum limit of the hierarchicalO(N) non-linear ?-model"", ""publication_year"": 1986, ""publication_date"": ""1986-12-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5059905087"", ""name"": ""Krzysztof Gawędzki"", ""orcid"": ""https://orcid.org/0000-0003-3634-908X"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Centre National de la Recherche Scientifique"", ""Institut des Hautes Études Scientifiques""]}, {""author_id"": ""A5081394703"", ""name"": ""Antti J. Kupiainen"", ""orcid"": ""https://orcid.org/0000-0002-5627-6853"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Helsinki""]}], ""source"": {""source_id"": ""S100601982"", ""display_name"": ""Communications in Mathematical Physics"", ""issn_l"": ""0010-3616"", ""type"": ""journal""}, ""biblio"": {""volume"": ""106"", ""issue"": ""4"", ""first_page"": ""533"", ""last_page"": ""550""}, ""cited_by_count"": 31, ""fwci"": 1.0802, ""referenced_works_count"": 16, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10591"", ""display_name"": ""Theoretical and Computational Physics"", ""field"": ""Physics and Astronomy"", ""subfield"": ""Condensed Matter Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""continuum limit"", ""asymptotic freedom"", ""Wilsonian renormalization group"", ""weak coupling""], ""abstract"": null, ""referenced_works"": [""W1965468324"", ""W1989659612"", ""W1993222918"", ""W2003008637"", ""W2004589772"", ""W2009177719"", ""W2027761034"", ""W2060457947"", ""W2079466094"", ""W2079889893"", ""W2121103309"", ""W2159303232"", ""W2483164457"", ""W2562559245"", ""W2578307220"", ""W4256321433""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R067,215,The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,Ginibre1970,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Ginibre1970,
  author = {Ginibre, J.},
  title = {General formulation of {Griffiths'} inequalities},
  journal = {Communications in Mathematical Physics}, volume = {16}, number = {4}, pages = {310--328}, year = {1970},
  doi = {10.1007/BF01646537}
}",172,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/references.bib#L172,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01646537"", ""title"": [""General formulation of Griffiths' inequalities""], ""authors"": [{""given"": ""J."", ""family"": ""Ginibre"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1970, 12]]}, ""container_title"": [""Communications in Mathematical Physics""], ""url"": ""https://doi.org/10.1007/bf01646537"", ""license"": [{""start"": {""date-parts"": [[1970, 12, 1]], ""date-time"": ""1970-12-01T00:00:00Z"", ""timestamp"": 28857600000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01646537""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01646537""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1974689275"", ""doi"": ""10.1007/bf01646537"", ""pmid"": null, ""title"": ""General formulation of Griffiths' inequalities"", ""publication_year"": 1970, ""publication_date"": ""1970-12-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5006142207"", ""name"": ""J. Ginibre"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Laboratoire de Physique Théorique et Hautes Energies""]}], ""source"": {""source_id"": ""S100601982"", ""display_name"": ""Communications in Mathematical Physics"", ""issn_l"": ""0010-3616"", ""type"": ""journal""}, ""biblio"": {""volume"": ""16"", ""issue"": ""4"", ""first_page"": ""310"", ""last_page"": ""328""}, ""cited_by_count"": 336, ""fwci"": 3.3801, ""referenced_works_count"": 13, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10591"", ""display_name"": ""Theoretical and Computational Physics"", ""field"": ""Physics and Astronomy"", ""subfield"": ""Condensed Matter Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""ferromagnetic spin systems"", ""spin correlations"", ""Ising model"", ""quantum spin systems"", ""model development""], ""abstract"": null, ""referenced_works"": [""W658745854"", ""W1500485857"", ""W1969442295"", ""W1975485534"", ""W1993648260"", ""W1994642941"", ""W2009386142"", ""W2038520523"", ""W2054096716"", ""W2060457947"", ""W2083178078"", ""W2088682706"", ""W2091939285""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 7}, {""year"": 2024, ""cited_by_count"": 6}, {""year"": 2023, ""cited_by_count"": 5}, {""year"": 2022, ""cited_by_count"": 9}, {""year"": 2021, ""cited_by_count"": 8}, {""year"": 2020, ""cited_by_count"": 9}, {""year"": 2019, ""cited_by_count"": 7}, {""year"": 2017, ""cited_by_count"": 2}, {""year"": 2016, ""cited_by_count"": 8}, {""year"": 2015, ""cited_by_count"": 2}, {""year"": 2014, ""cited_by_count"": 5}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R068,215,The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,OS75,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{OS75,
  author = {Osterwalder, Konrad and Schrader, Robert},
  title = {Axioms for {Euclidean} {Green}'s functions {II}},
  journal = {Communications in Mathematical Physics},
  volume = {42},
  number = {3},
  pages = {281--305},
  year = {1975},
  note = {With an Appendix by Stephen Summers},
  doi = {10.1007/BF01608978}
}",134,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/references.bib#L134,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01608978"", ""title"": [""Axioms for Euclidean Green's functions II""], ""authors"": [{""given"": ""Konrad"", ""family"": ""Osterwalder"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Robert"", ""family"": ""Schrader"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1975, 10]]}, ""container_title"": [""Communications in Mathematical Physics""], ""url"": ""https://doi.org/10.1007/bf01608978"", ""license"": [{""start"": {""date-parts"": [[1975, 10, 1]], ""date-time"": ""1975-10-01T00:00:00Z"", ""timestamp"": 181353600000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01608978""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01608978""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2002435891"", ""doi"": ""10.1007/bf01608978"", ""pmid"": null, ""title"": ""Axioms for Euclidean Green's functions II"", ""publication_year"": 1975, ""publication_date"": ""1975-10-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5061502850"", ""name"": ""Konrad Osterwalder"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Harvard University""]}, {""author_id"": ""A5043500537"", ""name"": ""Robert Schrader"", ""orcid"": ""https://orcid.org/0000-0001-6791-6740"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Freie Universität Berlin""]}], ""source"": {""source_id"": ""S100601982"", ""display_name"": ""Communications in Mathematical Physics"", ""issn_l"": ""0010-3616"", ""type"": ""journal""}, ""biblio"": {""volume"": ""42"", ""issue"": ""3"", ""first_page"": ""281"", ""last_page"": ""305""}, ""cited_by_count"": 982, ""fwci"": 41.8935, ""referenced_works_count"": 42, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11022"", ""display_name"": ""Spectral Theory in Mathematical Physics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""analytic continuation"", ""relativistic field theory"", ""axiomatic field theory"", ""necessary and sufficient conditions"", ""Euclidean quantum field theory""], ""abstract"": null, ""referenced_works"": [""W7957601"", ""W121037906"", ""W144893318"", ""W165822066"", ""W566348063"", ""W634568238"", ""W1490039160"", ""W1504171473"", ""W1542996551"", ""W1551762879"", ""W1554438420"", ""W1584607290"", ""W1594278185"", ""W1964717753"", ""W1968417685"", ""W1969916484"", ""W1985004078"", ""W1985425635"", ""W1988538428"", ""W2031075370"", ""W2032467209"", ""W2036856385"", ""W2043702877"", ""W2056244738"", ""W2066718121"", ""W2068856917"", ""W2078218449"", ""W2078278523"", ""W2085706864"", ""W2088553818"", ""W2097605098"", ""W2103079280"", ""W2125433609"", ""W2225673659"", ""W2316482036"", ""W2333206768"", ""W3089492008"", ""W3198666757"", ""W4236657544"", ""W4244247321"", ""W6610653970"", ""W6631356940""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 22}, {""year"": 2025, ""cited_by_count"": 38}, {""year"": 2024, ""cited_by_count"": 11}, {""year"": 2023, ""cited_by_count"": 24}, {""year"": 2022, ""cited_by_count"": 25}, {""year"": 2021, ""cited_by_count"": 38}, {""year"": 2020, ""cited_by_count"": 33}, {""year"": 2019, ""cited_by_count"": 27}, {""year"": 2018, ""cited_by_count"": 24}, {""year"": 2017, ""cited_by_count"": 19}, {""year"": 2016, ""cited_by_count"": 40}, {""year"": 2015, ""cited_by_count"": 34}, {""year"": 2014, ""cited_by_count"": 29}, {""year"": 2013, ""cited_by_count"": 27}, {""year"": 2012, ""cited_by_count"": 18}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R069,215,The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,OpenAI-O4,unverified,unverified,unverified,direct_records,OpenAI companion preprint: original repository retrieval blocked by network policy; the supplied attachment alone does not independently verify it.,"@misc{OpenAI-O4,
  author = {{OpenAI}},
  title = {{Sharp mass bounds for the two-dimensional O(4) model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf}{OAI:Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026}},
  year = {2026}
}",186,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/references.bib#L186,[],[],[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R070,215,The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026,Kupiainen,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Kupiainen,
  author = {Kupiainen, Antti J.},
  title = {On the {$1/n$} expansion},
  journal = {Communications in Mathematical Physics}, volume = {73}, pages = {273--294}, year = {1980},
  doi = {10.1007/BF01197703}
}",73,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/references.bib#L73,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01197703"", ""title"": [""On the 1/n expansion""], ""authors"": [{""given"": ""Antti J."", ""family"": ""Kupiainen"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1980, 10]]}, ""container_title"": [""Communications in Mathematical Physics""], ""url"": ""https://doi.org/10.1007/bf01197703"", ""license"": [{""start"": {""date-parts"": [[1980, 10, 1]], ""date-time"": ""1980-10-01T00:00:00Z"", ""timestamp"": 339206400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01197703""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01197703""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2047723697"", ""doi"": ""10.1007/bf01197703"", ""pmid"": null, ""title"": ""On the 1/n expansion"", ""publication_year"": 1980, ""publication_date"": ""1980-10-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5081394703"", ""name"": ""Antti J. Kupiainen"", ""orcid"": ""https://orcid.org/0000-0002-5627-6853"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Harvard University""]}], ""source"": {""source_id"": ""S100601982"", ""display_name"": ""Communications in Mathematical Physics"", ""issn_l"": ""0010-3616"", ""type"": ""journal""}, ""biblio"": {""volume"": ""73"", ""issue"": ""3"", ""first_page"": ""273"", ""last_page"": ""294""}, ""cited_by_count"": 42, ""fwci"": 2.6386, ""referenced_works_count"": 12, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10591"", ""display_name"": ""Theoretical and Computational Physics"", ""field"": ""Physics and Astronomy"", ""subfield"": ""Condensed Matter Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""1/N expansion"", ""nonlinear sigma model"", ""Heisenberg model"", ""lattice spin models"", ""correlation functions"", ""free energy"", ""asymptotic expansion"", ""spherical model"", ""mass gap"", ""critical temperature""], ""abstract"": null, ""referenced_works"": [""W1991501431"", ""W2004589772"", ""W2007513505"", ""W2042614414"", ""W2066512281"", ""W2069509099"", ""W2088593317"", ""W2143341543"", ""W2161155408"", ""W2169012202"", ""W2562559245"", ""W4253184530""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R071,230,An-exact-Hausdorff-gauge-for-SLE-September-25-2026,RohdeSchramm2005,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{RohdeSchramm2005,
  author = {Rohde, Steffen and Schramm, Oded},
  title = {Basic properties of {SLE}},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {161},
  number = {2},
  year = {2005},
  pages = {883--924},
  doi = {10.4007/annals.2005.161.883},
  url = {https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf},
  note = {\href{https://doi.org/10.4007/annals.2005.161.883}{doi:10.4007/annals.2005.161.883}. Published Theorem 5.1, p. 899; normalization in Section 2.1, pp. 886--887}
}",103,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/refs/sources.bib#L103,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4007/annals.2005.161.883"", ""title"": [""Basic properties of SLE""], ""authors"": [{""given"": ""Steffen"", ""family"": ""Rohde"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Oded"", ""family"": ""Schramm"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Annals of Mathematics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 3, 1]]}, ""container_title"": [""Annals of Mathematics""], ""url"": ""https://doi.org/10.4007/annals.2005.161.883"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4007%2Fannals.2005.161.883""}}]","[""https://api.crossref.org/works/10.4007%2Fannals.2005.161.883""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2949895073"", ""doi"": ""10.4007/annals.2005.161.883"", ""pmid"": null, ""title"": ""Basic properties of SLE"", ""publication_year"": 2005, ""publication_date"": ""2005-03-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5006765366"", ""name"": ""Steffen Rohde"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of Washington""]}, {""author_id"": ""A5056555920"", ""name"": ""Oded Schramm"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Microsoft (United States)""]}], ""source"": {""source_id"": ""S45918994"", ""display_name"": ""Annals of Mathematics"", ""issn_l"": ""0003-486X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""161"", ""issue"": ""2"", ""first_page"": ""883"", ""last_page"": ""924""}, ""cited_by_count"": 369, ""fwci"": 20.1618, ""referenced_works_count"": 25, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf""}, ""best_oa_pdf_url"": ""http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf"", ""primary_topic"": {""id"": ""T11152"", ""display_name"": ""Stochastic processes and statistical mechanics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Schramm-Loewner evolution"", ""simple paths"", ""self-intersection"", ""space-filling curves"", ""Hausdorff dimension"", ""scaling limits"", ""percolation clusters"", ""covering number""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4007/annals.2005.161.883"", ""referenced_works"": [""W1515452785"", ""W1529900298"", ""W1551131215"", ""W1580445772"", ""W1678619573"", ""W1972595847"", ""W1974270490"", ""W1979821081"", ""W1993160451"", ""W2003115626"", ""W2008137962"", ""W2021752571"", ""W2046308667"", ""W2048825720"", ""W2060865179"", ""W2064685239"", ""W2095272960"", ""W2120645838"", ""W2156895775"", ""W2949596418"", ""W2951944263"", ""W3099915674"", ""W3100740946"", ""W3124257919"", ""W4242893449""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 7}, {""year"": 2025, ""cited_by_count"": 15}, {""year"": 2024, ""cited_by_count"": 15}, {""year"": 2023, ""cited_by_count"": 22}, {""year"": 2022, ""cited_by_count"": 20}, {""year"": 2021, ""cited_by_count"": 19}, {""year"": 2020, ""cited_by_count"": 19}, {""year"": 2019, ""cited_by_count"": 22}, {""year"": 2018, ""cited_by_count"": 20}, {""year"": 2017, ""cited_by_count"": 19}, {""year"": 2016, ""cited_by_count"": 18}, {""year"": 2015, ""cited_by_count"": 16}, {""year"": 2014, ""cited_by_count"": 10}, {""year"": 2013, ""cited_by_count"": 15}, {""year"": 2012, ""cited_by_count"": 21}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R072,230,An-exact-Hausdorff-gauge-for-SLE-September-25-2026,OpenAI2026Explicit,unverified,unverified,unverified,direct_records,OpenAI companion preprint: original repository retrieval blocked by network policy; the supplied attachment alone does not independently verify it.,"@misc{OpenAI2026Explicit,
  author = {{OpenAI}},
  title = {{An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf}{OAI:An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026}},
  year = {2026}
}",214,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/refs/sources.bib#L214,[],[],[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R073,230,An-exact-Hausdorff-gauge-for-SLE-September-25-2026,PeresSolomyak2005,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{PeresSolomyak2005,
  author = {Peres, Yuval and Solomyak, Boris},
  title = {The sharp {Hausdorff} measure condition for length of projections},
  journal = {Proceedings of the American Mathematical Society},
  volume = {133},
  number = {11},
  year = {2005},
  pages = {3371--3379},
  doi = {10.1090/S0002-9939-05-08073-1},
  url = {https://arxiv.org/abs/math/0406375v1},
  note = {\href{https://doi.org/10.1090/S0002-9939-05-08073-1}{Published version}.
          Numbered references use \href{https://arxiv.org/abs/math/0406375v1}{arXiv:math/0406375v1},
          18 June 2004, Lemma 1.2 and Section 5}
}",126,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/refs/sources.bib#L126,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1090/s0002-9939-05-08073-1"", ""title"": [""The sharp Hausdorff measure condition for length of projections""], ""authors"": [{""given"": ""Yuval"", ""family"": ""Peres"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Boris"", ""family"": ""Solomyak"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""American Mathematical Society (AMS)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 6, 20]]}, ""container_title"": [""Proceedings of the American Mathematical Society""], ""url"": ""https://doi.org/10.1090/s0002-9939-05-08073-1"", ""license"": [{""start"": {""date-parts"": [[2006, 6, 20]], ""date-time"": ""2006-06-20T00:00:00Z"", ""timestamp"": 1150761600000}, ""content-version"": ""am"", ""delay-in-days"": 365, ""URL"": ""https://www.ams.org/publications/copyright-and-permissions""}], ""source_url"": ""https://api.crossref.org/works/10.1090%2Fs0002-9939-05-08073-1""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""math/0406375"", ""version"": 1, ""id_versioned"": ""math/0406375v1"", ""title"": ""The sharp Hausdorff measure condition for length of projections"", ""abstract"": ""In a recent paper, Pertti Mattila asked which gauge functions $φ$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $φ$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $φ(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $φ$ is regularly varying. Our proof is based on a random construction adapted to the gauge function."", ""authors"": [""Yuval Peres"", ""Boris Solomyak""], ""published"": ""2004-06-18T17:24:45Z"", ""updated"": ""2004-06-18T17:24:45Z"", ""primary_category"": ""math.CA"", ""categories"": [""math.CA""], ""doi"": null, ""journal_ref"": ""Proc. Amer. Math. Soc. 133 (2005), no. 11, 3371--3379"", ""comment"": ""11 pages, 1 figure"", ""abs_url"": ""http://arxiv.org/abs/math/0406375v1"", ""pdf_url"": ""https://arxiv.org/pdf/math/0406375v1""}}]","[""https://api.crossref.org/works/10.1090%2Fs0002-9939-05-08073-1"", ""http://arxiv.org/abs/math/0406375v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1708185612"", ""doi"": ""10.1090/s0002-9939-05-08073-1"", ""pmid"": null, ""title"": ""The sharp Hausdorff measure condition for length of projections"", ""publication_year"": 2005, ""publication_date"": ""2005-06-20"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5020836845"", ""name"": ""Yuval Peres"", ""orcid"": ""https://orcid.org/0000-0001-5456-6323"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""University of California, Berkeley""]}, {""author_id"": ""A5048740367"", ""name"": ""Boris Solomyak"", ""orcid"": ""https://orcid.org/0000-0003-1571-0606"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Washington""]}], ""source"": {""source_id"": ""S4210226894"", ""display_name"": ""Proceedings of the American Mathematical Society"", ""issn_l"": ""0002-9939"", ""type"": ""journal""}, ""biblio"": {""volume"": ""133"", ""issue"": ""11"", ""first_page"": ""3371"", ""last_page"": ""3379""}, ""cited_by_count"": 3, ""fwci"": 0, ""referenced_works_count"": 7, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.ams.org/proc/2005-133-11/S0002-9939-05-08073-1/S0002-9939-05-08073-1.pdf""}, ""best_oa_pdf_url"": ""https://www.ams.org/proc/2005-133-11/S0002-9939-05-08073-1/S0002-9939-05-08073-1.pdf"", ""primary_topic"": {""id"": ""T10588"", ""display_name"": ""Mathematical Dynamics and Fractals"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""gauge functions"", ""regularly varying functions"", ""necessary conditions"", ""random constructions"", ""Hausdorff dimension""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1090/s0002-9939-05-08073-1"", ""referenced_works"": [""W185079044"", ""W1967523109"", ""W1968412787"", ""W2072904618"", ""W2097783100"", ""W2120045167"", ""W2149123301""], ""counts_by_year"": [{""year"": 2018, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R074,230,An-exact-Hausdorff-gauge-for-SLE-September-25-2026,Zhan2019Decomposition,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Zhan2019Decomposition,
  author = {Zhan, Dapeng},
  title = {Decomposition of {Schramm--Loewner} evolution along its curve},
  journal = {Stochastic Processes and their Applications},
  volume = {129}, year = {2019}, pages = {129--152},
  doi = {10.1016/j.spa.2018.02.010},
  note = {\href{https://doi.org/10.1016/j.spa.2018.02.010}{doi:10.1016/j.spa.2018.02.010}.
          Numbered references use the \href{https://users.math.msu.edu/users/zhan/decomposition-SLE.pdf}{author-hosted published article}}
}",169,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/refs/sources.bib#L169,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.spa.2018.02.010"", ""title"": [""Decomposition of Schramm–Loewner evolution along its curve""], ""authors"": [{""ORCID"": ""https://orcid.org/0000-0001-5528-8142"", ""authenticated-orcid"": false, ""given"": ""Dapeng"", ""family"": ""Zhan"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2019, 1]]}, ""container_title"": [""Stochastic Processes and their Applications""], ""url"": ""https://doi.org/10.1016/j.spa.2018.02.010"", ""license"": [{""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/legal/tdmrep-license""}, {""start"": {""date-parts"": [[2023, 1, 1]], ""date-time"": ""2023-01-01T00:00:00Z"", ""timestamp"": 1672531200000}, ""content-version"": ""vor"", ""delay-in-days"": 1461, ""URL"": ""http://www.elsevier.com/open-access/userlicense/1.0/""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-017""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-037""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-012""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-029""}, {""start"": {""date-parts"": [[2019, 1, 1]], ""date-time"": ""2019-01-01T00:00:00Z"", ""timestamp"": 1546300800000}, ""content-version"": ""stm-asf"", ""delay-in-days"": 0, ""URL"": ""https://doi.org/10.15223/policy-004""}], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.spa.2018.02.010""}}]","[""https://api.crossref.org/works/10.1016%2Fj.spa.2018.02.010""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2963970729"", ""doi"": ""10.1016/j.spa.2018.02.010"", ""pmid"": null, ""title"": ""Decomposition of Schramm–Loewner evolution along its curve"", ""publication_year"": 2018, ""publication_date"": ""2018-03-28"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5065127781"", ""name"": ""Dapeng Zhan"", ""orcid"": ""https://orcid.org/0000-0001-5528-8142"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Michigan State University""]}], ""source"": {""source_id"": ""S182850260"", ""display_name"": ""Stochastic Processes and their Applications"", ""issn_l"": ""0304-4149"", ""type"": ""journal""}, ""biblio"": {""volume"": ""129"", ""issue"": ""1"", ""first_page"": ""129"", ""last_page"": ""152""}, ""cited_by_count"": 13, ""fwci"": 2.1469, ""referenced_works_count"": 38, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.sciencedirect.com/science/article/pii/S0304414918300279/pdf""}, ""best_oa_pdf_url"": ""https://www.sciencedirect.com/science/article/pii/S0304414918300279/pdf"", ""primary_topic"": {""id"": ""T11152"", ""display_name"": ""Stochastic processes and statistical mechanics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Schramm-Loewner evolution"", ""chordal SLE"", ""Girsanov theorem""], ""abstract"": null, ""referenced_works"": [""W1515452785"", ""W1545370368"", ""W1551488096"", ""W1584610335"", ""W1984749076"", ""W2005566665"", ""W2008137962"", ""W2013105826"", ""W2028248535"", ""W2035582733"", ""W2037440254"", ""W2048825720"", ""W2067662980"", ""W2099626568"", ""W2112136003"", ""W2148139524"", ""W2149234720"", ""W2150088264"", ""W2156895775"", ""W2161511058"", ""W2168122903"", ""W2210343391"", ""W2349285619"", ""W2949895073"", ""W2951938461"", ""W2951944263"", ""W2964228165"", ""W3100041526"", ""W3101760604"", ""W3104244307"", ""W3104479128"", ""W4229765068"", ""W6602499069"", ""W6636289968"", ""W6646262984"", ""W6681535915"", ""W6681797145"", ""W6684028003""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 2}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 2}, {""year"": 2020, ""cited_by_count"": 2}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}]}}]","[{""sample_id"": ""R074"", ""source_url"": ""https://arxiv.org/pdf/1509.05015"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""00402d7118f4c3e21f53efacc2360c925ce7eb93b9fc25076307f94242f9dc66"", ""locator"": ""PDF page 13, Theorem 4.1 and equations 4.3–4.8"", ""source_version"": ""arXiv:1509.05015v3; document dated August 26, 2018"", ""paraphrased_observations"": ""For kappa strictly between zero and eight, Theorem 4.1 relates Green-function-weighted two-sided radial SLE laws to natural-parametrization/Minkowski-content weighted chordal SLE. Equations 4.3–4.5 on the same page provide conditional remaining-mass and killing-measure identities. These passages support the cited core identity, but do not independently prove every subsequent derivation made in the citing paper."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R075,230,An-exact-Hausdorff-gauge-for-SLE-September-25-2026,RezaeiZhan2017,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{RezaeiZhan2017,
 author={Rezaei, Mohammad A. and Zhan, Dapeng},
 title={Higher moments of the natural parameterization for {SLE} curves},
 journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
 volume={53}, number={1}, year={2017}, pages={182--199},
 doi={10.1214/15-AIHP712}, url={https://arxiv.org/abs/1412.3700},
 note={\href{https://doi.org/10.1214/15-AIHP712}{doi:10.1214/15-AIHP712}. Numbered references use \href{https://arxiv.org/abs/1412.3700v2}{arXiv:1412.3700v2}}
}",150,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/refs/sources.bib#L150,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1214/15-aihp712"", ""title"": [""Higher moments of the natural parameterization for SLE curves""], ""authors"": [{""given"": ""Mohammad A."", ""family"": ""Rezaei"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Dapeng"", ""family"": ""Zhan"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Institute of Mathematical Statistics"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2017, 2, 1]]}, ""container_title"": [""Annales de l'Institut Henri Poincaré, Probabilités et Statistiques""], ""url"": ""https://doi.org/10.1214/15-aihp712"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1214%2F15-aihp712""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1412.3700"", ""version"": 2, ""id_versioned"": ""1412.3700v2"", ""title"": ""Higher moments of the natural parameterization for SLE curves"", ""abstract"": ""In this paper, we will show that the higher moments of the natural parametrization of SLE curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an SLE curve gets near n given points."", ""authors"": [""Mohammad A. Rezaei"", ""Dapeng Zhan""], ""published"": ""2014-12-11T16:08:29Z"", ""updated"": ""2017-07-21T15:40:35Z"", ""primary_category"": ""math.PR"", ""categories"": [""math.PR""], ""doi"": ""10.1214/15-AIHP712"", ""journal_ref"": ""Ann. IHP. 53(1), 182-199, 2017"", ""comment"": ""23 pages, 5 figures"", ""abs_url"": ""http://arxiv.org/abs/1412.3700v2"", ""pdf_url"": ""https://arxiv.org/pdf/1412.3700v2""}}]","[""https://api.crossref.org/works/10.1214%2F15-aihp712"", ""http://arxiv.org/abs/1412.3700v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1671294127"", ""doi"": ""10.1214/15-aihp712"", ""pmid"": null, ""title"": ""Higher moments of the natural parameterization for SLE curves"", ""publication_year"": 2017, ""publication_date"": ""2017-02-01"", ""type"": ""article"", ""language"": ""fr"", ""is_retracted"": false, ""authors"": [{""name"": ""Mohammad A. Rezaei"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Michigan State University""]}, {""name"": ""Dapeng Zhan"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Michigan State University""]}], ""source"": {""source_id"": ""S4210228960"", ""display_name"": ""Annales de l Institut Henri Poincaré Probabilités et Statistiques"", ""issn_l"": ""0246-0203"", ""type"": ""journal""}, ""biblio"": {""volume"": ""53"", ""issue"": ""1"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 5, ""fwci"": 0.9227, ""referenced_works_count"": 15, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://projecteuclid.org/journals/annales-de-linstitut-henri-poincare-probabilites-et-statistiques/volume-53/issue-1/Higher-moments-of-the-natural-parameterization-for-SLE-curves/10.1214/15-AIHP712.pdf""}, ""best_oa_pdf_url"": ""https://projecteuclid.org/journals/annales-de-linstitut-henri-poincare-probabilites-et-statistiques/volume-53/issue-1/Higher-moments-of-the-natural-parameterization-for-SLE-curves/10.1214/15-AIHP712.pdf"", ""primary_topic"": {""id"": ""T11152"", ""display_name"": ""Stochastic processes and statistical mechanics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Schramm-Loewner evolution"", ""higher-order moments"", ""finite moments"", ""upper half-plane"", ""bounded domains"", ""probability estimates""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1214/15-aihp712"", ""referenced_works"": [""W1974133453"", ""W2008137962"", ""W2048825720"", ""W2058737459"", ""W2064685239"", ""W2066695341"", ""W2156895775"", ""W2168122903"", ""W2210343391"", ""W2349285619"", ""W2609656529"", ""W2949895073"", ""W2951944263"", ""W2963070768"", ""W2964052128""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R076,245,Weak-and-strong-normalization-in-pure-type-systems-September-25-2026,Sorensen1997,unverified,unverified,match,supplied_notes_identity,Supplied title-page notes support the Sorensen dissertation identity and University of Copenhagen. The 1997/1998 date distinction remains unresolved.,"@phdthesis{Sorensen1997,
  author = {S{\o}rensen, M. H. B.},
  title = {Normalization in {$\lambda$}-Calculus and Type Theory},
  school = {University of Copenhagen},
  year = {1997},
  note = {Consulted revised edition, April 1998, DIKU Report 97/27},
  url = {https://di.ku.dk/forskning/Publikationer/tekniske_rapporter/tekniske-rapporter-1997/97-27.pdf}
}",20,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/references.bib#L20,[],[],[],"[{""sample_id"": ""R076"", ""source_url"": ""https://di.ku.dk/forskning/Publikationer/tekniske_rapporter/tekniske-rapporter-1997/97-27.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""fe69a78491ce8f91ef7f86f7f6c9be3435abf736b6bfc995208a3704fcc39756"", ""locator"": ""PDF pages 1–2"", ""source_version"": null, ""paraphrased_observations"": ""The title page identifies M. H. B. Sorensen, Normalization in lambda-Calculus and Type Theory, dissertation, Department of Computer Science, University of Copenhagen. The inspected first two pages do not establish the 1997/1998 date distinction."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R077,245,Weak-and-strong-normalization-in-pure-type-systems-September-25-2026,Poll1994,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@phdthesis{Poll1994,
  author = {Poll, Erik},
  title = {A Programming Logic Based on Type Theory},
  school = {Technische Universiteit Eindhoven},
  year = {1994},
  doi = {10.6100/IR423044},
  url = {https://pure.tue.nl/ws/files/1718573/423044.pdf}
}",128,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/references.bib#L128,"[{""provider"": ""DataCite original deposit"", ""record"": {""record"": {""doi"": ""10.6100/ir423044"", ""titles"": [{""title"": ""A programming logic based on type theory""}], ""creators"": [{""name"": ""Poll, E (Erik)"", ""nameType"": ""Personal"", ""affiliation"": [], ""nameIdentifiers"": []}], ""publisher"": ""Technische Universiteit Eindhoven"", ""publication_year"": 1994, ""resource_type"": {""ris"": ""GEN"", ""bibtex"": ""misc"", ""citeproc"": ""article"", ""schemaOrg"": ""CreativeWork""}, ""url"": ""https://research.tue.nl/en/publications/a-programming-logic-based-on-type-theory(c72f515d-fbbb-4b2e-af98-50ed5fb9b735).html"", ""rights"": [], ""related_identifiers"": [], ""version"": null}, ""source_url"": ""https://api.datacite.org/dois/10.6100%2Fir423044""}}]","[""https://api.datacite.org/dois/10.6100%2Fir423044""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R078,245,Weak-and-strong-normalization-in-pure-type-systems-September-25-2026,BartheCoquand2006,minor_difference,minor_difference,minor_difference,direct_records,"Same Barthe–Coquand work and DOI. Citation says 2006; Crossref published date is 2003-09-16. Volume 16(2), 137–155 is corroborated by OpenAlex; actual print-date reconciliation is unresolved.","@article{BartheCoquand2006,
  author = {Barthe, Gilles and Coquand, Thierry},
  title = {Remarks on the equational theory of non-normalizing pure type systems},
  journal = {Journal of Functional Programming},
  volume = {16},
  number = {2},
  pages = {137--155},
  year = {2006},
  doi = {10.1017/S0956796803004726}
}",107,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/references.bib#L107,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1017/s0956796803004726"", ""title"": [""Remarks on the equational theory of non-normalizing pure type systems""], ""authors"": [{""given"": ""GILLES"", ""family"": ""BARTHE"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""THIERRY"", ""family"": ""COQUAND"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Cambridge University Press (CUP)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2003, 9, 16]]}, ""container_title"": [""Journal of Functional Programming""], ""url"": ""https://doi.org/10.1017/s0956796803004726"", ""license"": [{""start"": {""date-parts"": [[2003, 9, 16]], ""date-time"": ""2003-09-16T00:00:00Z"", ""timestamp"": 1063670400000}, ""content-version"": ""unspecified"", ""delay-in-days"": 0, ""URL"": ""https://www.cambridge.org/core/terms""}], ""source_url"": ""https://api.crossref.org/works/10.1017%2Fs0956796803004726""}}]","[""https://api.crossref.org/works/10.1017%2Fs0956796803004726""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2168328957"", ""doi"": ""10.1017/s0956796803004726"", ""pmid"": null, ""title"": ""Remarks on the equational theory of non-normalizing pure type systems"", ""publication_year"": 2003, ""publication_date"": ""2003-09-16"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5075577852"", ""name"": ""Gilles Barthe"", ""orcid"": ""https://orcid.org/0000-0002-3853-1777"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Institut national de recherche en sciences et technologies du numérique""]}, {""author_id"": ""A5087100539"", ""name"": ""Thierry Coquand"", ""orcid"": ""https://orcid.org/0000-0002-5429-5153"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""Chalmers University of Technology""]}], ""source"": {""source_id"": ""S12266929"", ""display_name"": ""Journal of Functional Programming"", ""issn_l"": ""0956-7968"", ""type"": ""journal""}, ""biblio"": {""volume"": ""16"", ""issue"": ""2"", ""first_page"": ""137"", ""last_page"": ""155""}, ""cited_by_count"": 5, ""fwci"": 0.3542, ""referenced_works_count"": 16, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7802BAE4C650D853790158D63E0CC4FE/S0956796803004726a.pdf/div-class-title-remarks-on-the-equational-theory-of-non-normalizing-pure-type-systems-div.pdf""}, ""best_oa_pdf_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7802BAE4C650D853790158D63E0CC4FE/S0956796803004726a.pdf/div-class-title-remarks-on-the-equational-theory-of-non-normalizing-pure-type-systems-div.pdf"", ""primary_topic"": {""id"": ""T10126"", ""display_name"": ""Logic, programming, and type systems"", ""field"": ""Computer Science"", ""subfield"": ""Artificial Intelligence"", ""domain"": ""Physical Sciences""}, ""keywords"": [""pure type systems"", ""equational theories"", ""Curry-Howard correspondence""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1017/s0956796803004726"", ""referenced_works"": [""W145869046"", ""W1488210308"", ""W1496535287"", ""W1520936552"", ""W1533293692"", ""W1536208674"", ""W1903854703"", ""W1971128176"", ""W2093576032"", ""W2114705282"", ""W2120300141"", ""W2154273644"", ""W2160283637"", ""W2398461955"", ""W2586148739"", ""W4388310154""], ""counts_by_year"": [{""year"": 2023, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R079,245,Weak-and-strong-normalization-in-pure-type-systems-September-25-2026,Geuvers1993,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@phdthesis{Geuvers1993,
  author = {Geuvers, Jan Herman},
  title = {Logics and Type Systems},
  school = {Katholieke Universiteit Nijmegen},
  year = {1993},
  url = {https://www.cs.ru.nl/~herman/PUBS/Proefschrift.pdf}
}",1,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/references.bib#L1,"[{""provider"": ""DataCite original deposit"", ""record"": {""record"": {""doi"": ""10.6100/ir414311"", ""titles"": [{""title"": ""Logics and type systems""}], ""creators"": [{""name"": ""Geuvers, JH Herman"", ""nameType"": ""Personal"", ""givenName"": ""JH Herman"", ""familyName"": ""Geuvers"", ""affiliation"": [], ""nameIdentifiers"": []}], ""publisher"": ""Katholieke Universiteit Nijmegen"", ""publication_year"": 1993, ""resource_type"": {""ris"": ""GEN"", ""bibtex"": ""misc"", ""citeproc"": ""article"", ""schemaOrg"": ""CreativeWork""}, ""url"": ""https://research.tue.nl/en/publications/logics-and-type-systems(37f0d72a-56bc-41b8-9796-e67c1e1c18e7).html"", ""rights"": [], ""related_identifiers"": [], ""version"": null}, ""source_url"": ""https://api.datacite.org/dois/10.6100%2Fir414311""}}]","[""https://api.datacite.org/dois/10.6100%2Fir414311""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R080,245,Weak-and-strong-normalization-in-pure-type-systems-September-25-2026,Mull2023,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@inproceedings{Mull2023,
  author = {Mull, Nathan},
  title = {An Irrelevancy-Eliminating Translation of Pure Type Systems},
  booktitle = {28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  series = {Leibniz International Proceedings in Informatics},
  volume = {269},
  pages = {7:1--7:21},
  publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\""u}r Informatik},
  year = {2023},
  doi = {10.4230/LIPIcs.TYPES.2022.7}
}",43,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/references.bib#L43,"[{""provider"": ""DataCite original deposit"", ""record"": {""record"": {""doi"": ""10.4230/lipics.types.2022.7"", ""titles"": [{""title"": ""An Irrelevancy-Eliminating Translation of Pure Type Systems"", ""lang"": ""en""}], ""creators"": [{""name"": ""Mull, Nathan"", ""nameType"": ""Personal"", ""givenName"": ""Nathan"", ""familyName"": ""Mull"", ""affiliation"": [""University of Chicago, IL, USA""], ""nameIdentifiers"": []}], ""publisher"": ""Schloss Dagstuhl – Leibniz-Zentrum für Informatik"", ""publication_year"": 2023, ""resource_type"": {""resourceTypeGeneral"": ""ConferencePaper"", ""ris"": ""CPAPER"", ""bibtex"": ""inproceedings"", ""citeproc"": """", ""schemaOrg"": ""Article""}, ""url"": ""https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.7"", ""rights"": [{""lang"": ""en"", ""rights"": ""Creative Commons Attribution 4.0 International license"", ""rightsUri"": ""https://creativecommons.org/licenses/by/4.0/legalcode"", ""rightsIdentifier"": ""cc by 4.0""}, {""rights"": ""info:eu-repo/semantics/openAccess""}], ""related_identifiers"": [{""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""10.4230/LIPIcs.TYPES.2022"", ""relatedIdentifierType"": ""DOI""}, {""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""978-3-95977-285-3"", ""relatedIdentifierType"": ""ISBN""}, {""relationType"": ""IsPartOf"", ""relatedIdentifier"": ""1868-8969"", ""relatedIdentifierType"": ""ISSN""}], ""version"": null}, ""source_url"": ""https://api.datacite.org/dois/10.4230%2Flipics.types.2022.7""}}]","[""https://api.datacite.org/dois/10.4230%2Flipics.types.2022.7""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R081,296,Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026,DSSW2008,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{DSSW2008,
  author = {Dykema, Ken and Sinclair, Allan and Smith, Roger and White, Stuart},
  title = {Generators of {$\mathrm{II}_1$} factors},
  journal = {Operators and Matrices},
  volume = {2}, number = {4}, pages = {555--582}, year = {2008},
  doi = {10.7153/oam-02-35}, eprint = {0706.1953}, archivePrefix = {arXiv},
  url = {https://files.ele-math.com/articles/oam-02-35.pdf}
}",116,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/references.bib#L116,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.7153/oam-02-35"", ""title"": [""Generators of II_1 factors""], ""authors"": [{""given"": ""Ken"", ""family"": ""Dykema"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Allan"", ""family"": ""Sinclair"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Roger"", ""family"": ""Smith"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Stuart"", ""family"": ""White"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Element d.o.o."", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2008]]}, ""container_title"": [""Operators and Matrices""], ""url"": ""https://doi.org/10.7153/oam-02-35"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.7153%2Foam-02-35""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""0706.1953"", ""version"": 2, ""id_versioned"": ""0706.1953v2"", ""title"": ""Generators of II_1 Factors"", ""abstract"": ""In 2005, Shen introduced a new invariant, $\\mathcal G(N)$, of a diffuse von Neumann algebra $N$ with a fixed faithful trace, and he used this invariant to give a unified approach to showing that large classes of ${\\mathrm{II}}_1$ factors $M$ are singly generated. This paper focuses on properties of this invariant. We relate $\\mathcal G(M)$ to the number of self-adjoint generators of a ${\\mathrm{II}}_1$ factor $M$: if $\\mathcal G(M)<n/2$, then $M$ is generated by $n+1$ self-adjoint operators, whereas if $M$ is generated by $n+1$ self-adjoint operators, then $\\mathcal G(M)\\leq n/2$. The invariant $\\mathcal G(\\cdot)$ is well-behaved under amplification, satisfying $\\mathcal G(M_t)=t^{-2}\\mathcal G(M)$ for all $t>0$. In particular, if $\\mathcal G(\\mathcal L\\mathbb F_r)>0$ for any particular $r>1$, then the free group factors are pairwise non-isomorphic and are not singly generated for sufficiently large values of $r$. Estimates are given for forming free products and passing to finite index subfactors and the basic construction. We also examine a version of the invariant $\\mathcal G_{\\text{sa}}(M)$ defined only using self-adjoint operators; this is proved to satisfy $\\mathcal G_{\\text{sa}}(M)=2\\mathcal G(M)$. Finally we give inequalities relating a quantity involved in the calculation of $\\mathcal G(M)$ to the free-entropy dimension $δ_0$ of a collection of generators for $M$."", ""authors"": [""Ken Dykema"", ""Allan Sinclair"", ""Roger Smith"", ""Stuart White""], ""published"": ""2007-06-13T17:10:39Z"", ""updated"": ""2007-11-02T03:17:59Z"", ""primary_category"": ""math.OA"", ""categories"": [""math.OA""], ""doi"": null, ""journal_ref"": ""Oper. Matrices 2, no 4 (2008), 555-582."", ""comment"": ""36 Pages, section 8 rewritten"", ""abs_url"": ""http://arxiv.org/abs/0706.1953v2"", ""pdf_url"": ""https://arxiv.org/pdf/0706.1953v2""}}]","[""https://api.crossref.org/works/10.7153%2Foam-02-35"", ""http://arxiv.org/abs/0706.1953v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1996458343"", ""doi"": ""10.7153/oam-02-35"", ""pmid"": null, ""title"": ""Generators of II_1 factors"", ""publication_year"": 2008, ""publication_date"": ""2008-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5112524324"", ""name"": ""Ken Dykema"", ""orcid"": ""https://orcid.org/0000-0002-6786-3279"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5006640905"", ""name"": ""Allan M. Sinclair"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5103088531"", ""name"": ""Roger R. Smith"", ""orcid"": ""https://orcid.org/0000-0002-7175-8586"", ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": []}, {""author_id"": ""A5057544177"", ""name"": ""Stuart White"", ""orcid"": ""https://orcid.org/0000-0003-2264-8943"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": []}], ""source"": {""source_id"": ""S136112250"", ""display_name"": ""Operators and Matrices"", ""issn_l"": ""1846-3886"", ""type"": ""journal""}, ""biblio"": {""volume"": null, ""issue"": ""4"", ""first_page"": ""555"", ""last_page"": ""582""}, ""cited_by_count"": 1, ""fwci"": 0, ""referenced_works_count"": 3, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""http://files.ele-math.com/abstracts/oam-02-35-abs.pdf""}, ""best_oa_pdf_url"": ""http://files.ele-math.com/abstracts/oam-02-35-abs.pdf"", ""primary_topic"": {""id"": ""T10899"", ""display_name"": ""Advanced Operator Algebra Research"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""II_1 factors"", ""classification of von Neumann algebras""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.7153/oam-02-35"", ""referenced_works"": [""W2015411876"", ""W6601235630"", ""W6837690359""], ""counts_by_year"": [{""year"": 2017, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R082,296,Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026,Popa1995,minor_difference,minor_difference,match,direct_records_and_supplied_notes,"Supplied Numdam item/BibTeX reading identifies Asterisque 232 (1995), 187–202. The cited historical issue date is supported; the later Crossref date remains a registry discrepancy, not a demonstrated citation error. Numdam notes were not independently retrieved.","@incollection{Popa1995,
  author = {Popa, Sorin},
  title = {Free-independent sequences in type {$\mathrm{II}_1$} factors and related problems},
  booktitle = {Recent advances in operator algebras---Orl{\'e}ans, 1992},
  series = {Ast{\'e}risque},
  number = {232},
  publisher = {Soci{\'e}t{\'e} math{\'e}matique de France},
  pages = {187--202},
  year = {1995},
  url = {https://www.numdam.org/item/AST_1995__232__187_0/}
}",85,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/references.bib#L85,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.24033/ast.318"", ""title"": [""Free-independent sequences in type $II_1$ factors and related problems""], ""authors"": [{""given"": ""S."", ""family"": ""POPA"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Societe Mathematique de France"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2018, 11, 6]]}, ""container_title"": [""Astérisque""], ""url"": ""https://doi.org/10.24033/ast.318"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.24033%2Fast.318""}}]","[""https://api.crossref.org/works/10.24033%2Fast.318""]",[],"[{""sample_id"": ""R082"", ""source_url"": ""https://www.numdam.org/item/AST_1995__232__187_0/"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""16397130d68b0b8faa44681578e795c8c92d2b6c774024c990dd6ff879de0012"", ""locator"": ""Numdam item record and BibTeX"", ""source_version"": ""Historical issue catalogue"", ""paraphrased_observations"": ""Numdam explicitly identifies Sorin Popa, Free-independent sequences in type II_1 factors and related problems, Asterisque 232 (1995), pages 187–202. This resolves the 1995 versus later registry date for this issue."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R083,296,Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026,BrannanEtAl2023,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{BrannanEtAl2023,
  author = {Brannan, Michael and Elzinga, Floris and Harris, Samuel J. and Yamashita, Makoto},
  title = {Crossed product equivalence of quantum automorphism groups of finite dimensional {$C^*$}-algebras},
  journal = {International Mathematics Research Notices},
  volume = {2023}, number = {20}, pages = {17749--17787}, year = {2023},
  doi = {10.1093/imrn/rnad060},
  url = {https://doi.org/10.1093/imrn/rnad060}
}",190,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/references.bib#L190,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1093/imrn/rnad060"", ""title"": [""Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras""], ""authors"": [{""given"": ""Michael"", ""family"": ""Brannan"", ""sequence"": ""first"", ""affiliation"": [{""name"": ""Department of Pure Mathematics and The Institute for Quantum Computing, University of Waterloo , Waterloo, Ontario N2L 3G1, Canada""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Floris"", ""family"": ""Elzinga"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Department of Mathematics, University of Oslo , P.O box 1053, Blindern, 0316 Oslo, Norway""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Samuel J"", ""family"": ""Harris"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Department of Mathematics & Statistics, Northern Arizona University , PO Box 5717, Flagstaff, AZ, 86011 USA""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}, {""given"": ""Makoto"", ""family"": ""Yamashita"", ""sequence"": ""additional"", ""affiliation"": [{""name"": ""Department of Mathematics, University of Oslo , P.O box 1053, Blindern, 0316 Oslo, Norway""}], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Oxford University Press (OUP)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2023, 3, 30]]}, ""container_title"": [""International Mathematics Research Notices""], ""url"": ""https://doi.org/10.1093/imrn/rnad060"", ""license"": [{""start"": {""date-parts"": [[2023, 3, 31]], ""date-time"": ""2023-03-31T00:00:00Z"", ""timestamp"": 1680220800000}, ""content-version"": ""vor"", ""delay-in-days"": 1, ""URL"": ""https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model""}], ""source_url"": ""https://api.crossref.org/works/10.1093%2Fimrn%2Frnad060""}}]","[""https://api.crossref.org/works/10.1093%2Fimrn%2Frnad060""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4362192376"", ""doi"": ""10.1093/imrn/rnad060"", ""pmid"": null, ""title"": ""Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras"", ""publication_year"": 2023, ""publication_date"": ""2023-03-30"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5016789688"", ""name"": ""Michael Brannan"", ""orcid"": ""https://orcid.org/0000-0001-5710-8328"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Waterloo""]}, {""author_id"": ""A5031007177"", ""name"": ""Floris Elzinga"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""University of Oslo""]}, {""name"": ""Samuel J Harris"", ""orcid"": null, ""position"": ""middle"", ""is_corresponding"": false, ""institutions"": [""Northern Arizona University""]}, {""author_id"": ""A5031662115"", ""name"": ""Makoto Yamashita"", ""orcid"": ""https://orcid.org/0000-0002-1738-9652"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Oslo""]}], ""source"": {""source_id"": ""S111381568"", ""display_name"": ""International Mathematics Research Notices"", ""issn_l"": ""1073-7928"", ""type"": ""journal""}, ""biblio"": {""volume"": ""2023"", ""issue"": ""20"", ""first_page"": ""17749"", ""last_page"": ""17787""}, ""cited_by_count"": 0, ""fwci"": 0, ""referenced_works_count"": 78, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10899"", ""display_name"": ""Advanced Operator Algebra Research"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""quantum automorphism groups"", ""finite-dimensional C*-algebras"", ""crossed products"", ""quantum permutation groups"", ""finite abelian groups"", ""quantum groups""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See http://dx.doi.org/10.1093/imrn/rnad060"", ""referenced_works"": [""W153923367"", ""W645521785"", ""W1569992215"", ""W1610770237"", ""W1628689112"", ""W1641930121"", ""W1807929543"", ""W1889808743"", ""W1894480176"", ""W1926217137"", ""W1974267617"", ""W1974451370"", ""W1980921524"", ""W1981058161"", ""W1998156821"", ""W2002753250"", ""W2008565461"", ""W2010420931"", ""W2015079526"", ""W2019899074"", ""W2021318741"", ""W2026938829"", ""W2037260048"", ""W2060402408"", ""W2063099505"", ""W2063932211"", ""W2069955012"", ""W2079690201"", ""W2083474360"", ""W2094291513"", ""W2095814226"", ""W2101239948"", ""W2109491441"", ""W2123596803"", ""W2128203159"", ""W2145945838"", ""W2151575758"", ""W2170594311"", ""W2580975142"", ""W2611047923"", ""W2626040212"", ""W2785460059"", ""W2888672771"", ""W2962839795"", ""W2963181542"", ""W2963217391"", ""W2963291331"", ""W2963355705"", ""W2963358246"", ""W2963396315"", ""W2963440515"", ""W2963564912"", ""W2963769525"", ""W2963848779"", ""W2964195385"", ""W3000266829"", ""W3037444540"", ""W3038428577"", ""W3085876781"", ""W3098123020"", ""W3099103539"", ""W3100484668"", ""W3100513847"", ""W3100579360"", ""W3110307957"", ""W3128390254"", ""W3138172839"", ""W3208447640"", ""W4214821892"", ""W4297818187"", ""W6634133893"", ""W6636584688"", ""W6636677426"", ""W6638555867"", ""W6691685535"", ""W6695809791"", ""W6783465791"", ""W6802635130""], ""counts_by_year"": []}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R084,296,Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026,Wogen1969,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Wogen1969,
  author = {Wogen, Warren},
  title = {On generators for von {Neumann} algebras},
  journal = {Bulletin of the American Mathematical Society},
  volume = {75}, number = {1}, pages = {95--99}, year = {1969},
  doi = {10.1090/S0002-9904-1969-12157-9}
}",109,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/references.bib#L109,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1090/s0002-9904-1969-12157-9"", ""title"": [""On generators for von Neumann algebras""], ""authors"": [{""given"": ""Warren"", ""family"": ""Wogen"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""American Mathematical Society (AMS)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1969]]}, ""container_title"": [""Bulletin of the American Mathematical Society""], ""url"": ""https://doi.org/10.1090/s0002-9904-1969-12157-9"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1090%2Fs0002-9904-1969-12157-9""}}]","[""https://api.crossref.org/works/10.1090%2Fs0002-9904-1969-12157-9""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2027507010"", ""doi"": ""10.1090/s0002-9904-1969-12157-9"", ""pmid"": null, ""title"": ""On generators for von Neumann algebras"", ""publication_year"": 1969, ""publication_date"": ""1969-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5108394720"", ""name"": ""Warren R. Wogen"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S71584477"", ""display_name"": ""Bulletin of the American Mathematical Society"", ""issn_l"": ""0002-9904"", ""type"": ""journal""}, ""biblio"": {""volume"": ""75"", ""issue"": ""1"", ""first_page"": ""95"", ""last_page"": ""99""}, ""cited_by_count"": 39, ""fwci"": 5.9594, ""referenced_works_count"": 11, ""open_access"": {""is_oa"": true, ""oa_status"": ""diamond"", ""oa_url"": ""https://www.ams.org/bull/1969-75-01/S0002-9904-1969-12157-9/S0002-9904-1969-12157-9.pdf""}, ""best_oa_pdf_url"": ""https://www.ams.org/bull/1969-75-01/S0002-9904-1969-12157-9/S0002-9904-1969-12157-9.pdf"", ""primary_topic"": {""id"": ""T10899"", ""display_name"": ""Advanced Operator Algebra Research"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""von Neumann algebras"", ""finitely generated algebras"", ""centralizers"", ""separable Hilbert space""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1090/s0002-9904-1969-12157-9"", ""referenced_works"": [""W1530330303"", ""W1975956985"", ""W1984474434"", ""W1985616867"", ""W2069102059"", ""W2086760343"", ""W2485888813"", ""W4240296251"", ""W4243653473"", ""W6646344021"", ""W6722227699""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 4}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R085,296,Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026,Sherman2012,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Sherman2012,
  author = {Sherman, David},
  title = {On cardinal invariants and generators for von {Neumann} algebras},
  journal = {Canadian Journal of Mathematics},
  volume = {64},
  number = {2},
  pages = {455--480},
  year = {2012},
  doi = {10.4153/CJM-2011-048-2},
  eprint = {0908.4565},
  archivePrefix = {arXiv},
  primaryClass = {math.OA},
  url = {https://arxiv.org/abs/0908.4565}
}",28,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/references.bib#L28,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4153/cjm-2011-048-2"", ""title"": [""On Cardinal Invariants and Generators for von Neumann Algebras""], ""authors"": [{""given"": ""David"", ""family"": ""Sherman"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Canadian Mathematical Society"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2012, 4, 16]]}, ""container_title"": [""Canadian Journal of Mathematics""], ""url"": ""https://doi.org/10.4153/cjm-2011-048-2"", ""license"": [{""start"": {""date-parts"": [[2018, 11, 20]], ""date-time"": ""2018-11-20T00:00:00Z"", ""timestamp"": 1542672000000}, ""content-version"": ""unspecified"", ""delay-in-days"": 2409, ""URL"": ""https://www.cambridge.org/core/terms""}], ""source_url"": ""https://api.crossref.org/works/10.4153%2Fcjm-2011-048-2""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""0908.4565"", ""version"": 2, ""id_versioned"": ""0908.4565v2"", ""title"": ""On cardinal invariants and generators for von Neumann algebras"", ""abstract"": ""We demonstrate how virtually all common cardinal invariants associated to a von Neumann algebra M can be computed from the decomposability number, dec(M), and the minimal cardinality of a generating set, gen(M). Applications include the equivalence of the well-known generator problem, \""Is every separably-acting von Neumann algebra singly-generated?\"", with the formally stronger questions, \""Is every countably-generated von Neumann algebra singly-generated?\"" and \""Is the gen invariant monotone?\"" Modulo the generator problem, we determine the range of the invariant (gen(M), dec(M)), which is mostly governed by the inequality dec(M) leq c^{gen(M)}."", ""authors"": [""David Sherman""], ""published"": ""2009-08-31T15:53:12Z"", ""updated"": ""2009-12-18T16:46:13Z"", ""primary_category"": ""math.OA"", ""categories"": [""math.OA"", ""math.FA""], ""doi"": ""10.4153/CJM-2011-048-2"", ""journal_ref"": ""Can. J. Math.-J. Can. Math. 64 (2012) 455-480"", ""comment"": ""22 pages; the main additions are Theorem 3.8 and Section 8"", ""abs_url"": ""http://arxiv.org/abs/0908.4565v2"", ""pdf_url"": ""https://arxiv.org/pdf/0908.4565v2""}}]","[""https://api.crossref.org/works/10.4153%2Fcjm-2011-048-2"", ""http://arxiv.org/abs/0908.4565v2""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2038231956"", ""doi"": ""10.4153/cjm-2011-048-2"", ""pmid"": null, ""title"": ""On Cardinal Invariants and Generators for von Neumann Algebras"", ""publication_year"": 2011, ""publication_date"": ""2011-07-15"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""name"": ""David Sherman"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Virginia""]}], ""source"": {""source_id"": ""S131268793"", ""display_name"": ""Canadian Journal of Mathematics"", ""issn_l"": ""0008-414X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""64"", ""issue"": ""2"", ""first_page"": ""455"", ""last_page"": ""480""}, ""cited_by_count"": 51, ""fwci"": 0, ""referenced_works_count"": 4, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9740E3C57D76456C5D026F3E1BAABC26/S0008414X00009147a.pdf/div-class-title-on-cardinal-invariants-and-generators-for-von-neumann-algebras-div.pdf""}, ""best_oa_pdf_url"": ""https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9740E3C57D76456C5D026F3E1BAABC26/S0008414X00009147a.pdf/div-class-title-on-cardinal-invariants-and-generators-for-von-neumann-algebras-div.pdf"", ""primary_topic"": {""id"": ""T10899"", ""display_name"": ""Advanced Operator Algebra Research"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""von Neumann algebras"", ""cardinal invariants"", ""minimal generating sets""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4153/cjm-2011-048-2"", ""referenced_works"": [""W2003173901"", ""W4242603544"", ""W4246894044"", ""W7028268273""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 19}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 29}, {""year"": 2017, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R086,351,A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,GuZhu2008,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization. Identity verified at preprint level; claimed journal/chapter publication details are not independently certified by that record.,"\bibitem{GuZhu2008}
H.-L.~Gu and X.-P. Zhu,
\emph{The existence of Type II singularities for the Ricci flow on
\(S^{n+1}\)},
Comm. Anal. Geom. \textbf{16} (2008), no.~3, 467--494.
\href{https://arxiv.org/abs/0707.0033v1}{arXiv:0707.0033v1}.",36,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/references/bibliography.tex#L36,"[{""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""0707.0033"", ""version"": 1, ""id_versioned"": ""0707.0033v1"", ""title"": ""The Existence of Type II Singularities for the Ricci Flow on $S^{n+1}$"", ""abstract"": ""In this paper we prove the existence of Type II singularities for the Ricci flow on $S^{n+1}$ for all $n\\geq 2$."", ""authors"": [""Hui-Ling Gu"", ""Xi-Ping Zhu""], ""published"": ""2007-06-30T01:55:31Z"", ""updated"": ""2007-06-30T01:55:31Z"", ""primary_category"": ""math.DG"", ""categories"": [""math.DG"", ""math.AP""], ""doi"": null, ""journal_ref"": null, ""comment"": ""28 pages"", ""abs_url"": ""http://arxiv.org/abs/0707.0033v1"", ""pdf_url"": ""https://arxiv.org/pdf/0707.0033v1""}}]","[""http://arxiv.org/abs/0707.0033v1""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R087,351,A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,Stolarski2019,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Stolarski2019}
M.~Stolarski,
\emph{Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow},
\href{https://arxiv.org/abs/1905.00087v1}{arXiv:1905.00087v1}, 2019.
All numbered source references in this paper use this version.
Published in Mem. Amer. Math. Soc. \textbf{295} (2024), no.~1470.
\href{https://doi.org/10.1090/memo/1470}{doi:10.1090/memo/1470}.",80,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/references/bibliography.tex#L80,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1090/memo/1470"", ""title"": [""Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow""], ""authors"": [{""given"": ""Maxwell"", ""family"": ""Stolarski"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""American Mathematical Society (AMS)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2024, 3, 19]]}, ""container_title"": [""Memoirs of the American Mathematical Society""], ""url"": ""https://doi.org/10.1090/memo/1470"", ""license"": [{""start"": {""date-parts"": [[2025, 3, 19]], ""date-time"": ""2025-03-19T00:00:00Z"", ""timestamp"": 1742342400000}, ""content-version"": ""am"", ""delay-in-days"": 365, ""URL"": ""https://www.ams.org/publications/copyright-and-permissions""}], ""source_url"": ""https://api.crossref.org/works/10.1090%2Fmemo%2F1470""}}, {""provider"": ""arXiv original repository"", ""record"": {""arxiv_id"": ""1905.00087"", ""version"": 1, ""id_versioned"": ""1905.00087v1"", ""title"": ""Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow"", ""abstract"": ""For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \\times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\\| Rm \\|_{\\infty} (t) \\gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales."", ""authors"": [""Maxwell Stolarski""], ""published"": ""2019-04-30T20:07:41Z"", ""updated"": ""2019-04-30T20:07:41Z"", ""primary_category"": ""math.DG"", ""categories"": [""math.DG""], ""doi"": null, ""journal_ref"": null, ""comment"": ""134 pages. Comments welcome"", ""abs_url"": ""http://arxiv.org/abs/1905.00087v1"", ""pdf_url"": ""https://arxiv.org/pdf/1905.00087v1""}}]","[""https://api.crossref.org/works/10.1090%2Fmemo%2F1470"", ""http://arxiv.org/abs/1905.00087v1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4392943161"", ""doi"": ""10.1090/memo/1470"", ""pmid"": null, ""title"": ""Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow"", ""publication_year"": 2024, ""publication_date"": ""2024-03-19"", ""type"": ""article"", ""language"": ""lv"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5062575485"", ""name"": ""Maxwell Stolarski"", ""orcid"": ""https://orcid.org/0009-0002-1544-3827"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S104585175"", ""display_name"": ""Memoirs of the American Mathematical Society"", ""issn_l"": ""0065-9266"", ""type"": ""journal""}, ""biblio"": {""volume"": ""295"", ""issue"": ""1470"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 0, ""fwci"": 0, ""referenced_works_count"": 45, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10229"", ""display_name"": ""Geometric Analysis and Curvature Flows"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Ricci flow"", ""curvature blow-up"", ""conical singularities"", ""type II singularities"", ""Einstein metrics"", ""parabolic scaling""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1090/memo/1470"", ""referenced_works"": [""W45717556"", ""W79367134"", ""W135543600"", ""W334353125"", ""W349591683"", ""W1480964215"", ""W1481491273"", ""W1542957889"", ""W1546164573"", ""W1591254021"", ""W1966687081"", ""W1982629055"", ""W1990716079"", ""W2019138867"", ""W2028443198"", ""W2060241233"", ""W2072685235"", ""W2079303959"", ""W2093041805"", ""W2129013141"", ""W2227788618"", ""W2302148688"", ""W2316114213"", ""W2336002111"", ""W2500662118"", ""W2614902970"", ""W2789559908"", ""W2790423469"", ""W2950714836"", ""W2962951457"", ""W2963319380"", ""W2963366263"", ""W2963451452"", ""W2963544055"", ""W2964087133"", ""W2990439825"", ""W3039948875"", ""W3100261296"", ""W3100681271"", ""W4212978921"", ""W4231839916"", ""W4242522601"", ""W6611293017"", ""W6635132904"", ""W6681585578""], ""counts_by_year"": []}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R088,351,A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,Hamilton1995,minor_difference,minor_difference,minor_difference,direct_records,"Same Hamilton work/author/venue. Citation says “formation”, 1995; Crossref says “formations”, 1993. Publication/conference/volume dating unresolved; no evidence of a different work.","\bibitem{Hamilton1995}
R.~S. Hamilton,
\emph{The formation of singularities in the Ricci flow},
Surveys in Differential Geometry, Vol.~II (Cambridge, MA, 1993),
International Press, Cambridge, MA, 1995, 7--136.
\href{https://intlpress.com/site/pub/files/_fulltext/journals/sdg/1993/0002/0001/SDG-1993-0002-0001-a002.pdf}{Full text}.",43,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/references/bibliography.tex#L43,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4310/sdg.1993.v2.n1.a2"", ""title"": [""The formations of singularities in the Ricci Flow""], ""authors"": [{""given"": ""Richard"", ""family"": ""Hamilton"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""International Press of Boston"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1993]]}, ""container_title"": [""Surveys in Differential Geometry""], ""url"": ""https://doi.org/10.4310/sdg.1993.v2.n1.a2"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4310%2Fsdg.1993.v2.n1.a2""}}]","[""https://api.crossref.org/works/10.4310%2Fsdg.1993.v2.n1.a2""]",[],[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R089,351,A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,Shi1989,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Shi1989}
W.-X. Shi,
\emph{Deforming the metric on complete Riemannian manifolds},
J. Differential Geom. \textbf{30} (1989), no.~1, 223--301.
\href{https://doi.org/10.4310/jdg/1214443292}{doi:10.4310/jdg/1214443292}.",74,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/references/bibliography.tex#L74,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.4310/jdg/1214443292"", ""title"": [""Deforming the metric on complete Riemannian manifolds""], ""authors"": [{""given"": ""Wan-Xiong"", ""family"": ""Shi"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""International Press of Boston"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1989, 1, 1]]}, ""container_title"": [""Journal of Differential Geometry""], ""url"": ""https://doi.org/10.4310/jdg/1214443292"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.4310%2Fjdg%2F1214443292""}}]","[""https://api.crossref.org/works/10.4310%2Fjdg%2F1214443292""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1611092396"", ""doi"": ""10.4310/jdg/1214443292"", ""pmid"": null, ""title"": ""Deforming the metric on complete Riemannian manifolds"", ""publication_year"": 1989, ""publication_date"": ""1989-01-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5040960945"", ""name"": ""Wan-Xiong Shi"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S23326474"", ""display_name"": ""Journal of Differential Geometry"", ""issn_l"": ""0022-040X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""30"", ""issue"": ""1"", ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 564, ""fwci"": 7.6417, ""referenced_works_count"": 4, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10229"", ""display_name"": ""Geometric Analysis and Curvature Flows"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Ricci flow"", ""complete noncompact Riemannian manifolds"", ""Ricci curvature bounds"", ""short-time existence"", ""curvature estimates"", ""Ricci tensor""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.4310/jdg/1214443292"", ""referenced_works"": [""W1509364003"", ""W1530161281"", ""W1970224775"", ""W2042049082""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 5}, {""year"": 2025, ""cited_by_count"": 15}, {""year"": 2024, ""cited_by_count"": 10}, {""year"": 2023, ""cited_by_count"": 10}, {""year"": 2022, ""cited_by_count"": 13}, {""year"": 2021, ""cited_by_count"": 33}, {""year"": 2020, ""cited_by_count"": 33}, {""year"": 2019, ""cited_by_count"": 14}, {""year"": 2018, ""cited_by_count"": 20}, {""year"": 2017, ""cited_by_count"": 30}, {""year"": 2016, ""cited_by_count"": 22}, {""year"": 2015, ""cited_by_count"": 25}, {""year"": 2014, ""cited_by_count"": 22}, {""year"": 2013, ""cited_by_count"": 26}, {""year"": 2012, ""cited_by_count"": 44}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R090,351,A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,Sesum2005,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"\bibitem{Sesum2005}
N.~\v{S}e\v{s}um,
\emph{Curvature tensor under the Ricci flow},
Amer. J. Math. \textbf{127} (2005), no.~6, 1315--1324.
\href{https://doi.org/10.1353/ajm.2005.0042}{doi:10.1353/ajm.2005.0042}.",68,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/references/bibliography.tex#L68,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1353/ajm.2005.0042"", ""title"": [""Curvature tensor under the Ricci flow""], ""authors"": [{""given"": ""Nataša"", ""family"": ""Šešum"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Project MUSE"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 12]]}, ""container_title"": [""American Journal of Mathematics""], ""url"": ""https://doi.org/10.1353/ajm.2005.0042"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1353%2Fajm.2005.0042""}}]","[""https://api.crossref.org/works/10.1353%2Fajm.2005.0042""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2093041805"", ""doi"": ""10.1353/ajm.2005.0042"", ""pmid"": null, ""title"": ""Curvature tensor under the Ricci flow"", ""publication_year"": 2005, ""publication_date"": ""2005-12-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5036804648"", ""name"": ""Nataša Šešum"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S114406506"", ""display_name"": ""American Journal of Mathematics"", ""issn_l"": ""0002-9327"", ""type"": ""journal""}, ""biblio"": {""volume"": ""127"", ""issue"": ""6"", ""first_page"": ""1315"", ""last_page"": ""1324""}, ""cited_by_count"": 90, ""fwci"": 3.83, ""referenced_works_count"": 6, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T10229"", ""display_name"": ""Geometric Analysis and Curvature Flows"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Ricci flow"", ""curvature tensor"", ""Ricci curvature"", ""Banach-Steinhaus theorem"", ""Ricci operator"", ""Ricci flow singularities""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1353/ajm.2005.0042"", ""referenced_works"": [""W1530161281"", ""W1587685671"", ""W1970564715"", ""W1977333154"", ""W2043328930"", ""W2168915657""], ""counts_by_year"": [{""year"": 2025, ""cited_by_count"": 4}, {""year"": 2024, ""cited_by_count"": 6}, {""year"": 2023, ""cited_by_count"": 1}, {""year"": 2022, ""cited_by_count"": 4}, {""year"": 2021, ""cited_by_count"": 7}, {""year"": 2020, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 2}, {""year"": 2018, ""cited_by_count"": 6}, {""year"": 2017, ""cited_by_count"": 6}, {""year"": 2016, ""cited_by_count"": 4}, {""year"": 2015, ""cited_by_count"": 5}, {""year"": 2014, ""cited_by_count"": 4}, {""year"": 2013, ""cited_by_count"": 7}, {""year"": 2012, ""cited_by_count"": 4}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R091,369,Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,Filonov2005,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Filonov2005,
  author = {Filonov, N.},
  title = {On an inequality between {Dirichlet} and {Neumann} eigenvalues for the {Laplace} operator},
  journal = {St. Petersburg Mathematical Journal},
  volume = {16}, number = {2}, pages = {413--416}, year = {2005},
  doi = {10.1090/S1061-0022-05-00857-5},
  url = {https://doi.org/10.1090/S1061-0022-05-00857-5},
  note = {Russian original: Algebra i Analiz 16 (2004), no. 2, 172--176}
}",130,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/references/references.bib#L130,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1090/s1061-0022-05-00857-5"", ""title"": [""On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator""], ""authors"": [{""given"": ""N."", ""family"": ""Filonov"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""American Mathematical Society (AMS)"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2005, 3, 9]]}, ""container_title"": [""St. Petersburg Mathematical Journal""], ""url"": ""https://doi.org/10.1090/s1061-0022-05-00857-5"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1090%2Fs1061-0022-05-00857-5""}}]","[""https://api.crossref.org/works/10.1090%2Fs1061-0022-05-00857-5""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W1521793072"", ""doi"": ""10.1090/s1061-0022-05-00857-5"", ""pmid"": null, ""title"": ""On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator"", ""publication_year"": 2005, ""publication_date"": ""2005-03-09"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5084389198"", ""name"": ""Nikolai Dmitrievich Filonov"", ""orcid"": ""https://orcid.org/0000-0002-1586-3031"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""St Petersburg University""]}], ""source"": {""source_id"": ""S113566553"", ""display_name"": ""St Petersburg Mathematical Journal"", ""issn_l"": ""1061-0022"", ""type"": ""journal""}, ""biblio"": {""volume"": ""16"", ""issue"": ""2"", ""first_page"": ""413"", ""last_page"": ""416""}, ""cited_by_count"": 61, ""fwci"": 1.6459, ""referenced_works_count"": 7, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://www.ams.org/spmj/2005-16-02/S1061-0022-05-00857-5/S1061-0022-05-00857-5.pdf""}, ""best_oa_pdf_url"": ""https://www.ams.org/spmj/2005-16-02/S1061-0022-05-00857-5/S1061-0022-05-00857-5.pdf"", ""primary_topic"": {""id"": ""T11022"", ""display_name"": ""Spectral Theory in Mathematical Physics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Dirichlet eigenvalues"", ""Neumann eigenvalues"", ""Laplacian"", ""eigenvalue inequalities"", ""spectral theory"", ""bounded domains""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1090/s1061-0022-05-00857-5"", ""referenced_works"": [""W2057096837"", ""W2064679812"", ""W2313169515"", ""W2326586272"", ""W2335197257"", ""W2972471826"", ""W4238052107""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 3}, {""year"": 2025, ""cited_by_count"": 5}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 3}, {""year"": 2021, ""cited_by_count"": 4}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2019, ""cited_by_count"": 5}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 2}, {""year"": 2015, ""cited_by_count"": 4}, {""year"": 2014, ""cited_by_count"": 5}, {""year"": 2013, ""cited_by_count"": 4}, {""year"": 2012, ""cited_by_count"": 3}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R092,369,Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,BanuelosBurdzy1999,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{BanuelosBurdzy1999,
  author  = {Ba{\~n}uelos, Rodrigo and Burdzy, Krzysztof},
  title   = {On the ``hot spots'' conjecture of {J. Rauch}},
  journal = {Journal of Functional Analysis},
  volume  = {164},
  number  = {1},
  pages   = {1--33},
  year    = {1999},
  doi     = {10.1006/jfan.1999.3397},
  url     = {https://doi.org/10.1006/jfan.1999.3397}
}",16,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/references/references.bib#L16,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1006/jfan.1999.3397"", ""title"": [""On the “Hot Spots” Conjecture of J. Rauch""], ""authors"": [{""given"": ""Rodrigo"", ""family"": ""Banuelos"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Krzysztof"", ""family"": ""Burdzy"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Elsevier BV"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1999, 5]]}, ""container_title"": [""Journal of Functional Analysis""], ""url"": ""https://doi.org/10.1006/jfan.1999.3397"", ""license"": [{""start"": {""date-parts"": [[1999, 5, 1]], ""date-time"": ""1999-05-01T00:00:00Z"", ""timestamp"": 925516800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""https://www.elsevier.com/tdm/userlicense/1.0/""}, {""start"": {""date-parts"": [[2013, 7, 17]], ""date-time"": ""2013-07-17T00:00:00Z"", ""timestamp"": 1374019200000}, ""content-version"": ""vor"", ""delay-in-days"": 5191, ""URL"": ""https://www.elsevier.com/open-access/userlicense/1.0/""}], ""source_url"": ""https://api.crossref.org/works/10.1006%2Fjfan.1999.3397""}}]","[""https://api.crossref.org/works/10.1006%2Fjfan.1999.3397""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2005045710"", ""doi"": ""10.1006/jfan.1999.3397"", ""pmid"": null, ""title"": ""On the “Hot Spots” Conjecture of J. Rauch"", ""publication_year"": 1999, ""publication_date"": ""1999-05-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5066168271"", ""name"": ""Rodrigo Bañuelos"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Purdue University West Lafayette""]}, {""author_id"": ""A5016736709"", ""name"": ""Krzysztof Burdzy"", ""orcid"": ""https://orcid.org/0000-0003-0986-3622"", ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Washington""]}], ""source"": {""source_id"": ""S54953690"", ""display_name"": ""Journal of Functional Analysis"", ""issn_l"": ""0022-1236"", ""type"": ""journal""}, ""biblio"": {""volume"": ""164"", ""issue"": ""1"", ""first_page"": ""1"", ""last_page"": ""33""}, ""cited_by_count"": 175, ""fwci"": 8.597, ""referenced_works_count"": 20, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://doi.org/10.1006/jfan.1999.3397""}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11022"", ""display_name"": ""Spectral Theory in Mathematical Physics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Rauch"", ""heat equation"", ""heat kernel"", ""maximum principle"", ""eigenfunctions"", ""counterexample""], ""abstract"": null, ""referenced_works"": [""W1535130726"", ""W1567771969"", ""W1580561995"", ""W1592958377"", ""W1982537625"", ""W2007151587"", ""W2009155227"", ""W2025775059"", ""W2041355784"", ""W2052062972"", ""W2075085615"", ""W2077395514"", ""W2090224342"", ""W2091603956"", ""W2104898428"", ""W2144624397"", ""W2164094632"", ""W3209611505"", ""W4300793121"", ""W4302567662""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 3}, {""year"": 2025, ""cited_by_count"": 3}, {""year"": 2024, ""cited_by_count"": 5}, {""year"": 2023, ""cited_by_count"": 6}, {""year"": 2022, ""cited_by_count"": 6}, {""year"": 2021, ""cited_by_count"": 6}, {""year"": 2020, ""cited_by_count"": 8}, {""year"": 2019, ""cited_by_count"": 26}, {""year"": 2018, ""cited_by_count"": 5}, {""year"": 2017, ""cited_by_count"": 4}, {""year"": 2016, ""cited_by_count"": 4}, {""year"": 2015, ""cited_by_count"": 5}, {""year"": 2014, ""cited_by_count"": 12}, {""year"": 2013, ""cited_by_count"": 9}, {""year"": 2012, ""cited_by_count"": 5}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R093,369,Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,Rauch1975,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@incollection{Rauch1975,
  author    = {Rauch, Jeffrey},
  title     = {Lecture \#1. {Five} problems: An introduction to the qualitative theory of partial differential equations},
  editor    = {Goldstein, Jerome A.},
  booktitle = {Partial Differential Equations and Related Topics},
  series    = {Lecture Notes in Mathematics},
  volume    = {446},
  pages     = {355--369},
  publisher = {Springer},
  address   = {Berlin, Heidelberg},
  year      = {1975},
  doi       = {10.1007/BFb0070610},
  url       = {https://doi.org/10.1007/BFb0070610}
}",1,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/references/references.bib#L1,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bfb0070610"", ""title"": [""Lecture #1. Five problems: An introduction to the qualitative theory of partial differential equations""], ""authors"": [{""given"": ""Jeffrey"", ""family"": ""Rauch"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""role"": ""author"", ""vocabulary"": ""crossref""}]}], ""publisher"": ""Springer Berlin Heidelberg"", ""type"": ""book-chapter"", ""published"": {""date-parts"": [[1975]]}, ""container_title"": [""Lecture Notes in Mathematics"", ""Partial Differential Equations and Related Topics""], ""url"": ""https://doi.org/10.1007/bfb0070610"", ""license"": [{""start"": {""date-parts"": [[1975, 1, 1]], ""date-time"": ""1975-01-01T00:00:00Z"", ""timestamp"": 157766400000}, ""content-version"": ""unspecified"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbfb0070610""}}]","[""https://api.crossref.org/works/10.1007%2Fbfb0070610""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W999437016"", ""doi"": ""10.1007/bfb0070610"", ""pmid"": null, ""title"": ""Lecture #1. Five problems: An introduction to the qualitative theory of partial differential equations"", ""publication_year"": 1975, ""publication_date"": ""1975-01-01"", ""type"": ""book-chapter"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5113616831"", ""name"": ""Jeffrey B. Rauch"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Michigan""]}], ""source"": {""source_id"": ""S118039931"", ""display_name"": ""Lecture notes in mathematics"", ""issn_l"": ""0075-8434"", ""type"": ""book series""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": ""355"", ""last_page"": ""369""}, ""cited_by_count"": 28, ""fwci"": 2.1388, ""referenced_works_count"": 0, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T12589"", ""display_name"": ""Differential Equations and Boundary Problems"", ""field"": ""Mathematics"", ""subfield"": ""Applied Mathematics"", ""domain"": ""Physical Sciences""}, ""keywords"": [], ""abstract"": null, ""referenced_works"": [], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 2}, {""year"": 2021, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 1}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2013, ""cited_by_count"": 1}, {""year"": 2012, ""cited_by_count"": 2}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R094,369,Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,Pommerenke1992,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@book{Pommerenke1992,
  author = {Pommerenke, Christian},
  title = {Boundary Behaviour of Conformal Maps},
  series = {Grundlehren der mathematischen Wissenschaften},
  volume = {299}, publisher = {Springer}, address = {Berlin, Heidelberg},
  year = {1992}, doi = {10.1007/978-3-662-02770-7},
  url = {https://doi.org/10.1007/978-3-662-02770-7}
}",149,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/references/references.bib#L149,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/978-3-662-02770-7"", ""title"": [""Boundary Behaviour of Conformal Maps""], ""authors"": [{""given"": ""Christian"", ""family"": ""Pommerenke"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Berlin Heidelberg"", ""type"": ""book"", ""published"": {""date-parts"": [[1992]]}, ""container_title"": [""Grundlehren der mathematischen Wissenschaften""], ""url"": ""https://doi.org/10.1007/978-3-662-02770-7"", ""license"": [{""start"": {""date-parts"": [[1992, 1, 1]], ""date-time"": ""1992-01-01T00:00:00Z"", ""timestamp"": 694224000000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2F978-3-662-02770-7""}}]","[""https://api.crossref.org/works/10.1007%2F978-3-662-02770-7""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2064685239"", ""doi"": ""10.1007/978-3-662-02770-7"", ""pmid"": null, ""title"": ""Boundary Behaviour of Conformal Maps"", ""publication_year"": 1992, ""publication_date"": ""1992-01-01"", ""type"": ""book"", ""language"": ""de"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5110824357"", ""name"": ""Christian Pommerenke"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""Technische Universität Berlin""]}], ""source"": {""source_id"": ""S4210203205"", ""display_name"": ""Grundlehren der mathematischen Wissenschaften"", ""issn_l"": ""0072-7830"", ""type"": ""book series""}, ""biblio"": {""volume"": null, ""issue"": null, ""first_page"": null, ""last_page"": null}, ""cited_by_count"": 1987, ""fwci"": 6.3443, ""referenced_works_count"": 2, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11822"", ""display_name"": ""Analytic and geometric function theory"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""conformal mapping"", ""boundary behavior"", ""unit disk"", ""univalent functions"", ""boundary correspondence"", ""geometric function theory""], ""abstract"": null, ""referenced_works"": [""W88299743"", ""W122400612""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 35}, {""year"": 2025, ""cited_by_count"": 64}, {""year"": 2024, ""cited_by_count"": 70}, {""year"": 2023, ""cited_by_count"": 66}, {""year"": 2022, ""cited_by_count"": 85}, {""year"": 2021, ""cited_by_count"": 81}, {""year"": 2020, ""cited_by_count"": 83}, {""year"": 2019, ""cited_by_count"": 81}, {""year"": 2018, ""cited_by_count"": 81}, {""year"": 2017, ""cited_by_count"": 91}, {""year"": 2016, ""cited_by_count"": 82}, {""year"": 2015, ""cited_by_count"": 93}, {""year"": 2014, ""cited_by_count"": 88}, {""year"": 2013, ""cited_by_count"": 83}, {""year"": 2012, ""cited_by_count"": 73}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R095,369,Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026,Miyamoto2007,unverified,unverified,match,supplied_notes_identity,Supplied notes identify Miyamoto and the RIMS1591 nearly circular domains preprint. Its publication date and stated revision date remain unverified.,"@misc{Miyamoto2007,
  author = {Miyamoto, Yasuhito},
  title = {The ``hot spots'' conjecture for nearly circular planar convex domains},
  year = {2007},
  howpublished = {RIMS Preprint 1591},
  note = {Revised 30 May 2007},
  url = {https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf}
}",295,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/references/references.bib#L295,[],[],[],"[{""sample_id"": ""R095"", ""source_url"": ""https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""f0327dfdbbd01a05e860a7cde3e5b088bbd5f28f0ae029d4c1fd8a972c20c9b5"", ""locator"": ""PDF pages 1–2, Theorem A, Lemma B, Lemma 1.2"", ""source_version"": ""RIMS1591 author preprint"", ""paraphrased_observations"": ""The title and author identify Yasuhito Miyamoto and the nearly circular planar convex domains preprint. Theorem A uses a strict diameter-squared/area bound near 1.378. Lemma B uses the strict spectral-diameter condition sqrt(lambda_2) < j_0 / diameter. Lemma 1.2 excludes interior critical points under those hypotheses. Page 1 defines an open bounded smooth domain and a nonzero second Neumann eigenfunction. Page 2 expressly does not require symmetry. The preprint URL is RIMS1591; the inspected pages do not by themselves establish the publication date."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R096,372,Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,Grauert1958,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Grauert1958,
  author  = {Grauert, Hans},
  title   = {Analytische {Faserungen} {\""u}ber holomorph-vollst{\""a}ndigen {R{\""a}umen}},
  journal = {Mathematische Annalen},
  volume  = {135},
  year    = {1958},
  pages   = {263--273},
  doi     = {10.1007/BF01351803},
  note    = {\href{https://doi.org/10.1007/BF01351803}{doi:10.1007/BF01351803}}
}",100,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/references.bib#L100,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/bf01351803"", ""title"": [""Analytische Faserungen �ber holomorph-vollst�ndigen R�umen""], ""authors"": [{""given"": ""Hans"", ""family"": ""Grauert"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[1958, 6]]}, ""container_title"": [""Mathematische Annalen""], ""url"": ""https://doi.org/10.1007/bf01351803"", ""license"": [{""start"": {""date-parts"": [[1958, 6, 1]], ""date-time"": ""1958-06-01T00:00:00Z"", ""timestamp"": -365644800000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fbf01351803""}}]","[""https://api.crossref.org/works/10.1007%2Fbf01351803""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2914074126"", ""doi"": ""10.1007/bf01351803"", ""pmid"": null, ""title"": ""Analytische Faserungen �ber holomorph-vollst�ndigen R�umen"", ""publication_year"": 1958, ""publication_date"": ""1958-06-01"", ""type"": ""article"", ""language"": ""de"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5012532416"", ""name"": ""Hans Grauert"", ""orcid"": null, ""position"": ""first"", ""is_corresponding"": true, ""institutions"": []}], ""source"": {""source_id"": ""S192217950"", ""display_name"": ""Mathematische Annalen"", ""issn_l"": ""0025-5831"", ""type"": ""journal""}, ""biblio"": {""volume"": ""135"", ""issue"": ""3"", ""first_page"": ""263"", ""last_page"": ""273""}, ""cited_by_count"": 282, ""fwci"": 5.7682, ""referenced_works_count"": 13, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T12351"", ""display_name"": ""Geometry and complex manifolds"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""holomorphic mappings""], ""abstract"": null, ""referenced_works"": [""W1509036934"", ""W1570939055"", ""W1970104469"", ""W1973576673"", ""W1985038052"", ""W1991290444"", ""W2012015894"", ""W2021304735"", ""W2029138857"", ""W2082916946"", ""W2525125412"", ""W4230588464"", ""W4243298300""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 4}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2023, ""cited_by_count"": 6}, {""year"": 2022, ""cited_by_count"": 4}, {""year"": 2021, ""cited_by_count"": 5}, {""year"": 2020, ""cited_by_count"": 5}, {""year"": 2019, ""cited_by_count"": 8}, {""year"": 2018, ""cited_by_count"": 8}, {""year"": 2017, ""cited_by_count"": 4}, {""year"": 2016, ""cited_by_count"": 8}, {""year"": 2015, ""cited_by_count"": 3}, {""year"": 2014, ""cited_by_count"": 9}, {""year"": 2013, ""cited_by_count"": 7}, {""year"": 2012, ""cited_by_count"": 5}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R097,372,Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,Eskin2001,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Eskin2001,
  author  = {Eskin, Gregory},
  title   = {Global uniqueness in the inverse scattering problem for the {Schr{\""o}dinger} operator with external {Yang--Mills} potentials},
  journal = {Communications in Mathematical Physics},
  volume  = {222},
  number  = {3},
  year    = {2001},
  pages   = {503--531},
  doi     = {10.1007/s002200100522},
  note    = {\href{https://doi.org/10.1007/s002200100522}{doi:10.1007/s002200100522}}
}",73,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/references.bib#L73,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s002200100522"", ""title"": [""Global Uniqueness in the Inverse Scattering Problem¶for the Schrödinger Operator¶with External Yang-Mills Potentials""], ""authors"": [{""given"": ""G."", ""family"": ""Eskin"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2001, 9, 1]]}, ""container_title"": [""Communications in Mathematical Physics""], ""url"": ""https://doi.org/10.1007/s002200100522"", ""license"": [{""start"": {""date-parts"": [[2001, 9, 1]], ""date-time"": ""2001-09-01T00:00:00Z"", ""timestamp"": 999302400000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs002200100522""}}]","[""https://api.crossref.org/works/10.1007%2Fs002200100522""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2053745339"", ""doi"": ""10.1007/s002200100522"", ""pmid"": null, ""title"": ""Global Uniqueness in the Inverse Scattering Problem¶for the Schrödinger Operator¶with External Yang-Mills Potentials"", ""publication_year"": 2001, ""publication_date"": ""2001-09-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5038696251"", ""name"": ""Gregory Eskin"", ""orcid"": ""https://orcid.org/0000-0001-6837-8470"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of California, Los Angeles""]}], ""source"": {""source_id"": ""S100601982"", ""display_name"": ""Communications in Mathematical Physics"", ""issn_l"": ""0010-3616"", ""type"": ""journal""}, ""biblio"": {""volume"": ""222"", ""issue"": ""3"", ""first_page"": ""503"", ""last_page"": ""531""}, ""cited_by_count"": 52, ""fwci"": 2.0136, ""referenced_works_count"": 15, ""open_access"": {""is_oa"": false, ""oa_status"": ""closed"", ""oa_url"": null}, ""best_oa_pdf_url"": null, ""primary_topic"": {""id"": ""T11022"", ""display_name"": ""Spectral Theory in Mathematical Physics"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""inverse scattering"", ""Schrödinger operators"", ""Yang-Mills potential"", ""gauge transformations"", ""global uniqueness"", ""inverse boundary value problem"", ""compactly supported potentials"", ""bounded convex domains"", ""fixed energy scattering""], ""abstract"": null, ""referenced_works"": [""W1601637945"", ""W1676548716"", ""W1976602763"", ""W2020048703"", ""W2035089934"", ""W2043315668"", ""W2047621588"", ""W2048309834"", ""W2049130415"", ""W2052824783"", ""W2061570073"", ""W2104152993"", ""W2289765423"", ""W2326571829"", ""W2608960539""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 2}, {""year"": 2022, ""cited_by_count"": 2}, {""year"": 2021, ""cited_by_count"": 3}, {""year"": 2020, ""cited_by_count"": 3}, {""year"": 2019, ""cited_by_count"": 1}, {""year"": 2018, ""cited_by_count"": 2}, {""year"": 2017, ""cited_by_count"": 5}, {""year"": 2015, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 1}, {""year"": 2013, ""cited_by_count"": 6}, {""year"": 2012, ""cited_by_count"": 4}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R098,372,Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,Weck1969,unverified,unverified,match,supplied_notes_identity,"Supplied EuDML catalogue notes support Weck, title, Mathematische Zeitschrift 111 (1969), 387–398. This is an externally read digital-library catalogue, not a directly checked publisher DOI record.","@article{Weck1969,
  author  = {Weck, Norbert},
  title   = {Au{\ss}enraumaufgaben in der {Theorie} station{\""a}rer {Schwingungen} inhomogener elastischer {K{\""o}rper}},
  journal = {Mathematische Zeitschrift},
  volume  = {111},
  year    = {1969},
  pages   = {387--398},
  url     = {https://eudml.org/doc/171213},
  note    = {\href{https://eudml.org/doc/171213}{EuDML record}}
}",123,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/references.bib#L123,[],[],[],"[{""sample_id"": ""R098"", ""source_url"": ""https://eudml.org/doc/171213"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""f2d3217568a5a2549de7bdd48ecab16bad4a53004b04b99a4d1faf69fff395eb"", ""locator"": ""EuDML document 171213, bibliography record"", ""source_version"": null, ""paraphrased_observations"": ""The record identifies Norbert Weck, Aussenraumaufgaben in der Theorie stationarer Schwingungen inhomogener elastischer Korper, Mathematische Zeitschrift 111 (1969), pages 387–398. Treat EuDML as a digital-library catalogue; distinguish it from a publisher-deposited DOI record."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R099,372,Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,NU2003,minor_difference,minor_difference,minor_difference,direct_records,Same Nakamura–Uhlmann DOI/authors and 2003 publication. Citation prefixes “Erratum:”; Crossref omits that label from the title. OpenAlex pages 205–207 agree; registry title alone cannot certify correction scope.,"@article{NU2003,
  author  = {Nakamura, Gen and Uhlmann, Gunther},
  title   = {Erratum: {Global} uniqueness for an inverse boundary value problem arising in elasticity},
  journal = {Inventiones Mathematicae},
  volume  = {152},
  number  = {1},
  year    = {2003},
  pages   = {205--207},
  doi     = {10.1007/s00222-002-0276-1},
  note    = {\href{https://doi.org/10.1007/s00222-002-0276-1}{doi:10.1007/s00222-002-0276-1}}
}",12,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/references.bib#L12,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1007/s00222-002-0276-1"", ""title"": [""Global uniqueness for an inverse boundary value problem arising in elasticity""], ""authors"": [{""given"": ""Gen"", ""family"": ""Nakamura"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}, {""given"": ""Gunther"", ""family"": ""Uhlmann"", ""sequence"": ""additional"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""Springer Science and Business Media LLC"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2003, 4, 1]]}, ""container_title"": [""Inventiones Mathematicae""], ""url"": ""https://doi.org/10.1007/s00222-002-0276-1"", ""license"": [{""start"": {""date-parts"": [[2003, 4, 1]], ""date-time"": ""2003-04-01T00:00:00Z"", ""timestamp"": 1049155200000}, ""content-version"": ""tdm"", ""delay-in-days"": 0, ""URL"": ""http://www.springer.com/tdm""}], ""source_url"": ""https://api.crossref.org/works/10.1007%2Fs00222-002-0276-1""}}]","[""https://api.crossref.org/works/10.1007%2Fs00222-002-0276-1""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W2061570073"", ""doi"": ""10.1007/s00222-002-0276-1"", ""pmid"": null, ""title"": ""Global uniqueness for an inverse boundary value problem arising in elasticity"", ""publication_year"": 2003, ""publication_date"": ""2003-04-01"", ""type"": ""article"", ""language"": ""en"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5061165395"", ""name"": ""Gen Nakamura"", ""orcid"": ""https://orcid.org/0000-0002-7911-8612"", ""position"": ""first"", ""is_corresponding"": false, ""institutions"": [""Hokkaido University""]}, {""author_id"": ""A5109171815"", ""name"": ""Gunther A. Uhlmann"", ""orcid"": null, ""position"": ""last"", ""is_corresponding"": false, ""institutions"": [""University of Washington""]}], ""source"": {""source_id"": ""S98347115"", ""display_name"": ""Inventiones mathematicae"", ""issn_l"": ""0020-9910"", ""type"": ""journal""}, ""biblio"": {""volume"": ""152"", ""issue"": ""1"", ""first_page"": ""205"", ""last_page"": ""207""}, ""cited_by_count"": 170, ""fwci"": 5.388, ""referenced_works_count"": 11, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://link.springer.com/content/pdf/10.1007/s00222-002-0276-1.pdf""}, ""best_oa_pdf_url"": ""https://link.springer.com/content/pdf/10.1007/s00222-002-0276-1.pdf"", ""primary_topic"": {""id"": ""T11205"", ""display_name"": ""Numerical methods in inverse problems"", ""field"": ""Mathematics"", ""subfield"": ""Mathematical Physics"", ""domain"": ""Physical Sciences""}, ""keywords"": [""inverse boundary value problem"", ""global uniqueness"", ""Lamé parameters"", ""boundary measurements"", ""elasticity""], ""abstract"": null, ""referenced_works"": [""W1586402789"", ""W2012485390"", ""W2020596016"", ""W2023638857"", ""W2027842332"", ""W2048309834"", ""W2049176462"", ""W2055133633"", ""W2087318848"", ""W2326571829"", ""W4248913130""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 2}, {""year"": 2025, ""cited_by_count"": 2}, {""year"": 2024, ""cited_by_count"": 3}, {""year"": 2023, ""cited_by_count"": 10}, {""year"": 2022, ""cited_by_count"": 6}, {""year"": 2021, ""cited_by_count"": 13}, {""year"": 2020, ""cited_by_count"": 5}, {""year"": 2019, ""cited_by_count"": 13}, {""year"": 2018, ""cited_by_count"": 3}, {""year"": 2017, ""cited_by_count"": 11}, {""year"": 2016, ""cited_by_count"": 5}, {""year"": 2015, ""cited_by_count"": 9}, {""year"": 2014, ""cited_by_count"": 11}, {""year"": 2013, ""cited_by_count"": 7}, {""year"": 2012, ""cited_by_count"": 4}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
R100,372,Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,Forstneric2009,match,match,match,direct_records,Original record supports work identity (title and authors); available venue/date information is consistent after ordinary formatting normalization.,"@article{Forstneric2009,
  author  = {Forstneri{\v c}, Franc},
  title   = {Oka manifolds},
  journal = {Comptes Rendus. Math\'{e}matique},
  volume  = {347},
  number  = {17--18},
  year    = {2009},
  pages   = {1017--1020},
  doi     = {10.1016/j.crma.2009.07.005},
  note    = {\href{https://doi.org/10.1016/j.crma.2009.07.005}{doi:10.1016/j.crma.2009.07.005}}
}",111,https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/references.bib#L111,"[{""provider"": ""Crossref publisher deposit"", ""record"": {""doi"": ""10.1016/j.crma.2009.07.005"", ""title"": [""Oka manifolds""], ""authors"": [{""given"": ""Franc"", ""family"": ""Forstnerič"", ""sequence"": ""first"", ""affiliation"": [], ""role"": [{""vocabulary"": ""crossref"", ""role"": ""author""}]}], ""publisher"": ""MathDoc/Centre Mersenne"", ""type"": ""journal-article"", ""published"": {""date-parts"": [[2009, 8, 5]]}, ""container_title"": [""Comptes Rendus. Mathématique""], ""url"": ""https://doi.org/10.1016/j.crma.2009.07.005"", ""license"": [], ""source_url"": ""https://api.crossref.org/works/10.1016%2Fj.crma.2009.07.005""}}]","[""https://api.crossref.org/works/10.1016%2Fj.crma.2009.07.005""]","[{""provider"": ""OpenAlex (secondary corroboration)"", ""record"": {""openalex_id"": ""W4235939950"", ""doi"": ""10.1016/j.crma.2009.07.005"", ""pmid"": null, ""title"": ""Oka manifolds"", ""publication_year"": 2009, ""publication_date"": ""2009-08-05"", ""type"": ""article"", ""language"": ""fr"", ""is_retracted"": false, ""authors"": [{""author_id"": ""A5043857437"", ""name"": ""Franc Forstnerič"", ""orcid"": ""https://orcid.org/0000-0002-7975-0212"", ""position"": ""first"", ""is_corresponding"": true, ""institutions"": [""University of Ljubljana"", ""Institute of Mathematics, Physics, and Mechanics""]}], ""source"": {""source_id"": ""S128122872"", ""display_name"": ""Comptes Rendus Mathématique"", ""issn_l"": ""1631-073X"", ""type"": ""journal""}, ""biblio"": {""volume"": ""347"", ""issue"": ""17-18"", ""first_page"": ""1017"", ""last_page"": ""1020""}, ""cited_by_count"": 24, ""fwci"": 1.2532, ""referenced_works_count"": 9, ""open_access"": {""is_oa"": true, ""oa_status"": ""bronze"", ""oa_url"": ""https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2009.07.005.pdf""}, ""best_oa_pdf_url"": ""https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2009.07.005.pdf"", ""primary_topic"": {""id"": ""T12351"", ""display_name"": ""Geometry and complex manifolds"", ""field"": ""Mathematics"", ""subfield"": ""Geometry and Topology"", ""domain"": ""Physical Sciences""}, ""keywords"": [""Oka manifolds"", ""Gromov's question"", ""holomorphic mappings"", ""convex sets"", ""weak homotopy equivalence"", ""Stein spaces"", ""holomorphic function spaces"", ""complex manifolds""], ""abstract"": null, ""abstract_license"": null, ""abstract_policy"": ""Abstract withheld: license 'undeclared' is not verified-open (only cc-by/cc-by-sa/cc0/public-domain reconstruct). See https://doi.org/10.1016/j.crma.2009.07.005"", ""referenced_works"": [""W1789934502"", ""W1974198746"", ""W1977539191"", ""W2019386427"", ""W2034370106"", ""W2098243695"", ""W2914074126"", ""W2962701743"", ""W4205111176""], ""counts_by_year"": [{""year"": 2026, ""cited_by_count"": 1}, {""year"": 2025, ""cited_by_count"": 1}, {""year"": 2024, ""cited_by_count"": 1}, {""year"": 2023, ""cited_by_count"": 3}, {""year"": 2022, ""cited_by_count"": 1}, {""year"": 2021, ""cited_by_count"": 1}, {""year"": 2020, ""cited_by_count"": 1}, {""year"": 2017, ""cited_by_count"": 1}, {""year"": 2016, ""cited_by_count"": 5}, {""year"": 2015, ""cited_by_count"": 1}, {""year"": 2014, ""cited_by_count"": 3}, {""year"": 2013, ""cited_by_count"": 2}, {""year"": 2012, ""cited_by_count"": 1}]}}]",[],Identity classification only; missing edition/date/locator fields are not certified. Direct records are preserved separately from operator-supplied reading notes; source-document hashes supplied in notes were not recomputed.
