{"id":"https://openalex.org/W7223063752","doi":"https://doi.org/10.48550/arxiv.2610.24080","title":"An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps","display_name":"An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps","publication_year":2026,"publication_date":"2026-10-16","ids":{"openalex":"https://openalex.org/W7223063752","doi":"https://doi.org/10.48550/arxiv.2610.24080"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.24080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24080","pdf_url":"https://arxiv.org/pdf/2610.24080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.24080","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5124920886","display_name":"Bangjie Wang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Bangjie","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5017897375","display_name":"Igor Zelenko","orcid":"https://orcid.org/0000-0001-7900-2567","observed_orcids":["https://orcid.org/0000-0001-7900-2567"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zelenko, Igor","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.5889000296592712,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.5889000296592712,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.301800012588501,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.013299999758601189,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/sub-riemannian-geometry","display_name":"sub-Riemannian geometry","score":0.8809999823570251},{"id":"https://openalex.org/keywords/growth-vector","display_name":"growth vector","score":0.6859999895095825},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.42899999022483826}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8616999983787537},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.7226999998092651},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.585099995136261},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5062000155448914},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.461899995803833},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.4368000030517578},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3869999945163727},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38499999046325684},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.373199999332428},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3659999966621399},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.34700000286102295},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.34139999747276306},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.33320000767707825},{"id":"https://openalex.org/C95857938","wikidata":"https://www.wikidata.org/wiki/Q658429","display_name":"Vector bundle","level":2,"score":0.3330000042915344},{"id":"https://openalex.org/C46581059","wikidata":"https://www.wikidata.org/wiki/Q116301","display_name":"Periodic point","level":2,"score":0.31139999628067017},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.31060001254081726},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30889999866485596},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.30399999022483826},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.29159998893737793},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.28519999980926514},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C200917960","wikidata":"https://www.wikidata.org/wiki/Q3847057","display_name":"Singular point of a curve","level":2,"score":0.27469998598098755},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.26429998874664307},{"id":"https://openalex.org/C2777727622","wikidata":"https://www.wikidata.org/wiki/Q5251772","display_name":"Degeneracy (biology)","level":2,"score":0.25360000133514404},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.25279998779296875}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.24080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24080","pdf_url":"https://arxiv.org/pdf/2610.24080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.24080","doi":"https://doi.org/10.48550/arxiv.2610.24080","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.24080","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.24080","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24080","pdf_url":"https://arxiv.org/pdf/2610.24080","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223063752.pdf","grobid_xml":"https://content.openalex.org/works/W7223063752.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W2017942945","https://openalex.org/W2078359100","https://openalex.org/W2795802300","https://openalex.org/W7163193747","https://openalex.org/W7168397998","https://openalex.org/W7212061285"],"related_works":["https://openalex.org/W7212061285","https://openalex.org/W4289785367","https://openalex.org/W4388184607","https://openalex.org/W4406486946","https://openalex.org/W2801873482","https://openalex.org/W4319322638","https://openalex.org/W3110818605","https://openalex.org/W2894806444","https://openalex.org/W1513745936","https://openalex.org/W4400375078"],"abstract_inverted_index":{"Lerario,":[0],"Rizzi":[1],"and":[2,41,82,130,172,218,242,259,262],"Tiberio":[3],"disproved":[4],"the":[5,22,42,71,91,94,104,110,115,120,131,205,210,219,267,276],"sub-Riemannian":[6],"Sard":[7,95],"conjecture":[8],"with":[9,62,162,185,234],"a":[10,135,155,189,198,213,247,263,273],"rank-two":[11,52,85],"distribution":[12,35,53,124],"on":[13,54],"$\\mathbb":[14,60],"R^4$,":[15],"of":[16,18,93,103,109,134,147,164,166,200,216,221,266,275],"degree":[17],"nonholonomy":[19],"five":[20],"at":[21],"origin,":[23],"whose":[24,67],"endpoint":[25,68],"map":[26,69],"from":[27,70,114,212],"that":[28],"point":[29,215],"has":[30,45,73],"no":[31,74],"regular":[32,75],"values.":[33,76],"Their":[34],"is":[36,80,97,125,138],"neither":[37],"equiregular":[38,43,51,229],"nor":[39],"real-analytic,":[40],"case":[44],"remained":[46],"open.":[47],"We":[48,182],"construct":[49],"an":[50,55,139,144,169,173,228],"open":[56,145],"set":[57,146],"diffeomorphic":[58],"to":[59,100,127,154,188,243,257,272],"R^9$,":[61],"small":[63],"growth":[64,78,105],"vector":[65,79],"$(2,3,5,8,9)$,":[66],"origin":[72],"This":[77],"maximal":[81],"generic":[83],"for":[84],"distributions":[86],"in":[87,204,231,246],"dimension":[88,232],"nine,":[89],"so":[90],"failure":[92],"property":[96],"not":[98],"tied":[99],"any":[101],"degeneracy":[102],"vector.":[106],"Singular":[107],"curves":[108],"Lerario--Rizzi--Tiberio-type":[111],"structure":[112],"built":[113],"same":[116],"coefficient":[117],"lift,":[118],"inside":[119],"strip":[121,141],"where":[122],"our":[123,196],"defined,":[126],"singular":[128,151],"curves,":[129],"vertical":[132,148],"displacement":[133],"horizontal":[136],"loop":[137],"explicit":[140],"integral.":[142],"Realising":[143],"displacements":[149],"by":[150],"loops":[152],"reduces":[153],"graded":[156],"moment":[157],"problem,":[158],"which":[159,208,269],"we":[160],"solve":[161],"pairs":[163],"bumps":[165],"opposite":[167],"winding,":[168],"anisotropic":[170],"rescaling,":[171],"inverse":[174],"function":[175],"theorem":[176],"uniform":[177],"over":[178],"infinitely":[179],"many":[180],"scales.":[181],"also":[183,254,270],"state,":[184],"proofs":[186],"deferred":[187],"planned":[190],"longer":[191,251],"version,":[192],"two":[193],"results":[194],"about":[195],"distribution:":[197],"description":[199],"its":[201,222],"abnormal":[202],"dynamics":[203],"cotangent":[206],"bundle,":[207],"explains":[209],"example":[211],"dynamical":[214],"view,":[217],"non-constancy":[220],"Tanaka":[223,236],"symbol.":[224],"The":[225,250],"latter":[226],"motivates":[227],"counterexample":[230],"twelve":[233],"constant":[235],"symbol":[237],"(constant":[238],"nilpotentisation),":[239],"announced":[240],"here":[241],"be":[244],"proved":[245],"separate":[248],"paper.":[249],"version":[252],"will":[253],"contain":[255],"obstructions":[256],"real-analytic":[258],"lower-dimensional":[260],"versions,":[261],"transversal":[264],"variant":[265],"construction,":[268],"leads":[271],"modification":[274],"Lerario--Rizzi--Tiberio":[277],"counterexample.":[278]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-21T00:00:00"}