{"id":"https://openalex.org/W7163996125","doi":"https://doi.org/10.48550/arxiv.2606.09449","title":"Reasoning without Gold Standards: A Proxy-Judge Theory of Autoformalization","display_name":"Reasoning without Gold Standards: A Proxy-Judge Theory of Autoformalization","publication_year":2026,"publication_date":"2026-06-08","ids":{"openalex":"https://openalex.org/W7163996125","doi":"https://doi.org/10.48550/arxiv.2606.09449"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.09449","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09449","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.09449","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138204676","display_name":"Lei Xu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Lei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138254691","display_name":"Xin Quan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Quan, Xin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5138259477","display_name":"Andr\u00e9 Freitas","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Freitas, Andr\u00e9","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":false,"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/T10142","display_name":"Formal Methods in Verification","score":0.4674000144004822,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.4674000144004822,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.05389999970793724,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.04740000143647194,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/correctness","display_name":"Correctness","score":0.7131999731063843},{"id":"https://openalex.org/keywords/proxy","display_name":"Proxy (statistics)","score":0.680400013923645},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5821999907493591},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.5016999840736389},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.3912999927997589},{"id":"https://openalex.org/keywords/argument","display_name":"Argument (complex analysis)","score":0.3869999945163727},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.35260000824928284},{"id":"https://openalex.org/keywords/unary-operation","display_name":"Unary operation","score":0.3517000079154968}],"concepts":[{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.7131999731063843},{"id":"https://openalex.org/C2780148112","wikidata":"https://www.wikidata.org/wiki/Q1432581","display_name":"Proxy (statistics)","level":2,"score":0.680400013923645},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6381000280380249},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5821999907493591},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.5016999840736389},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4652000069618225},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3991999924182892},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.3912999927997589},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.3869999945163727},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.35260000824928284},{"id":"https://openalex.org/C78023250","wikidata":"https://www.wikidata.org/wiki/Q657596","display_name":"Unary operation","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C159718280","wikidata":"https://www.wikidata.org/wiki/Q5526353","display_name":"Gas meter prover","level":3,"score":0.3472000062465668},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.336899995803833},{"id":"https://openalex.org/C28427503","wikidata":"https://www.wikidata.org/wiki/Q13580300","display_name":"Internal model","level":3,"score":0.31540000438690186},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3000999987125397},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C160697094","wikidata":"https://www.wikidata.org/wiki/Q1233197","display_name":"Oblique case","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C134752490","wikidata":"https://www.wikidata.org/wiki/Q374182","display_name":"Logical consequence","level":2,"score":0.27480000257492065},{"id":"https://openalex.org/C21847791","wikidata":"https://www.wikidata.org/wiki/Q191081","display_name":"Logical conjunction","level":2,"score":0.2741999924182892},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C65059942","wikidata":"https://www.wikidata.org/wiki/Q270105","display_name":"Argumentation theory","level":2,"score":0.2678000032901764}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.09449","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09449","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.09449","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09449","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Complex":[0],"reasoning":[1,37],"tasks":[2],"increasingly":[3],"require":[4],"systems":[5],"to":[6,31,115,124,151,175,215],"produce":[7],"outputs":[8],"whose":[9],"correctness":[10],"cannot":[11],"be":[12],"judged":[13,163],"by":[14],"exact":[15,74,234],"match":[16],"against":[17],"a":[18,24,29,39,53,78,89,133,139,145,152,176,204,223,228],"single":[19,54],"reference.":[20],"Autoformalization":[21],"(AF)":[22],"is":[23,162],"representative":[25],"example;":[26],"it":[27,123],"asks":[28],"model":[30],"translate":[32],"informal":[33,55,126],"mathematical":[34],"or":[35],"logical":[36],"into":[38,132],"formally":[40],"checkable":[41],"object,":[42,111],"yet":[43],"expert-validated":[44],"formalizations":[45],"do":[46],"not":[47],"scale":[48],"beyond":[49],"toy":[50],"cases":[51],"and":[52,118,128,187,199,227],"argument":[56],"can":[57,71],"admit":[58],"many":[59],"valid":[60],"formal":[61],"renderings.":[62],"Progress":[63],"therefore":[64,220],"depends":[65],"on":[66,183,208,231],"whether":[67],"partial,":[68],"structured":[69],"proxies":[70],"substitute":[72],"for":[73,82],"references.":[75],"We":[76],"introduce":[77],"reference-free":[79],"proxy-judge":[80],"framework":[81,96],"AF":[83],"that":[84,104,121],"replaces":[85],"gold-standard":[86],"matching":[87,153],"with":[88],"vector":[90,137],"of":[91,108],"per-axis":[92,201],"property":[93],"checks.":[94],"The":[95,136],"organizes":[97],"the":[98,109,125,149,169,195,200,211],"proxy":[99,202,207,218],"along":[100],"three":[101],"structural":[102],"scopes":[103],"cover":[105],"global":[106],"properties":[107,113,120],"elicited":[110],"per-module":[112],"internal":[114],"its":[116],"sub-components,":[117],"cross-domain":[119],"re-align":[122],"source,":[127],"aggregates":[129],"each":[130,157],"axis":[131],"verdict":[134],"vector.":[135],"drives":[138],"reflective":[140],"refinement":[141,189,225],"loop":[142],"in":[143],"which":[144],"violated":[146],"coordinate":[147],"routes":[148],"controller":[150],"repair":[154],"target,":[155],"so":[156],"iteration":[158],"changes":[159],"only":[160],"what":[161],"wrong.":[164],"Under":[165],"bounded":[166],"judge":[167],"noise,":[168],"expected":[170],"intrinsic":[171],"gap":[172],"contracts":[173],"geometrically":[174],"noise-dependent":[177],"plateau.":[178],"Across":[179],"seven":[180],"formalization":[181],"backbones":[182],"miniF2F,":[184],"ProofNet,":[185],"e-SNLI,":[186],"ProntoQA,":[188],"consistently":[190],"lifts":[191],"Pass":[192],"Rate":[193],"over":[194],"single-shot":[196],"ICL":[197],"baseline,":[198],"outperforms":[203],"matched":[205],"scalar":[206],"benchmarks":[209],"where":[210],"baseline":[212],"has":[213],"room":[214],"improve.":[216],"Structured":[217],"judgments":[219],"provide":[221],"both":[222],"practical":[224],"signal":[226],"theoretical":[229],"handle":[230],"convergence":[232],"when":[233],"references":[235],"are":[236],"unavailable.":[237]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-10T00:00:00"}
