{"id":"https://openalex.org/W7164831380","doi":"https://doi.org/10.48550/arxiv.2606.14688","title":"Flood and Harvest: The Provable Necessity of Trivia for Generating Valuable Mathematics via the Lens of Language Generation in the Limit","display_name":"Flood and Harvest: The Provable Necessity of Trivia for Generating Valuable Mathematics via the Lens of Language Generation in the Limit","publication_year":2026,"publication_date":"2026-06-12","ids":{"openalex":"https://openalex.org/W7164831380","doi":"https://doi.org/10.48550/arxiv.2606.14688"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.14688","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.14688","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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.2606.14688","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138649892","display_name":"Xiaoyu Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Xiaoyu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031625303","display_name":"Andi Han","orcid":"https://orcid.org/0000-0003-4655-655X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Han, Andi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138671679","display_name":"Dai Shi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shi, Dai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138688026","display_name":"Zheng Gao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gao, Zheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138681340","display_name":"Jiaojiao Jiang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jiang, Jiaojiao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5112625169","display_name":"Gao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gao, Junbin","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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.28299999237060547,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.28299999237060547,"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":0.17710000276565552,"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/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.04190000146627426,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/generator","display_name":"Generator (circuit theory)","score":0.625},{"id":"https://openalex.org/keywords/oracle","display_name":"Oracle","score":0.5733000040054321},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5680999755859375},{"id":"https://openalex.org/keywords/verifiable-secret-sharing","display_name":"Verifiable secret sharing","score":0.5422000288963318},{"id":"https://openalex.org/keywords/gas-meter-prover","display_name":"Gas meter prover","score":0.4507000148296356},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.41839998960494995},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.36410000920295715},{"id":"https://openalex.org/keywords/proof-assistant","display_name":"Proof assistant","score":0.353300005197525},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.3531999886035919}],"concepts":[{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.625},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.5733000040054321},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5680999755859375},{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.5422000288963318},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4909000098705292},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46140000224113464},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4537000060081482},{"id":"https://openalex.org/C159718280","wikidata":"https://www.wikidata.org/wiki/Q5526353","display_name":"Gas meter prover","level":3,"score":0.4507000148296356},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4357999861240387},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.41839998960494995},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.37959998846054077},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C203265346","wikidata":"https://www.wikidata.org/wiki/Q11387554","display_name":"Proof assistant","level":3,"score":0.353300005197525},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3531999886035919},{"id":"https://openalex.org/C131672422","wikidata":"https://www.wikidata.org/wiki/Q852594","display_name":"Provable security","level":3,"score":0.3522000014781952},{"id":"https://openalex.org/C52370388","wikidata":"https://www.wikidata.org/wiki/Q752532","display_name":"Regular language","level":3,"score":0.3490000069141388},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.3409999907016754},{"id":"https://openalex.org/C137293760","wikidata":"https://www.wikidata.org/wiki/Q3621696","display_name":"Language model","level":2,"score":0.33980000019073486},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.334199994802475},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.30300000309944153},{"id":"https://openalex.org/C94284585","wikidata":"https://www.wikidata.org/wiki/Q228184","display_name":"Random oracle","level":4,"score":0.29840001463890076},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2955000102519989},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2946000099182129},{"id":"https://openalex.org/C146072743","wikidata":"https://www.wikidata.org/wiki/Q192161","display_name":"Formal language","level":2,"score":0.29440000653266907},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.27059999108314514},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.2662000060081482},{"id":"https://openalex.org/C201866948","wikidata":"https://www.wikidata.org/wiki/Q228206","display_name":"Random number generation","level":2,"score":0.26589998602867126},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.265500009059906},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2621000111103058},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.25459998846054077},{"id":"https://openalex.org/C110251889","wikidata":"https://www.wikidata.org/wiki/Q1569697","display_name":"Model checking","level":2,"score":0.2513999938964844},{"id":"https://openalex.org/C43091099","wikidata":"https://www.wikidata.org/wiki/Q1067788","display_name":"Through-the-lens metering","level":3,"score":0.2502000033855438}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.14688","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.14688","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.14688","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.14688","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"score":0.7920428514480591,"display_name":"Quality Education","id":"https://metadata.un.org/sdg/4"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"AI":[0],"systems":[1],"coupled":[2],"to":[3,158,191],"proof":[4,57],"assistants":[5],"now":[6],"generate":[7],"formal":[8,48],"mathematics":[9,38,259],"at":[10,185],"scale,":[11],"and":[12,21,161,202],"the":[13,29,34,44,106,111,121,130,167,189,199,216,220,240],"gap":[14,217],"between":[15],"what":[16,22],"a":[17,23,46,53,74,97,163,228,252],"checker":[18],"can":[19],"verify":[20],"mathematician":[24],"would":[25],"value":[26],"has":[27],"become":[28],"binding":[30],"constraint.":[31],"We":[32,101],"model":[33,230],"generation":[35,42,114],"of":[36,73,79,120,231,243,267],"valuable":[37,62,88,139,258],"as":[39,198],"nested":[40],"language":[41,49,63],"in":[43,211,227],"limit:":[45],"verifiable":[47],"$F$,":[50],"accessed":[51],"through":[52,69],"membership":[54],"oracle":[55],"(the":[56,83],"checker),":[58],"contains":[59],"an":[60,70,248,261],"unknown":[61],"$H":[64],"\\in":[65],"\\mathcal{H}$":[66],"revealed":[67],"only":[68,143],"adversarial":[71],"enumeration":[72],"core":[75],"$C":[76],"\\subseteq":[77],"H$":[78],"exact":[80],"density":[81],"$\u03b1$":[82],"literature).":[84],"Every":[85],"output":[86],"is":[87,108,210,219,246],"($\\in":[89,92],"H$),":[90,95],"trivial":[91],"F":[93],"\\setminus":[94],"or":[96],"hallucination":[98],"($\\notin":[99],"F$).":[100],"settle":[102],"four":[103],"questions.":[104],"First,":[105],"verifier":[107,131,235],"not":[109,214,247],"taste:":[110,239],"collections":[112],"admitting":[113],"with":[115,147],"breadth":[116],"are":[117],"exactly":[118],"those":[119],"oracle-free":[122],"model,":[123],"characterized":[124],"fiber-wise":[125],"by":[126],"Angluin's":[127],"condition.":[128],"Second,":[129],"does":[132],"buy":[133],"sound":[134],"coverage,":[135],"covering":[136,256],"all":[137],"unseen":[138],"statements":[140,245],"while":[141,179],"asserting":[142],"valid":[144],"ones:":[145],"possible":[146],"it,":[148],"impossible":[149],"without":[150],"it;":[151],"it":[152],"relocates":[153],"unavoidable":[154],"errors":[155],"from":[156],"false":[157],"trivial.":[159],"Third,":[160],"centrally,":[162],"sharp":[164],"dichotomy":[165],"on":[166],"tight":[168],"family:":[169],"generators":[170],"emitting":[171],"finitely":[172],"many":[173],"trivia":[174,182,212],"achieve":[175],"optimal":[176],"coverage":[177],"$\u03b1/2$,":[178],"any":[180],"infinite":[181],"allowance,":[183],"even":[184],"vanishing":[186],"rate,":[187],"jumps":[188],"optimum":[190],"$1-\u03b1/2$":[192],"(both":[193],"tight,":[194],"for":[195,238],"cores":[196],"presented":[197],"candidate":[200],"intersection),":[201],"one":[203],"generator":[204],"attains":[205],"both":[206,224],"ends.":[207],"The":[208],"transition":[209],"count,":[213],"rate;":[215],"$1-\u03b1$":[218],"unrecorded":[221,257],"mass.":[222],"Fourth,":[223],"regimes":[225],"instantiate":[226],"compression":[229],"mathematics.":[232],"A":[233],"perfect":[234],"cannot":[236],"substitute":[237],"unbounded":[241],"stream":[242,266],"correct-but-worthless":[244],"engineering":[249],"accident":[250],"but":[251,263],"provable":[253],"necessity,":[254],"since":[255],"requires":[260],"infinite,":[262],"asymptotically":[264],"negligible,":[265],"certified":[268],"trivia.":[269]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-16T00:00:00"}
