{"id":"https://openalex.org/W7225774453","doi":"https://doi.org/10.5281/zenodo.24134501","title":"Investigating the Banach-Tarski Paradox Using a Generative Sequence Engine","display_name":"Investigating the Banach-Tarski Paradox Using a Generative Sequence Engine","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225774453","doi":"https://doi.org/10.5281/zenodo.24134501"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24134501","doi":"https://doi.org/10.5281/zenodo.24134501","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134501","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24134501","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153484223","display_name":"Stein Michael","orcid":"https://orcid.org/0009-0001-0801-1957","observed_orcids":["https://orcid.org/0009-0001-0801-1957"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Stein, Michael","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0001-0801-1957","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153484223"],"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.249099999666214,"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"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.249099999666214,"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":0.17270000278949738,"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"}},{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.14920000731945038,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6140999794006348},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6068000197410583},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.57669997215271},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5121999979019165},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48890000581741333},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.48820000886917114},{"id":"https://openalex.org/C39890363","wikidata":"https://www.wikidata.org/wiki/Q36108","display_name":"Generative grammar","level":2,"score":0.4830999970436096},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4749999940395355},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46050000190734863},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.41510000824928284},{"id":"https://openalex.org/C167966045","wikidata":"https://www.wikidata.org/wiki/Q5532625","display_name":"Generative model","level":3,"score":0.41499999165534973},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.39980000257492065},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.3853999972343445},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36910000443458557},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3544999957084656},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.3393999934196472},{"id":"https://openalex.org/C2776235265","wikidata":"https://www.wikidata.org/wiki/Q18392052","display_name":"Fragment (logic)","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C130327152","wikidata":"https://www.wikidata.org/wiki/Q748349","display_name":"Mathematical structure","level":2,"score":0.32659998536109924},{"id":"https://openalex.org/C2779231336","wikidata":"https://www.wikidata.org/wiki/Q7534724","display_name":"Sketch","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.3009999990463257},{"id":"https://openalex.org/C123657996","wikidata":"https://www.wikidata.org/wiki/Q12271","display_name":"Architecture","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27079999446868896},{"id":"https://openalex.org/C125773388","wikidata":"https://www.wikidata.org/wiki/Q792542","display_name":"Axiomatic system","level":3,"score":0.25540000200271606},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24134501","doi":"https://doi.org/10.5281/zenodo.24134501","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134501","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.24134502","doi":"https://doi.org/10.5281/zenodo.24134502","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134502","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24134501","doi":"https://doi.org/10.5281/zenodo.24134501","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134501","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0,35,79,207,289],"Banach\u2013Tarski":[1,71,218,290],"paradox":[2],"is":[3,42,89,124,150,155,230,284],"a":[4,12,74,105,128,173,203,295,302],"theorem":[5,219],"of":[6,64,216,268,298,304,307],"classical":[7,179,222],"set":[8],"theory":[9],"in":[10,104],"which":[11],"solid":[13],"three-dimensional":[14],"ball":[15],"can":[16],"be":[17,139,144,187],"partitioned":[18],"into":[19,28],"finitely":[20],"many":[21],"subsets":[22],"and":[23,41,120,247,251,301,311],"reassembled,":[24],"using":[25],"rigid":[26],"motions,":[27],"two":[29,82],"balls":[30],"congruent":[31],"to":[32,69,93,118,157,186],"the":[33,59,70,87,109,113,159,166,170,182,213,217,228,234,254,269,280,299,305,308,312],"original.":[34],"construction":[36,72,229],"depends":[37,273],"upon":[38,274],"nonmeasurable":[39],"sets":[40],"formally":[43],"valid":[44],"within":[45,220],"its":[46,51,121,221],"accepted":[47],"mathematical":[48,191,245,281],"framework,":[49],"yet":[50],"result":[52],"remains":[53],"profoundly":[54],"counterintuitive.":[55],"This":[56],"paper":[57],"applies":[58],"Generative":[60],"Sequence":[61],"Engine":[62],"(GSE)":[63],"Whole":[65],"Architecture":[66],"Theory":[67],"(WAT)":[68],"through":[73],"deliberately":[75],"coarse-grained":[76],"functional":[77,249],"mapping.":[78],"analysis":[80,171,208],"produces":[81],"candidate":[83,175],"architectural":[84,260],"diagnoses.":[85],"First,":[86],"decomposition":[88],"represented":[90,140],"as":[91,100,127,189,196,202,259,264],"purporting":[92],"generate":[94],"fragments":[95],"that":[96,137],"must":[97],"subsequently":[98],"function":[99],"extrinsic":[101,160],"Input":[102,161],"Structures":[103],"reconstruction":[106,167],"layer.":[107],"Under":[108],"present":[110,255],"WAT":[111,193,236],"mapping,":[112],"intended":[114],"Fragment-Configuration":[115],"Pattern":[116],"fails":[117],"instantiate,":[119],"purported":[122],"output":[123],"therefore":[125,209],"classified":[126],"Class":[129],"0":[130],"non-Pattern,":[131],"or":[132],"phantom":[133],"Pattern:":[134],"an":[135],"entity":[136,154],"may":[138],"symbolically":[141],"but":[142],"cannot":[143],"generatively":[145,231],"instantiated.":[146],"If":[147],"this":[148],"assignment":[149],"sustained,":[151],"no":[152,239],"instantiated":[153,205],"available":[156],"fulfil":[158],"Structure":[162],"role":[163],"required":[164],"by":[165],"Mechanism.":[168],"Second,":[169],"identifies":[172],"distinct":[174],"infinity-architecture":[176],"problem.":[177],"Whereas":[178],"mathematics":[180],"permits":[181],"relevant":[183],"infinite":[184],"collections":[185],"treated":[188,258],"completed":[190,204],"objects,":[192],"models":[194],"infinity":[195],"unbounded":[197],"generative":[198],"potential":[199],"rather":[200,262],"than":[201,263],"Pattern.":[206],"does":[210],"not":[211],"challenge":[212],"formal":[214],"validity":[215],"axiomatic":[223],"framework.":[224],"It":[225],"investigates":[226],"whether":[227,275],"instantiable":[232],"under":[233],"proposed":[235],"architecture.":[237],"Because":[238],"standardized":[240],"correspondence":[241],"currently":[242],"exists":[243],"between":[244],"structures":[246],"GSE":[248,300,313],"roles":[250],"layer":[252],"boundaries,":[253],"diagnoses":[256],"are":[257],"hypotheses":[261],"uniquely":[265],"established":[266],"causes":[267],"paradox.":[270],"Their":[271],"significance":[272],"they":[276],"remain":[277],"defensible":[278],"when":[279],"dependency":[282],"structure":[283],"mapped":[285],"at":[286],"higher":[287],"resolution.":[288],"case":[291],"thereby":[292],"provides":[293],"both":[294],"substantive":[296],"application":[297],"test":[303],"adequacy":[306],"mapping":[309],"methodology":[310],"architecture":[314],"itself.":[315]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}