{"id":"https://openalex.org/W7224371980","doi":"https://doi.org/10.5281/zenodo.23755764","title":"BB(5) = 47,176,870 Proven; BB(6) Remains Open \u2014 E8 Intelligence Research","display_name":"BB(5) = 47,176,870 Proven; BB(6) Remains Open \u2014 E8 Intelligence Research","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224371980","doi":"https://doi.org/10.5281/zenodo.23755764"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23755764","doi":"https://doi.org/10.5281/zenodo.23755764","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23755764","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","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23755764","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153285007","display_name":"Andrew Stewart Caldin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Caldin, Andrew Stewart","raw_affiliation_strings":["Independent Researcher, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher, United Kingdom","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153285007"],"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":0.9970999956130981,"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":0.9970999956130981,"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/T11567","display_name":"semigroups and automata theory","score":0.0013000000035390258,"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/T11166","display_name":"Analytic Number Theory Research","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/busy-beaver-function","display_name":"Busy Beaver function","score":0.9879999756813049},{"id":"https://openalex.org/keywords/turing-machines","display_name":"Turing machines","score":0.7039999961853027},{"id":"https://openalex.org/keywords/self-similar-growth","display_name":"self-similar growth","score":0.5339999794960022}],"concepts":[{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.6992999911308289},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.6280999779701233},{"id":"https://openalex.org/C521332185","wikidata":"https://www.wikidata.org/wiki/Q185816","display_name":"Analogy","level":2,"score":0.559499979019165},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.5401999950408936},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5041000247001648},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.4966999888420105},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.46779999136924744},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.42160001397132874},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4174000024795532},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3882000148296356},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3747999966144562},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.3495999872684479},{"id":"https://openalex.org/C119350121","wikidata":"https://www.wikidata.org/wiki/Q1091134","display_name":"Commensurability (mathematics)","level":2,"score":0.3352000117301941},{"id":"https://openalex.org/C48415503","wikidata":"https://www.wikidata.org/wiki/Q5251122","display_name":"Time hierarchy theorem","level":5,"score":0.27970001101493835},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27300000190734863},{"id":"https://openalex.org/C577917","wikidata":"https://www.wikidata.org/wiki/Q189223","display_name":"Turing test","level":2,"score":0.27230000495910645},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.2619999945163727},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C171268870","wikidata":"https://www.wikidata.org/wiki/Q1486676","display_name":"GRASP","level":2,"score":0.25699999928474426},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23755764","doi":"https://doi.org/10.5281/zenodo.23755764","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23755764","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.23755765","doi":"https://doi.org/10.5281/zenodo.23755765","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23755765","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.23755764","doi":"https://doi.org/10.5281/zenodo.23755764","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23755764","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7223806449","https://openalex.org/W7208739465","https://openalex.org/W7210289023","https://openalex.org/W7221564783","https://openalex.org/W7208831203","https://openalex.org/W7211936312","https://openalex.org/W7204871285","https://openalex.org/W7212448504","https://openalex.org/W7212195190","https://openalex.org/W7203896502"],"abstract_inverted_index":{"FINDING:":[0],"BB(5)":[1,137],"=":[2,138],"47,176,870":[3,139],"was":[4,140],"proven":[5],"(2024,":[6],"Busy":[7,104],"Beaver":[8,105],"Challenge)":[9],"as":[10],"the":[11,67,81,156],"true":[12],"value":[13],"for":[14,116],"5-state":[15],"2-symbol":[16],"Turing":[17,111],"machines;":[18],"BB(6)":[19,49],"remains":[20],"open":[21],"and":[22],"is":[23,37,73,90,108],"conjectured":[24],"to":[25,60],"vastly":[26],"exceed":[27],"it.":[28],"|":[29,55,94],"MATH:":[30],"BB(5)=47,176,870;":[31],"BB(4)=13;":[32],"BB(3)=21;":[33],"BB(2)=6;":[34],"BB(1)=1.":[35],"Growth":[36],"non-computable":[38],"(\u03a3(n)":[39],"grows":[40,120,128],"faster":[41,121,129],"than":[42,122,130],"any":[43,123,131],"computable":[44,124],"function).":[45],"Known":[46],"lower":[47],"bounds:":[48],"\u2265":[50],"10\u2191\u219115":[51],"(via":[52],"Antihydra/Collatz-like":[53],"machines).":[54],"CONNECTION:":[56],"No":[57],"direct":[58],"link":[59],"\u03c6-ratios,":[61],"base-60,":[62],"or":[63],"crystallographic":[64],"symmetry":[65],"in":[66,85,142],"evidence.":[68],"The":[69,103],"relevant":[70],"structural":[71],"analogy":[72],"**hierarchical/recursive":[74],"growth**":[75],"(oracle":[76],"hierarchy":[77],"BB^(k)),":[78],"which":[79],"mirrors":[80],"*self-similar":[82],"nesting*":[83],"seen":[84],"\u03c6-scaling":[86],"\u2014":[87],"but":[88],"this":[89],"analogy,":[91],"not":[92],"identity.":[93],"DEPTH:":[95],"8":[96],"---":[97],"**1.":[98],"Key":[99],"mathematical":[100],"insight**":[101],"-":[102,136],"function":[106,125,132],"\u03a3(n)":[107],"**non-computable**:":[109],"no":[110],"machine":[112],"can":[113,134],"compute":[114],"it":[115],"all":[117],"n.":[118],"It":[119],"(Aaronson:":[126],"\"BB":[127],"you":[133],"name\").":[135],"**resolved":[141],"2024**":[143],"by":[144],"a":[145],"distrib":[146],"Author:":[147],"Andrew":[148],"Stewart":[149],"Caldin,":[150],"Independent":[151],"Researcher,":[152],"UK.":[153],"Part":[154],"of":[155],"E8":[157],"Intelligence":[158],"Research":[159],"series.":[160],"Platform:":[161],"e8intelligence.com":[162]},"counts_by_year":[],"updated_date":"2026-10-31T12:09:11.096373","created_date":"2026-10-30T00:00:00"}