{"id":"https://openalex.org/W7225835700","doi":"https://doi.org/10.5281/zenodo.24134831","title":"Complete Nim-Words and Sharp Preperiods for the Four-Move Subtraction Family {5,11,15,c}","display_name":"Complete Nim-Words and Sharp Preperiods for the Four-Move Subtraction Family {5,11,15,c}","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225835700","doi":"https://doi.org/10.5281/zenodo.24134831"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24134831","doi":"https://doi.org/10.5281/zenodo.24134831","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134831","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":null,"license_id":null,"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.24134831","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148078153","display_name":"Ryutaro Yonezu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Yonezu, Ryutaro","raw_affiliation_strings":["Independent Researcher"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5148078153"],"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.8349999785423279,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.8349999785423279,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11567","display_name":"semigroups and automata theory","score":0.032600000500679016,"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":0.013500000350177288,"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":[],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5336999893188477},{"id":"https://openalex.org/C65045869","wikidata":"https://www.wikidata.org/wiki/Q81365","display_name":"Decimal","level":2,"score":0.4880000054836273},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.45339998602867126},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4032999873161316},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.4025999903678894},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4009000062942505},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.400299996137619},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.3917999863624573},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3785000145435333},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.35929998755455017},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3434000015258789},{"id":"https://openalex.org/C43364308","wikidata":"https://www.wikidata.org/wiki/Q8799","display_name":"Byte","level":2,"score":0.32600000500679016},{"id":"https://openalex.org/C9395851","wikidata":"https://www.wikidata.org/wiki/Q177929","display_name":"Stack (abstract data type)","level":2,"score":0.3118000030517578},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.30799999833106995},{"id":"https://openalex.org/C2777826928","wikidata":"https://www.wikidata.org/wiki/Q3745713","display_name":"Fingerprint (computing)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C2778738651","wikidata":"https://www.wikidata.org/wiki/Q16546687","display_name":"Novelty","level":2,"score":0.30140000581741333},{"id":"https://openalex.org/C68060419","wikidata":"https://www.wikidata.org/wiki/Q40754","display_name":"Subtraction","level":2,"score":0.29989999532699585},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.29789999127388},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.28600001335144043},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.2831000089645386},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27320000529289246},{"id":"https://openalex.org/C2777855551","wikidata":"https://www.wikidata.org/wiki/Q12310021","display_name":"Subject (documents)","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.25780001282691956},{"id":"https://openalex.org/C96147967","wikidata":"https://www.wikidata.org/wiki/Q190686","display_name":"Subroutine","level":2,"score":0.25699999928474426},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.2533999979496002},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.25110000371932983}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24134831","doi":"https://doi.org/10.5281/zenodo.24134831","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134831","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24134832","doi":"https://doi.org/10.5281/zenodo.24134832","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134832","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24134831","doi":"https://doi.org/10.5281/zenodo.24134831","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24134831","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":null,"license_id":null,"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":[],"abstract_inverted_index":{"AI-assisted":[0],"mathematical":[1],"preprint,":[2],"No.64,":[3],"version":[4],"1.0.":[5],"Ryutaro":[6,288],"Yonezu":[7],"\u2014":[8],"Independent":[9],"Researcher.":[10],"Published":[11],"11":[12],"November":[13],"2026":[14,287],"(JST).":[15],"For":[16,38],"the":[17,28,81,101,124,144,173,207,210,229],"normal-play":[18],"one-pile":[19],"subtraction":[20],"family":[21],"S_c={5,11,15,c},":[22],"odd":[23,40,55],"c>15,":[24],"this":[25],"paper":[26],"classifies":[27],"complete":[29],"Sprague\u2013Grundy":[30],"sequences":[31],"and":[32,36,49,92,131,189,222,249,257],"sharp":[33],"eventual":[34],"periods":[35],"preperiods.":[37],"every":[39],"c>=59,":[41],"a":[42],"uniform":[43],"explicit":[44],"nim-word":[45],"has":[46],"least":[47,50],"T=c+29":[48],"p=2.":[51],"The":[52,63],"remaining":[53],"21":[54],"c=17..57":[56],"are":[57,193,261],"handled":[58],"by":[59,100],"finite":[60,258],"recurrence":[61],"certificates.":[62],"general":[64],"all-odd-a":[65],"quadratic":[66],"formula":[67],"T=c+2a^2-4a-1":[68],"is":[69,121,238,277],"an":[70],"unproved":[71],"computational":[72],"conjecture.":[73],"External":[74],"independent":[75],"AI":[76],"audit:":[77],"MODEL/PROOF/COMPUTATION/MINIMALITY/PACKAGE":[78],"PASS":[79,109,136],"within":[80],"reviewed":[82],"scope,":[83],"reported":[84,208],"in":[85,206],"author-supplied":[86],"review":[87,154],"text.":[88],"Reviewer":[89],"raw":[90,264],"code":[91],"logs":[93],"were":[94,204],"not":[95,114,194],"provided":[96],"or":[97,273],"independently":[98,115],"rerun":[99,130],"publication":[102,138],"editor.":[103],"Author-run":[104],"Windows":[105,119,265],"PowerShell":[106],"5.1":[107],"reproduction":[108],"on":[110],"canonical":[111],"frozen":[112,254],"v0.1;":[113],"witnessed.":[116],"No":[117,263],"new":[118],"execution":[120],"claimed":[122,195],"for":[123],"editorial":[125],"v1.0":[126],"package.":[127],"Linux":[128],"Python/C++17":[129],"regenerated":[132],"CSV":[133,259],"byte":[134],"checks":[135],"during":[137],"preparation.":[139],"Finite":[140],"numerical":[141],"tests":[142],"supplement":[143],"analytical":[145],"infinite-family":[146],"proof.":[147],"NOVELTY":[148],"INCONCLUSIVE.":[149],"REFERENCES":[150],"PARTIAL.":[151],"Human":[152],"peer":[153],"NOT":[155,160],"PERFORMED.":[156,161],"Formal":[157],"proof-assistant":[158],"verification":[159],"All":[162,290],"rights":[163,291],"reserved.":[164,292],"Known":[165],"prior":[166],"results:":[167],"Ho":[168],"(2015),":[169],"Theorem":[170],"5,":[171],"gives":[172],"base":[174],"triple":[175],"{5,11,15}":[176],"with":[177],"T=44,p=2.":[178],"Zhang":[179],"(2024),":[180],"Remark":[181],"7,":[182],"previously":[183],"reports":[184],"bounded":[185],"observations,":[186],"including":[187],"c=21":[188],"c=29.":[190],"Those":[191],"results":[192],"as":[196,225],"new.":[197],"Relevant":[198],"portions":[199],"of":[200,235],"Flammenkamp\u2019s":[201],"original":[202],"dissertation":[203],"compared":[205,230],"audit;":[209],"expanded":[211],"1997":[212],"book":[213],"ISBN":[214],"3-932136-10-1":[215],"remains":[216],"unread.":[217],"Moriwaki":[218],"(2026),":[219],"Theorems":[220],"2":[221],"4,":[223],"fail":[224],"direct":[226],"applications":[227],"under":[228],"conditions;":[231],"no":[232],"exhaustive":[233],"exclusion":[234],"related":[236],"work":[237],"claimed.":[239],"Public":[240],"assets:":[241],"preprint":[242],"PDF,":[243],"corresponding":[244],"LaTeX,":[245],"reproducibility":[246],"ZIP,":[247],"README":[248],"SHA256":[250],"checksums.":[251],"Mathematical":[252],"core,":[253],"Python/C++17/PowerShell":[255],"sources":[256],"certificates":[260],"unchanged.":[262],"log,":[266],"local":[267],"absolute":[268],"path,":[269],"private":[270],"audit":[271],"archive":[272],"third-party":[274],"full":[275],"text":[276],"redistributed.":[278],"GitHub":[279],"repository:":[280],"https://github.com/yonezaemon1-hub/no64-four-move-subtraction-nimwordsGitHub":[281],"Release":[282],"v1.0:":[283],"https://github.com/yonezaemon1-hub/no64-four-move-subtraction-nimwords/releases/tag/v1.0":[284],"Copyright":[285],"(c)":[286],"Yonezu.":[289]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}