{"id":"https://openalex.org/W7149994244","doi":"https://doi.org/10.5281/zenodo.19424832","title":"The Collatz Conjecture as a Consequence of the 720/360 Degree Regime Incommensurability","display_name":"The Collatz Conjecture as a Consequence of the 720/360 Degree Regime Incommensurability","publication_year":2026,"publication_date":"2026-04-05","ids":{"openalex":"https://openalex.org/W7149994244","doi":"https://doi.org/10.5281/zenodo.19424832"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.19424832","doi":"https://doi.org/10.5281/zenodo.19424832","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.19424832","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.19424832","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059872525","display_name":"Caitlin Dempsey McKoy","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"McKoy, Caitlin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5059872525"],"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/T11166","display_name":"Analytic Number Theory Research","score":0.39399999380111694,"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"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.39399999380111694,"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"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.31839999556541443,"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/T14500","display_name":"Advanced Mathematical Theories","score":0.18150000274181366,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/collatz-conjecture","display_name":"Collatz conjecture","score":1.0},{"id":"https://openalex.org/keywords/2-adic-valuation","display_name":"2-adic valuation","score":0.8190000057220459},{"id":"https://openalex.org/keywords/geometric-distribution","display_name":"geometric distribution","score":0.7310000061988831},{"id":"https://openalex.org/keywords/angular-displacement","display_name":"angular displacement","score":0.5220000147819519},{"id":"https://openalex.org/keywords/incommensurability","display_name":"incommensurability","score":0.7580000162124634},{"id":"https://openalex.org/keywords/zeros-of-the-riemann-zeta-function","display_name":"Riemann zeta function zeros","score":0.753000020980835},{"id":"https://openalex.org/keywords/critical-line","display_name":"critical line","score":0.6779999732971191}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8664000034332275},{"id":"https://openalex.org/C128814272","wikidata":"https://www.wikidata.org/wiki/Q837314","display_name":"Collatz conjecture","level":3,"score":0.6972000002861023},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.558899998664856},{"id":"https://openalex.org/C19086982","wikidata":"https://www.wikidata.org/wiki/Q2115856","display_name":"Pseudorandomness","level":3,"score":0.4812999963760376},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.47859999537467957},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.46720001101493835},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4498000144958496},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.4401000142097473},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.42080000042915344},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.3887999951839447},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.351500004529953},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.34630000591278076},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.3321000039577484},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.3174999952316284},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.3131999969482422},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.30550000071525574},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3027999997138977},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.29420000314712524},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.29339998960494995},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29159998893737793},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.28209999203681946},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2775999903678894},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.26809999346733093},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.26570001244544983},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2556000053882599},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.2538999915122986}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.19424832","doi":"https://doi.org/10.5281/zenodo.19424832","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.19424832","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.19424833","doi":"https://doi.org/10.5281/zenodo.19424833","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.19424833","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.19424832","doi":"https://doi.org/10.5281/zenodo.19424832","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.19424832","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/W4412571807","https://openalex.org/W7128706950","https://openalex.org/W7163168572","https://openalex.org/W7125589804","https://openalex.org/W6931054988","https://openalex.org/W7116931550","https://openalex.org/W7137041200","https://openalex.org/W7085812638","https://openalex.org/W7156093491","https://openalex.org/W7212160125"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2,236],"every":[3],"positive":[4],"integer":[5],"reaches":[6],"1":[7],"under":[8],"the":[9,33,62,71,75,83,98,107,116,135,156,180,193,224,228,238,242],"Collatz":[10,41,221],"iteration":[11],"T(n)":[12],"=":[13,60,87,89,95,104,126,139,145,151,161,177,246],"n/2":[14],"if":[15,20],"n":[16,21,47,81,138,144,150],"is":[17,22,114,128,187,223],"even,":[18],"3n+1":[19,111,160],"odd.":[23],"The":[24,185,218],"proof":[25,186],"proceeds":[26],"in":[27,197,205,215],"three":[28],"steps,":[29],"each":[30,66],"exact.":[31],"First,":[32],"gap":[34],"ratio":[35],"(4n+2)/n":[36],"between":[37,231],"odd":[38,67,80,199],"and":[39,49,171,208],"even":[40,209],"steps":[42,123,133],"converges":[43],"to":[44,143,149,169,202,212],"4":[45,51,57],"as":[46],"grows":[48],"exceeds":[50,58,70],"for":[52,79],"all":[53],"finite":[54],"n;":[55],"since":[56],"log2(3)":[59],"1.585,":[61],"angular":[63],"displacement":[64],"of":[65,110,122,175,220,227],"step":[68,158],"always":[69,129],"convergence":[72,219],"threshold.":[73],"Second,":[74],"2-adic":[76],"valuation":[77],"v2(3n+1)":[78],"follows":[82],"geometric":[84,191],"distribution":[85],"P(v2":[86],"k)":[88],"1/2^k,":[90],"giving":[91,163],"expected":[92],"value":[93],"E[v2]":[94],"2":[96],"by":[97,134,189],"exact":[99],"algebraic":[100],"identity":[101],"sum":[102],"k/2^k":[103],"2.":[105],"Third,":[106],"residue":[108],"arithmetic":[109,225],"modulo":[112],"8":[113,142,148],"self-correcting:":[115],"only":[117],"dangerous":[118],"configuration,":[119],"a":[120,172,190],"run":[121],"with":[124],"v2":[125,164],"1,":[127],"resolved":[130],"within":[131],"two":[132,233],"forced":[136],"cycle":[137],"7":[140],"mod":[141,147,153],"3":[146],"5":[152],"8,":[154],"where":[155],"third":[157],"satisfies":[159],"8(3k+2),":[162],"greater":[165],"than":[166],"or":[167],"equal":[168],"3,":[170],"minimum":[173],"mean":[174],"5/3":[176],"1.667":[178],"over":[179],"three-step":[181],"cycle,":[182],"exceeding":[183],"log2(3).":[184],"motivated":[188],"framework,":[192],"One-Object":[194],"Universe":[195],"(OOU),":[196],"which":[198],"integers":[200,210],"correspond":[201,211],"fermionic":[203],"windings":[204,214],"720-degree":[206],"space":[207],"bosonic":[213],"360-degree":[216],"space.":[217],"trajectories":[222],"statement":[226],"same":[229],"incommensurability":[230],"these":[232],"topological":[234],"regimes":[235],"places":[237],"Riemann":[239],"zeros":[240],"on":[241],"critical":[243],"line":[244],"Re(s)":[245],"1/2.":[247]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-04-06T00:00:00"}
