{"id":"https://openalex.org/W7204159646","doi":"https://doi.org/10.5281/zenodo.22100466","title":"The Single-Generator Theory of Cycle-Decomposed Digit-Subtraction Dynamics","display_name":"The Single-Generator Theory of Cycle-Decomposed Digit-Subtraction Dynamics","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204159646","doi":"https://doi.org/10.5281/zenodo.22100466"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.22100466","doi":"https://doi.org/10.5281/zenodo.22100466","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22100466","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.22100466","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107529050","display_name":"Anshveer Bindra","orcid":"https://orcid.org/0009-0008-3593-1450","observed_orcids":["https://orcid.org/0009-0008-3593-1450"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Bindra, Anshveer","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0008-3593-1450","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5107529050"],"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9574999809265137,"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"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9574999809265137,"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":0.007799999788403511,"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/T13234","display_name":"advanced mathematical theories","score":0.0031999999191612005,"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/permutation-cycles","display_name":"permutation cycles","score":0.6909999847412109},{"id":"https://openalex.org/keywords/semi-conjugacy","display_name":"semi-conjugacy","score":0.7900000214576721},{"id":"https://openalex.org/keywords/multiplicative-order","display_name":"multiplicative order","score":0.5630000233650208},{"id":"https://openalex.org/keywords/fixed-points","display_name":"fixed points","score":0.5370000004768372},{"id":"https://openalex.org/keywords/attracting-cycles","display_name":"attracting cycles","score":0.8040000200271606},{"id":"https://openalex.org/keywords/direct-product-decomposition","display_name":"direct product decomposition","score":0.6589999794960022}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8565999865531921},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6413000226020813},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.510699987411499},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.5037999749183655},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.5027999877929688},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49219998717308044},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4909000098705292},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46459999680519104},{"id":"https://openalex.org/C125257309","wikidata":"https://www.wikidata.org/wiki/Q131752","display_name":"Greatest common divisor","level":2,"score":0.41440001130104065},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.40380001068115234},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.4036000072956085},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.40299999713897705},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.4027000069618225},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4009000062942505},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3930000066757202},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.36070001125335693},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.34779998660087585},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.34619998931884766},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.34450000524520874},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3314000070095062},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.32260000705718994},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3052999973297119},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.30300000309944153},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.30239999294281006},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.30000001192092896},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.2971999943256378},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C204160518","wikidata":"https://www.wikidata.org/wiki/Q122653","display_name":"Numeral system","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C190290938","wikidata":"https://www.wikidata.org/wiki/Q387015","display_name":"Trie","level":3,"score":0.25589999556541443},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2558000087738037},{"id":"https://openalex.org/C102607178","wikidata":"https://www.wikidata.org/wiki/Q102761","display_name":"Least common multiple","level":2,"score":0.2547000050544739}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.22100466","doi":"https://doi.org/10.5281/zenodo.22100466","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22100466","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":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.22100467","doi":"https://doi.org/10.5281/zenodo.22100467","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22100467","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.22100466","doi":"https://doi.org/10.5281/zenodo.22100466","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22100466","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":[],"sustainable_development_goals_aurora":[],"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":{"For":[0,212],"a":[1,47,52,138,198,213,225],"base":[2],"$B\\ge2$,":[3],"digit":[4],"length":[5],"$k$,":[6],"and":[7,35,60,77,99,114],"permutation":[8],"$\\pi\\in":[9],"S_k$,":[10],"we":[11],"study":[12],"the":[13,26,37,105,109,115,132,144,148,164,175,187,202,220,229],"arithmetic":[14,102],"dynamical":[15],"system":[16],"$F_\\pi$":[17,72,206],"obtained":[18],"by":[19,131,142],"subtracting":[20],"each":[21,78],"local":[22,74,120],"base-$B$":[23],"numeral":[24],"along":[25],"disjoint":[27],"cycles":[28],"of":[29,71,95,135,168,205,228],"$\\pi$":[30],"from":[31],"its":[32],"cyclic":[33],"rotation":[34],"scattering":[36],"resulting":[38],"digits":[39],"back":[40],"to":[41,85],"their":[42],"original":[43],"positions.":[44],"We":[45],"establish":[46],"complete":[48,116],"closed-form":[49],"theory":[50],"for":[51,111,137,201,207],"single":[53],"generator":[54],"$\\pi$,":[55],"uniformly":[56],"across":[57],"all":[58,169],"bases":[59],"cycle":[61,64,210],"types.":[62],"The":[63,158,193],"structure":[65,134],"yields":[66,197],"an":[67,208],"exact":[68,106],"direct-product":[69,194],"decomposition":[70,195],"into":[73],"maps":[75],"$g_t$,":[76],"$g_t$":[79],"is":[80,147,163,186,217],"semiconjugate,":[81],"after":[82,178],"identifying":[83],"$x\\sim-x$,":[84],"multiplication":[86],"by\\[c=(B-1)B^{t-1}\\]on":[87],"$\\mathbb":[88],"Z/(B^t-1)\\mathbb":[89],"Z$.":[90],"Writing\\[B^t-1=N_1N_2,\\]where":[91],"every":[92],"prime":[93],"divisor":[94,133],"$N_1$":[96],"divides":[97],"$B-1$":[98],"$\\gcd(N_2,B-1)=1$,":[100],"this":[101,215],"reduction":[103],"determines":[104],"fixed-point":[107],"count,":[108],"criterion":[110],"total":[112],"collapse,":[113],"attracting-cycle":[117],"structure.":[118],"Each":[119],"map":[121],"has":[122],"$(N_2+1)/2$":[123],"periodic":[124,139,176],"states.":[125],"Their":[126],"individual":[127],"periods":[128],"are":[129],"determined":[130],"$N_2$:":[136],"class":[140],"represented":[141],"$N_1r$,":[143],"minimal":[145],"period":[146],"least":[149,165,188],"positive":[150],"integer":[151,189],"$j$":[152],"such":[153],"that":[154],"$c^j\\equiv\\pm1$":[155],"modulo":[156],"$N_2/\\gcd(r,N_2)$.":[157],"quantity":[159],"$P_*:=\\operatorname{ord}_{(\\mathbb":[160],"Z/N_2\\mathbb":[161],"Z)^\\times/\\{\\pm1\\}}(c)$":[162],"common":[166],"multiple":[167],"periodic-orbit":[170],"lengths.":[171],"Every":[172],"state":[173],"reaches":[174],"set":[177],"exactly":[179],"$a_0$":[180,185],"or":[181],"$a_0+1$":[182],"iterations,":[183],"where":[184],"satisfying":[190],"$N_1\\mid":[191],"c^{a_0}$.":[192],"then":[196],"closed":[199],"formula":[200],"eventual":[203],"rank":[204,216],"arbitrary":[209],"type.":[211],"two-cycle,":[214],"$(\\operatorname{odd}(B+1)+1)/2$,":[218],"giving":[219],"identification":[221],"with":[222],"OEIS~A003602":[223],"as":[224],"direct":[226],"consequence":[227],"underlying":[230],"$2$-adic":[231],"decomposition.":[232]},"counts_by_year":[],"updated_date":"2026-10-07T09:51:55.459714","created_date":"2026-08-26T00:00:00"}