{"id":"https://openalex.org/W7166325234","doi":"https://doi.org/10.5281/zenodo.20947546","title":"Hexagonal Confinement and Stochastic Dynamics of Prime Distribution : A Structural Study via Digital Root Projection","display_name":"Hexagonal Confinement and Stochastic Dynamics of Prime Distribution : A Structural Study via Digital Root Projection","publication_year":2026,"publication_date":"2026-06-27","ids":{"openalex":"https://openalex.org/W7166325234","doi":"https://doi.org/10.5281/zenodo.20947546"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.20947546","doi":"https://doi.org/10.5281/zenodo.20947546","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.20947546","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","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.20947546","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140748774","display_name":"Zakaria CHARRAT","orcid":"https://orcid.org/0009-0009-3714-8416","observed_orcids":["https://orcid.org/0009-0009-3714-8416"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"CHARRAT, Zakaria","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5140748774"],"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":0.9980000257492065,"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":0.9980000257492065,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.0010000000474974513,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14500","display_name":"Advanced Mathematical Theories","score":0.0003000000142492354,"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/prime-number-distribution","display_name":"prime number distribution","score":0.9279999732971191},{"id":"https://openalex.org/keywords/digital-root","display_name":"digital root","score":0.6859999895095825},{"id":"https://openalex.org/keywords/prime-residue-classes","display_name":"prime residue classes","score":0.6010000109672546},{"id":"https://openalex.org/keywords/stationary-distribution","display_name":"stationary distribution","score":0.9269999861717224},{"id":"https://openalex.org/keywords/prime-gaps","display_name":"prime gaps","score":0.5289999842643738}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7839999794960022},{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.770799994468689},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5217000246047974},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4717999994754791},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4465999901294708},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4375},{"id":"https://openalex.org/C49555168","wikidata":"https://www.wikidata.org/wiki/Q176583","display_name":"Stochastic matrix","level":3,"score":0.43479999899864197},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.41350001096725464},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.37880000472068787},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.3752000033855438},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.36500000953674316},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.35429999232292175},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3330000042915344},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.3140000104904175},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.31380000710487366},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29649999737739563},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.29100000858306885},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.29089999198913574},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2824000120162964},{"id":"https://openalex.org/C128765274","wikidata":"https://www.wikidata.org/wiki/Q663314","display_name":"Hexagonal crystal system","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C47600822","wikidata":"https://www.wikidata.org/wiki/Q7825032","display_name":"Topological dynamics","level":5,"score":0.27250000834465027},{"id":"https://openalex.org/C35908053","wikidata":"https://www.wikidata.org/wiki/Q3063631","display_name":"Hexagonal tiling","level":3,"score":0.2599000036716461},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.25459998846054077}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.20947546","doi":"https://doi.org/10.5281/zenodo.20947546","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.20947546","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":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.20947547","doi":"https://doi.org/10.5281/zenodo.20947547","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.20947547","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":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.20985705","doi":"https://doi.org/10.5281/zenodo.20985705","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.20985705","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.20947546","doi":"https://doi.org/10.5281/zenodo.20947546","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.20947546","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":[],"sustainable_development_goals_aurora":[{"display_name":"Reduced inequalities","score":0.8030346632003784,"id":"https://metadata.un.org/sdg/10"}],"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":{"This":[0],"paper":[1],"presents":[2],"a":[3,92,110],"rigorous":[4],"structural":[5],"and":[6],"probabilistic":[7,94],"analysis":[8,66],"of":[9,59,67,123,140,151],"prime":[10,29,73],"number":[11],"distribution":[12,122,139],"within":[13,87],"the":[14,21,38,60,105,136,143,152,159],"Modulo-9":[15],"projection":[16],"framework,":[17],"referred":[18],"to":[19,37,76,128],"as":[20],"Numerical":[22],"Origin":[23],"Theory.":[24],"We":[25],"establish":[26],"that":[27,56,82],"all":[28],"numbers":[30,74],"greater":[31],"than":[32],"3":[33],"are":[34,102],"topologically":[35],"confined":[36],"hexagonal":[39,89],"orbit":[40],"S":[41],"=":[42,69,78],"{1,":[43],"2,":[44],"4,":[45],"8,":[46],"7,":[47],"5}":[48],"\u2282":[49],"Z\u2089,":[50],"strictly":[51],"obeying":[52],"an":[53],"Exclusion":[54],"Principle":[55],"forbids":[57],"occupancy":[58],"36-9":[61],"axis.":[62],"Through":[63],"exhaustive":[64],"statistical":[65],"N":[68],"100":[70],"000":[71],"consecutive":[72],"(up":[75],"p":[77],"1,299,709),":[79],"we":[80],"demonstrate":[81],"transitions":[83],"between":[84],"successive":[85],"primes":[86,141],"this":[88,124],"domain":[90],"follow":[91],"structured":[93],"law":[95],"governed":[96],"by":[97],"topological":[98],"distance.":[99],"These":[100],"observations":[101],"formalized":[103],"through":[104],"Symmetric":[106,111],"Hexagonal":[107],"Kernel":[108],"\u2014":[109,115],"Circulant":[112],"Transition":[113],"Matrix":[114],"validated":[116],"against":[117],"empirical":[118],"data.":[119,160],"The":[120],"stationary":[121],"matrix":[125],"converges":[126],"exactly":[127],"1/6":[129],"per":[130],"node,":[131],"in":[132],"perfect":[133],"agreement":[134],"with":[135],"observed":[137],"uniform":[138],"across":[142],"six":[144],"admissible":[145],"residue":[146],"classes.":[147],"An":[148],"empirically-corrected":[149],"version":[150],"kernel":[153],"is":[154],"also":[155],"derived":[156],"directly":[157],"from":[158]},"counts_by_year":[],"updated_date":"2026-10-08T09:06:37.294493","created_date":"2026-06-28T00:00:00"}
