{"id":"https://openalex.org/W7225428769","doi":"https://doi.org/10.68366/90kyra95","title":"The Dual Cipher, Structurally Induced GF(4) Multiplication, and Real Spectral Structure: Companion-Matrix Realization of the Register Automorphism, Frobenius Reversal, and the Order-One Commutant Condition","display_name":"The Dual Cipher, Structurally Induced GF(4) Multiplication, and Real Spectral Structure: Companion-Matrix Realization of the Register Automorphism, Frobenius Reversal, and the Order-One Commutant Condition","publication_year":2026,"publication_date":"2026-11-06","ids":{"openalex":"https://openalex.org/W7225428769","doi":"https://doi.org/10.68366/90kyra95"},"language":"en","primary_location":{"id":"doi:10.68366/90kyra95","doi":"https://doi.org/10.68366/90kyra95","is_oa":true,"landing_page_url":"https://doi.org/10.68366/90kyra95","pdf_url":"https://chrysene.com/index.php/acf/article/download/92/225","source":{"id":"https://openalex.org/S7407068434","display_name":"Annals of the Chrysene Formalism","issn_l":"3144-2021","issn":["3144-2021"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annals of the Chrysene Formalism","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://chrysene.com/index.php/acf/article/download/92/225","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153411284","display_name":"Charles D. Schaper","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Charles Schaper","raw_affiliation_strings":["Annals of the Chrysene Formalism San Francisco , California , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Annals of the Chrysene Formalism San Francisco , California , USA","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153411284"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"1","issue":"4","first_page":"1847","last_page":"1855"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.1444000005722046,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.1444000005722046,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.13379999995231628,"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/T12037","display_name":"Algebraic and Geometric Analysis","score":0.07069999724626541,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/companion-matrix","display_name":"companion matrix","score":0.8820000290870667},{"id":"https://openalex.org/keywords/finite-field-multiplication","display_name":"finite field multiplication","score":0.41100001335144043},{"id":"https://openalex.org/keywords/spectral-realization","display_name":"spectral realization","score":0.49799999594688416},{"id":"https://openalex.org/keywords/lean-formal-verification","display_name":"Lean formal verification","score":0.4880000054836273}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6988000273704529},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5516999959945679},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.521399974822998},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.4431000053882599},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4309999942779541},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4284000098705292},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.40939998626708984},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4047999978065491},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40290001034736633},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.39969998598098755},{"id":"https://openalex.org/C75174853","wikidata":"https://www.wikidata.org/wiki/Q190629","display_name":"Centralizer and normalizer","level":2,"score":0.37209999561309814},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36970001459121704},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.33980000019073486},{"id":"https://openalex.org/C149364088","wikidata":"https://www.wikidata.org/wiki/Q185917","display_name":"Translation (biology)","level":4,"score":0.3391999900341034},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3379000127315521},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.33709999918937683},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.33559998869895935},{"id":"https://openalex.org/C135658033","wikidata":"https://www.wikidata.org/wiki/Q2903821","display_name":"Bimodule","level":3,"score":0.3343999981880188},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.33329999446868896},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.32350000739097595},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.31790000200271606},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.31529998779296875},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.3050999939441681},{"id":"https://openalex.org/C2778740170","wikidata":"https://www.wikidata.org/wiki/Q7575210","display_name":"Spectral space","level":2,"score":0.30309998989105225},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.28700000047683716},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.2777000069618225},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.27559998631477356},{"id":"https://openalex.org/C128489963","wikidata":"https://www.wikidata.org/wiki/Q1187724","display_name":"Filtration (mathematics)","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.2547000050544739},{"id":"https://openalex.org/C84316537","wikidata":"https://www.wikidata.org/wiki/Q36255","display_name":"Unit vector","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.68366/90kyra95","doi":"https://doi.org/10.68366/90kyra95","is_oa":true,"landing_page_url":"https://doi.org/10.68366/90kyra95","pdf_url":"https://chrysene.com/index.php/acf/article/download/92/225","source":{"id":"https://openalex.org/S7407068434","display_name":"Annals of the Chrysene Formalism","issn_l":"3144-2021","issn":["3144-2021"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annals of the Chrysene Formalism","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.68366/90kyra95","doi":"https://doi.org/10.68366/90kyra95","is_oa":true,"landing_page_url":"https://doi.org/10.68366/90kyra95","pdf_url":"https://chrysene.com/index.php/acf/article/download/92/225","source":{"id":"https://openalex.org/S7407068434","display_name":"Annals of the Chrysene Formalism","issn_l":"3144-2021","issn":["3144-2021"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annals of the Chrysene Formalism","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7225428769.pdf","grobid_xml":"https://content.openalex.org/works/W7225428769.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W2143726250","https://openalex.org/W2161471984","https://openalex.org/W7214955900","https://openalex.org/W7222796287","https://openalex.org/W7223702849","https://openalex.org/W7225006476"],"related_works":["https://openalex.org/W4234019166","https://openalex.org/W7206162447","https://openalex.org/W7224871404","https://openalex.org/W2789182819","https://openalex.org/W7154746581","https://openalex.org/W2321894187","https://openalex.org/W2053799487","https://openalex.org/W2779952407","https://openalex.org/W7222958881","https://openalex.org/W2517610151"],"abstract_inverted_index":{"The":[0,50,63],"two-bit":[1],"register":[2],"map":[3],"$\\chi(a,b)=(b,a+b)$":[4],"determines":[5],"an":[6],"explicit":[7,119],"companion-matrix":[8],"construction":[9],"of":[10],"$\\mathrm{GF}(4)$.":[11],"Its":[12],"matrix":[13,138],"satisfies":[14,142],"$M_\\chi^2+M_\\chi+I=0$,":[15],"and":[16,44,60,113,127,146,149,165,173,184,192],"evaluation":[17],"at":[18],"a":[19,27,87,92,100,105,114],"selected":[20],"cyclic":[21],"unit":[22],"transports":[23],"operator":[24,108,140],"composition":[25],"to":[26],"multiplication":[28],"on":[29,53],"the":[30,37,47,70,81,97,125,131,136,143,159,170,180,197],"structural":[31],"alphabet.":[32],"Register":[33],"reversal":[34],"becomes":[35,40],"Frobenius,":[36],"simultaneous":[38],"flip":[39],"translation":[41],"by":[42],"one,":[43],"alignment":[45],"is":[46,66,84],"field":[48,172],"trace.":[49],"resulting":[51],"action":[52],"three":[54],"registers":[55],"has":[56],"fixed-point":[57],"counts":[58],"$(64,0,8,8)$":[59],"20":[61],"orbits.":[62],"RNA":[64],"dictionary":[65],"evaluated":[67],"consistently":[68],"through":[69],"same":[71],"map.":[72],"A":[73],"perimeter":[74],"locant":[75],"correspondence":[76],"commutes":[77],"with":[78,91,109],"inversion,":[79],"while":[80,178],"dimer-to-node":[82],"identification":[83],"formulated":[85],"as":[86,187],"unitary":[88],"coordinate":[89],"isomorphism":[90],"separate":[93],"molecular-realization":[94],"hypothesis.":[95],"At":[96],"spectral":[98,176],"level,":[99],"symmetrized":[101,128],"support":[102],"filtration":[103],"yields":[104],"self-adjoint":[106,153],"compact-resolvent":[107],"bounded":[110],"algebraic":[111],"commutators":[112],"compatible":[115],"real":[116,144,175],"involution.":[117],"An":[118],"local-unitary":[120],"counterexample":[121],"shows":[122],"that":[123],"both":[124],"original":[126],"filtrations":[129],"fail":[130],"global":[132,181],"order-one":[133,147,182],"condition.":[134],"Separately,":[135],"finite":[137,171,193],"bimodule":[139],"$L(H)+\\epsilon'R(H)$":[141],"sign":[145],"relations":[148],"retains":[150],"them":[151],"under":[152],"inner":[154],"fluctuations.":[155],"Even":[156],"doubling":[157],"realizes":[158],"KO-sign":[160],"patterns":[161],"for":[162],"dimensions":[163],"0":[164],"6.":[166],"These":[167],"results":[168],"establish":[169],"internal":[174],"constructions":[177],"identifying":[179],"extension":[183],"molecular":[185],"implementation":[186],"distinct":[188],"targets.":[189],"Exact":[190],"verification":[191],"Lean":[194],"certificates":[195],"accompany":[196],"proofs.":[198]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-11-07T00:00:00"}
