{"id":"https://openalex.org/W7225629989","doi":"https://doi.org/10.5281/zenodo.24088196","title":"Prime-Composite Hilbert Space Decomposition, Schur Complement Completeness, and Spectral Localizer Analysis of the B\u00e1ez\u2013Duarte Criterion","display_name":"Prime-Composite Hilbert Space Decomposition, Schur Complement Completeness, and Spectral Localizer Analysis of the B\u00e1ez\u2013Duarte Criterion","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225629989","doi":"https://doi.org/10.5281/zenodo.24088196"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24088196","doi":"https://doi.org/10.5281/zenodo.24088196","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24088196","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","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24088196","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073401422","display_name":"Qamar Munawer","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Munawer, Qamar","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5073401422"],"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":0.9994999766349792,"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":0.9994999766349792,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":9.999999747378752e-05,"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":9.999999747378752e-05,"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/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.9070000052452087},{"id":"https://openalex.org/keywords/baez-duarte-criterion","display_name":"B\u00e1ez-Duarte criterion","score":0.9670000076293945},{"id":"https://openalex.org/keywords/riemann-zeta-function","display_name":"Riemann zeta function","score":0.8529999852180481},{"id":"https://openalex.org/keywords/prime-numbers","display_name":"prime numbers","score":0.5460000038146973},{"id":"https://openalex.org/keywords/composite-numbers","display_name":"composite numbers","score":0.8090000152587891},{"id":"https://openalex.org/keywords/hilbert-space-decomposition","display_name":"Hilbert space decomposition","score":0.8870000243186951},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.8169999718666077},{"id":"https://openalex.org/keywords/spectral-localizer","display_name":"spectral localizer","score":0.7129999995231628},{"id":"https://openalex.org/keywords/mobius-function","display_name":"M\u00f6bius function","score":0.8930000066757202},{"id":"https://openalex.org/keywords/square-free-integers","display_name":"square-free integers","score":0.4819999933242798}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8830000162124634},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.44999998807907104},{"id":"https://openalex.org/C2731191","wikidata":"https://www.wikidata.org/wiki/Q1778169","display_name":"Schur complement","level":3,"score":0.4496999979019165},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44670000672340393},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.4138000011444092},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.3822000026702881},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.38029998540878296},{"id":"https://openalex.org/C63356602","wikidata":"https://www.wikidata.org/wiki/Q1986013","display_name":"Lacunary function","level":2,"score":0.3725999891757965},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3671000003814697},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.3662000000476837},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3203999996185303},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3156999945640564},{"id":"https://openalex.org/C77246614","wikidata":"https://www.wikidata.org/wiki/Q1409400","display_name":"Gramian matrix","level":3,"score":0.31139999628067017},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.30570000410079956},{"id":"https://openalex.org/C134466208","wikidata":"https://www.wikidata.org/wiki/Q1520657","display_name":"Weight function","level":2,"score":0.29420000314712524},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2881999909877777},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C95439218","wikidata":"https://www.wikidata.org/wiki/Q1666173","display_name":"Interlacing","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.2734000086784363},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.27250000834465027},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.2718000113964081},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C92788228","wikidata":"https://www.wikidata.org/wiki/Q3777923","display_name":"Multiplicative group","level":3,"score":0.2700999975204468},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.2660999894142151},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.26579999923706055},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.257999986410141},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.25459998846054077}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24088196","doi":"https://doi.org/10.5281/zenodo.24088196","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24088196","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.24088197","doi":"https://doi.org/10.5281/zenodo.24088197","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24088197","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.24088196","doi":"https://doi.org/10.5281/zenodo.24088196","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24088196","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7225586524","https://openalex.org/W7223037696","https://openalex.org/W7164923588","https://openalex.org/W7225005937","https://openalex.org/W3012185539","https://openalex.org/W7153528370","https://openalex.org/W1697683484","https://openalex.org/W7210273147","https://openalex.org/W7159939858","https://openalex.org/W7204015783"],"abstract_inverted_index":{"The":[0],"B\u00e1ez\u2013Duarte":[1,64,277],"criterion":[2],"strengthens":[3],"the":[4,8,26,33,63,72,107,125,134,189,213,230,238,257,275,292,312,333,350,354,394,410,417,421],"Beurling\u2013Nyman":[5],"formulation":[6],"of":[7,37,133,171,256,274,353,416],"Riemann":[9,108,422],"Hypothesis":[10,423],"by":[11],"establishing":[12],"that":[13,180,212,237,252,369],"\u03b6(s)":[14,116],"\u2260":[15],"0":[16,51,251,268,381,383],"on":[17,130,202,329],"Re(s)":[18],"greater":[19,157,249,364],"than":[20,158,164,167,250,365],"1/2":[21],"if":[22,25],"and":[23,87,127,160,367,420],"only":[24],"indicator":[27],"function":[28,110,192],"\u03c7_(0,":[29],"1)":[30,187,309],"belongs":[31],"to":[32,142,272,403],"closed":[34],"linear":[35],"span":[36,185],"fractional":[38],"dilation":[39],"functions":[40],"in":[41,174,307],"L\u00b2(0,":[42,175,186,308],"1),":[43],"with":[44,269],"squared":[45],"projection":[46],"residuals":[47],"satisfying":[48],"d_N\u00b2":[49],"\u2192":[50,54,267,324,414],"as":[52],"N":[53,261,404,413],"\u221e.":[55],"In":[56],"this":[57],"master":[58],"monograph,":[59],"we":[60,348],"systematically":[61],"investigate":[62],"Hilbert":[65],"space":[66],"through":[67],"an":[68,112,151,218,244,319,425],"arithmetic":[69,152],"decomposition:":[70],"partitioning":[71],"natural":[73],"numbers":[74],"\u2115":[75],"=":[76,117,194,262,339,358,376,380,384,405],"{1}":[77],"\u222a":[78,80],"\u2119":[79,86],"\u2102_comp":[81],"into":[82,111],"multiplicative":[83],"prime":[84,181],"atoms":[85],"non-prime":[88],"composites":[89],"\u2102_comp.":[90],"We":[91,105,123,177,290],"establish":[92,349],"five":[93],"principal":[94],"findings:":[95],"Prime-Composite":[96],"Zeta":[97],"Duality":[98],"&":[99,388],"50/50":[100],"Critical":[101],"Parity":[102],"Exponent":[103],"(s*):":[104],"decompose":[106],"zeta":[109,191],"exact":[113,351],"additive":[114],"tripartition":[115],"1":[118],"+":[119,121,281,341,344,378,382],"P(s)":[120],"\u03b6_comp(s).":[122],"prove":[124,178,291,368],"existence":[126],"strict":[128],"uniqueness":[129],"(1,":[131],"\u221e)":[132],"critical":[135],"parity":[136],"exponent":[137],"s*":[138],"\u2248":[139,223],"1.446537622476006285931438514306":[140],"(certified":[141],"30":[143],"decimal":[144],"digits":[145],"using":[146],"70-digit":[147],"ball":[148],"arithmetic),":[149],"defining":[150],"boundary":[153],"between":[154],"prime-dominated":[155],"(\u03c3":[156],"s*)":[159,168],"composite-dominated":[161],"(1":[162,297],"less":[163,166],"\u03c3":[165],"arithmetic.":[169],"Incompleteness":[170],"Prime-Only":[172],"Dilations":[173],"1):":[176],"analytically":[179],"dilations":[182],"alone":[183],"cannot":[184],"because":[188],"reciprocal":[190],"1/\u03b6(s)":[193],"\u2211":[195],"\u03bc(n)":[196],"n^(-s)":[197],"carries":[198],"non-zero":[199],"M\u00f6bius":[200],"weight":[201],"composite":[203,239],"square-free":[204],"integers":[205],"(\u03bc(6)=1,":[206],"\u03bc(10)=1).":[207],"Certified":[208],"numerical":[209],"projections":[210],"confirm":[211],"prime-only":[214],"residual":[215,258],"stagnates":[216],"at":[217,260],"asymptotic":[219,320],"floor":[220],"d_(\u221e,":[221],"\u2119)\u00b2":[222],"0.028514.":[224],"Composite":[225],"Schur":[226,240],"Variance":[227],"Reduction:":[228],"Block-partitioning":[229],"centered":[231],"Vasyunin":[232],"Gram":[233,314],"matrix":[234,315],"Q_N":[235],"reveals":[236],"complement":[241],"S_N":[242],"contributes":[243],"orthogonal":[245],"variance":[246],"reduction":[247,259],"\u0394d_N\u00b2":[248],"accounts":[253],"for":[254],"66.08%":[255],"128,":[263],"driving":[264],"d_(N,":[265,299],"full)\u00b2":[266],"ratio":[270],"converging":[271],"1.0158":[273],"theoretical":[276],"decay":[278],"rate":[279],"(2":[280],"\u03b3":[282],"-":[283,298],"log(4\u03c0))":[284],"/":[285,301],"log(N).":[286],"Courant\u2013Fischer":[287],"Variational":[288],"Bridge:":[289],"Rayleigh":[293],"bound":[294],"\u03bb\u2081(Q_N)":[295],"\u2264":[296],"full)\u00b2)":[300],"||c_N||\u2082\u00b2,":[302],"demonstrating":[303],"mathematically":[304],"why":[305],"completeness":[306],"necessarily":[310],"forces":[311],"unperturbed":[313],"ground":[316],"state":[317],"toward":[318],"zero":[321],"mode":[322],"(\u03bb\u2081(Q_N)":[323],"0).":[325,399],"Clifford":[326],"Spectral":[327],"Localizer":[328],"Regularized":[330],"Models:":[331],"On":[332],"regularized":[334],"Hamiltonian":[335],"family":[336],"H^(N)(\u03b1,":[337],"\u03b8)":[338],"Q":[340],"\u03b1":[342],"D_s":[343],"\u03b8":[345],"(i":[346],"\u03a9),":[347],"block-diagonality":[352],"cusp":[355],"coordinate":[356],"(||Y_PC||":[357],"0),":[359,366],"bilateral":[360],"shift":[361],"cross-coupling":[362],"(||X_PC||":[363],"Haynsworth's":[370],"inertia":[371],"additivity":[372],"holds":[373],"identically:":[374],"sig(L_full)":[375],"sig(L_PP)":[377],"sig(S_L)":[379],"0.":[385],"Epistemic":[386],"Scope":[387],"Governance:":[389],"All":[390],"calculations":[391],"strictly":[392],"enforce":[393],"Zero-Free":[395],"Firewall":[396],"(ZERO_INPUT_COUNT":[397],"==":[398],"Finite-scale":[400],"computations":[401],"up":[402],"128":[406],"do":[407],"not":[408],"settle":[409],"infinite-dimensional":[411],"limit":[412],"\u221e":[415],"unregularized":[418],"system,":[419],"remains":[424],"open":[426],"mathematical":[427],"conjecture.":[428],"Accompanying":[429],"author":[430],"corrigendum":[431],"formally":[432],"rectifies":[433],"earlier":[434],"preprint":[435],"claims":[436]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}
