{"id":"https://openalex.org/W7225596179","doi":"https://doi.org/10.5281/zenodo.24047297","title":"Positive Realization and Hidden-State Complexity: Spectral Completion, Exact Minima, and the Necessity of Hidden Dynamics","display_name":"Positive Realization and Hidden-State Complexity: Spectral Completion, Exact Minima, and the Necessity of Hidden Dynamics","publication_year":2026,"publication_date":"2026-11-07","ids":{"openalex":"https://openalex.org/W7225596179","doi":"https://doi.org/10.5281/zenodo.24047297"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24047297","doi":"https://doi.org/10.5281/zenodo.24047297","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24047297","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":"article-journal"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24047297","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076858945","display_name":"K. Fathi","orcid":"https://orcid.org/0009-0001-5546-1475","observed_orcids":["https://orcid.org/0009-0001-5546-1475"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Fathi, Kevin","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0001-5546-1475","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5076858945"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"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":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.353300005197525,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.353300005197525,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.18019999563694,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.04430000111460686,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/positive-realization","display_name":"positive realization","score":0.8080000281333923},{"id":"https://openalex.org/keywords/rational-transfer-functions","display_name":"rational transfer functions","score":0.6129999756813049},{"id":"https://openalex.org/keywords/spectral-multiplicity","display_name":"spectral multiplicity","score":0.6809999942779541},{"id":"https://openalex.org/keywords/stationary-modes","display_name":"stationary modes","score":0.5210000276565552},{"id":"https://openalex.org/keywords/companion-matrix","display_name":"companion matrix","score":0.6019999980926514},{"id":"https://openalex.org/keywords/cyclotomic-polynomials","display_name":"cyclotomic polynomials","score":0.5979999899864197},{"id":"https://openalex.org/keywords/ramanujan-sums","display_name":"Ramanujan sums","score":0.9629999995231628},{"id":"https://openalex.org/keywords/fourier-interpolation","display_name":"Fourier interpolation","score":0.4909999966621399},{"id":"https://openalex.org/keywords/periodic-response","display_name":"periodic response","score":0.41499999165534973}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8665000200271606},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.4738999903202057},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.4722000062465668},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.46889999508857727},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.46779999136924744},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.45010000467300415},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.44429999589920044},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4438000023365021},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4230000078678131},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3995000123977661},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.39640000462532043},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.39160001277923584},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3743000030517578},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36489999294281006},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.36329999566078186},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.34209999442100525},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3321000039577484},{"id":"https://openalex.org/C45542137","wikidata":"https://www.wikidata.org/wiki/Q381040","display_name":"Complex conjugate","level":2,"score":0.33000001311302185},{"id":"https://openalex.org/C87698059","wikidata":"https://www.wikidata.org/wiki/Q1808960","display_name":"LTI system theory","level":3,"score":0.30230000615119934},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C168773769","wikidata":"https://www.wikidata.org/wiki/Q1350299","display_name":"Satisfiability","level":2,"score":0.2985999882221222},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.2888999879360199},{"id":"https://openalex.org/C192065140","wikidata":"https://www.wikidata.org/wiki/Q1147928","display_name":"Multinomial distribution","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2786000072956085},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.27799999713897705},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.26980000734329224},{"id":"https://openalex.org/C195956108","wikidata":"https://www.wikidata.org/wiki/Q7268362","display_name":"Quadratic growth","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.26589998602867126},{"id":"https://openalex.org/C135007989","wikidata":"https://www.wikidata.org/wiki/Q643156","display_name":"Rotation number","level":3,"score":0.26339998841285706},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.5281/zenodo.24047297","doi":"https://doi.org/10.5281/zenodo.24047297","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24047297","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":"article-journal"},{"id":"doi:10.5281/zenodo.24047298","doi":"https://doi.org/10.5281/zenodo.24047298","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24047298","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":"article-journal"},{"id":"doi:10.5281/zenodo.24054047","doi":"https://doi.org/10.5281/zenodo.24054047","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24054047","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":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24047297","doi":"https://doi.org/10.5281/zenodo.24047297","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24047297","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":"article-journal"},"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/W7221330690","https://openalex.org/W4232235567","https://openalex.org/W2992732831","https://openalex.org/W7123336921","https://openalex.org/W2160718734","https://openalex.org/W2004549766","https://openalex.org/W2132292620","https://openalex.org/W4281669218","https://openalex.org/W2084427838","https://openalex.org/W2960361513"],"abstract_inverted_index":{"How":[0],"many":[1,101],"nonnegative":[2,170],"internal":[3],"states":[4],"are":[5,276],"needed":[6],"to":[7,15],"reproduce":[8],"a":[9,92,128,136,227],"rational":[10,51,207],"response":[11],"exactly?":[12],"This":[13],"sequel":[14],"Diagonal":[16],"Transport":[17],"and":[18,27,38,135,160,167,184,198,263,269,282,312,316],"Spectral":[19,237],"Fibers":[20],"studies":[21],"minimum":[22,290],"state":[23,216],"number,":[24],"transition":[25],"rank,":[26],"the":[28,70,124,177,204,296,303],"distinct":[29],"roles":[30],"of":[31,123,230],"hidden":[32,57,81,178,301],"reachable":[33,48,108],"recurrences,":[34],"unobserved":[35],"spectral":[36],"multiplicities,":[37],"positive":[39,61,89],"polyhedral":[40],"lifts.":[41],"A":[42,83],"sharp":[43],"duality":[44],"bound":[45,192,299],"controls":[46],"excess":[47],"dimension.":[49],"Explicit":[50],"families":[52],"show":[53,252],"that":[54,218],"one":[55,80],"compulsory":[56],"mode":[58],"can":[59,149],"reduce":[60],"order":[62,115,159],"by":[63,181],"an":[64,168],"unbounded":[65],"factor.":[66],"Positive-circuit":[67],"elimination":[68],"bounds":[69,232],"corresponding":[71],"ordinary-dimensional":[72],"cone":[73,310],"cost,":[74],"with":[75,244,278,295],"optimal":[76,169],"quadratic":[77],"growth":[78],"for":[79,94,131,300],"coordinate.":[82],"short":[84],"trace":[85,140],"argument":[86],"identifies":[87,176],"strictly":[88],"responses":[90,156,243],"approaching":[91],"constant":[93,242],"which":[95],"every":[96,114],"minimum-order":[97,147],"realization":[98],"requires":[99],"arbitrarily":[100],"stationary":[102,144],"directions,":[103],"even":[104],"though":[105],"no":[106],"additional":[107],"recurrence":[109],"is":[110,239],"needed.":[111],"Equality":[112],"classifies":[113],"optimum.":[116],"Nonnegative":[117],"companion":[118],"constructions":[119],"determine":[120,258],"exact":[121,158,205,240,284],"portions":[122],"damping":[125,315],"staircase,":[126],"including":[127],"complete":[129],"classification":[130],"two":[132],"conjugate":[133],"pairs":[134],"divisibility":[137],"criterion":[138],"at":[139,195,209],"thresholds.":[141],"The":[142,287],"least":[143,206],"dimension":[145],"among":[146],"realizations":[148],"be":[150],"nonmonotone":[151],"along":[152],"damping.":[153],"Eventually":[154],"periodic":[155,193],"admit":[157],"rank":[161],"formulas":[162],"through":[163],"their":[164],"transient":[165],"length":[166],"cycle":[171],"decomposition.":[172],"Fourier":[173],"interval":[174],"interpolation":[175],"modes":[179],"required":[180],"overlapping":[182],"cycles":[183],"supplies":[185],"sparse":[186],"separation":[187,307],"certificates.":[188],"Cyclotomic":[189],"orbit":[190],"budgets":[191],"complexity":[194],"fixed":[196,245],"ordinary":[197],"coefficient-field":[199],"degrees;":[200],"Ramanujan":[201],"sums":[202],"give":[203,266],"degree":[208],"any":[210],"prescribed":[211],"period.":[212],"Digit":[213],"arithmetic":[214],"certifies":[215],"costs":[217],"fixed-window":[219],"winding":[220],"observations":[221],"cannot":[222],"control.":[223],"Puiseux":[224],"valuations":[225],"provide":[226],"separate":[228],"transfer":[229],"lower":[231],"from":[233,308],"tropical":[234],"weighted":[235],"automata.":[236],"completion":[238],"near":[241],"simple":[246],"poles,":[247],"while":[248],"cubic":[249,306],"residue":[250,317],"examples":[251],"why":[253],"poles":[254],"alone":[255],"do":[256],"not":[257],"minimality.":[259],"Compression,":[260],"resolvent":[261],"normalization,":[262],"shift-compatible":[264],"factorizations":[265],"finite":[267],"certificates":[268],"closed":[270],"semialgebraic":[271],"strata.":[272],"All":[273],"theorem":[274],"proofs":[275],"analytical,":[277],"explanatory":[279],"vector":[280],"figures":[281],"supplementary":[283],"identity":[285],"checks.":[286],"unrestricted":[288],"structural":[289],"problem":[291],"remains":[292],"open,":[293],"together":[294],"general":[297,313],"ordinary-degree":[298],"complexity,":[302],"proposed":[304],"five-state":[305],"six":[309],"facets,":[311],"intermediate":[314],"boundaries.":[318]},"counts_by_year":[],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2026-11-09T00:00:00"}