{"id":"https://openalex.org/W7224134063","doi":"https://doi.org/10.5281/zenodo.23734084","title":"Theorem U\u2032-Local: The Local Exponent Law of the Syntonic Window","display_name":"Theorem U\u2032-Local: The Local Exponent Law of the Syntonic Window","publication_year":2026,"publication_date":"2026-10-28","ids":{"openalex":"https://openalex.org/W7224134063","doi":"https://doi.org/10.5281/zenodo.23734084"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23734084","doi":"https://doi.org/10.5281/zenodo.23734084","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23734084","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.23734084","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133867601","display_name":"Jean-Pierre Bronsard","orcid":"https://orcid.org/0009-0008-6639-7553","observed_orcids":["https://orcid.org/0009-0008-6639-7553"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Bronsard, Jean-Pierre","raw_affiliation_strings":["SyntonicAI Recherche"],"raw_orcid":"https://orcid.org/0009-0008-6639-7553","affiliations":[{"raw_affiliation_string":"SyntonicAI Recherche","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5133867601"],"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/T11325","display_name":"Inertial Sensor and Navigation","score":0.3082999885082245,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11325","display_name":"Inertial Sensor and Navigation","score":0.3082999885082245,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10711","display_name":"Target Tracking and Data Fusion in Sensor Networks","score":0.14480000734329224,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T13487","display_name":"Statistical and numerical algorithms","score":0.07339999824762344,"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/fractional-brownian-motion","display_name":"fractional Brownian motion","score":0.6869999766349792},{"id":"https://openalex.org/keywords/structure-functions","display_name":"structure functions","score":0.6069999933242798},{"id":"https://openalex.org/keywords/power-law-noise","display_name":"power-law noise","score":0.5759999752044678},{"id":"https://openalex.org/keywords/allan-deviation","display_name":"Allan deviation","score":0.722000002861023},{"id":"https://openalex.org/keywords/crossover-regime","display_name":"crossover regime","score":0.4309999942779541},{"id":"https://openalex.org/keywords/regular-variation","display_name":"regular variation","score":0.6909999847412109},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.4429999887943268},{"id":"https://openalex.org/keywords/low-frequency-noise","display_name":"low-frequency noise","score":0.6940000057220459},{"id":"https://openalex.org/keywords/gyroscope-noise","display_name":"gyroscope noise","score":0.5529999732971191},{"id":"https://openalex.org/keywords/nonparametric-estimation","display_name":"nonparametric estimation","score":0.46399998664855957}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7789000272750854},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.5882999897003174},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.5088000297546387},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.45750001072883606},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40470001101493835},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.40230000019073486},{"id":"https://openalex.org/C2778751112","wikidata":"https://www.wikidata.org/wiki/Q835016","display_name":"Window (computing)","level":2,"score":0.392300009727478},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.38670000433921814},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.3718999922275543},{"id":"https://openalex.org/C87040749","wikidata":"https://www.wikidata.org/wiki/Q428971","display_name":"Power law","level":2,"score":0.34299999475479126},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3359000086784363},{"id":"https://openalex.org/C108819105","wikidata":"https://www.wikidata.org/wiki/Q1143293","display_name":"Fractional Brownian motion","level":3,"score":0.32910001277923584},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.31929999589920044},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3165000081062317},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C140101238","wikidata":"https://www.wikidata.org/wiki/Q1404885","display_name":"Window function","level":3,"score":0.2971999943256378},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.29429998993873596},{"id":"https://openalex.org/C102366305","wikidata":"https://www.wikidata.org/wiki/Q1097688","display_name":"Nonparametric statistics","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.28349998593330383},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.28209999203681946},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C112633086","wikidata":"https://www.wikidata.org/wiki/Q381287","display_name":"White noise","level":2,"score":0.27630001306533813}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23734084","doi":"https://doi.org/10.5281/zenodo.23734084","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23734084","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.23734086","doi":"https://doi.org/10.5281/zenodo.23734086","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23734086","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.23734084","doi":"https://doi.org/10.5281/zenodo.23734084","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23734084","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7167329439","https://openalex.org/W7169845162","https://openalex.org/W3149846267","https://openalex.org/W3006759444","https://openalex.org/W2095277511","https://openalex.org/W2186746128","https://openalex.org/W2900336952","https://openalex.org/W2971544157","https://openalex.org/W1673760037","https://openalex.org/W2023889966"],"abstract_inverted_index":{"Theorem":[0],"U\u2032":[1],"gives":[2,70],"the":[3,10,26,32,46,58,71,88,100,103,108,112,125,129,133,142,151,155,166,191,208,213,221,226,229,233,243,253,264,273,285,296,306,312,325,338,342,345,359,368,374,379,393,397,402,408,433,444,447,459,482,520],"optimal":[4,130,244,346],"window":[5,59,102,131,245,347],"of":[6,49,55,111,135,141,161,168,179,197,207,228,240,249,324,344,356,367,480,513],"a":[7,50,53,66,138,158,188,205,279,290,308,328,336,383,415,462,506],"first-order":[8],"tracker":[9],"exponent":[11,156,255,274,314],"\\(\\alpha=1/(2H+1-\\gamma)\\).":[12],"The":[13,172,441,524],"statement":[14,420,529],"is":[15,30,41,52,90,106,132,157,175,204,218,237,246,299,478,485,510,517,531,538],"exact":[16],"for":[17,177,533],"two-power":[18],"losses":[19],"and":[20,57,150,182,252,283,372,391,410,418,436,450,457,468,495,516,530],"holds":[21],"at":[22,170,275,305,364],"leading":[23],"order":[24],"on":[25,263],"regular-variation":[27],"class,":[28],"which":[29,268],"how":[31],"Corpus":[33,521],"Status":[34,522],"Register":[35],"records":[36],"it.":[37],"Real":[38],"low-frequency":[39],"noise":[40,387],"rarely":[42],"one":[43],"power":[44,180],"law:":[45],"Allan":[47],"curve":[48],"gyroscope":[51],"sum":[54],"regimes,":[56],"an":[60,511,539],"engine":[61],"chooses":[62],"can":[63],"sit":[64],"in":[65,269,352,423,438,487,505,519],"crossover.":[67],"This":[68],"note":[69],"local":[72,254,286,313,339],"law.":[73],"For":[74,99,194],"any":[75],"separable":[76],"loss":[77],"\\(J(\\tau)=A(\\tau)+\\sigma^2\\beta(\\tau)\\)":[78],"whose":[79],"balance":[80,174,375],"ratio":[81],"\\(R=A'/(-\\beta')\\)":[82],"increases":[83],"from":[84,335,401],"\\(0\\)":[85],"to":[86,257,293,303,316],"\\(\\infty\\),":[87],"optimum":[89,192,380],"unique":[91],"and\\[\\alpha_{\\mathrm{loc}}(\\sigma^2)=\\frac{d\\log\\tau^\\star}{d\\log\\sigma^2}=\\frac{1}{\\mathsf{s}[A'](\\tau^\\star)-\\mathsf{s}[-\\beta'](\\tau^\\star)},\\qquad":[92],"\\mathsf{s}[f](\\tau)=\\frac{\\tau":[93],"f'(\\tau)}{f(\\tau)},\\]an":[94],"identity":[95],"with":[96,117,124,259,289,332],"no":[97,147,260,535],"asymptotics.":[98],"exponential":[101],"tracking":[104],"term":[105],"half":[107],"Laplace":[109],"average":[110],"structure":[113,230],"function,":[114],"\\(A(\\tau)=\\tfrac12\\mathbb{E}[D(\\tau":[115],"E)]\\)":[116],"\\(E\\)":[118],"standard":[119,350],"exponential.":[120],"Under":[121],"white":[122],"noise,":[123],"same":[126],"monotone":[127,173,378],"balance,":[128],"root":[134],"\\(\\tau\\,\\mathbb{E}[(E-1)D(\\tau":[136],"E)]=\\sigma^2\\),":[137],"nonparametric":[139],"form":[140],"syntonic":[143],"law":[144,287,340],"that":[145,153],"needs":[146],"memory":[148],"index,":[149],"index":[152,203,241,250],"governs":[154],"Laguerre-weighted":[159],"functional":[160],"\\(D\\)":[162,169,236],"around":[163],"\\(\\tau\\),":[164],"not":[165,183,414],"log-slope":[167],"\\(\\tau\\).":[171],"automatic":[176],"sums":[178,196],"laws":[181,403],"beyond:":[184],"Brownian":[185],"motion":[186],"plus":[187],"sinusoid":[189],"makes":[190],"jump.":[193],"finite":[195,276],"independent":[198,479],"fractional":[199],"components":[200],"this":[201,406,424,514],"effective":[202],"mean":[206],"component":[209],"indices":[210],"tilted":[211],"towards":[212],"smoother":[214],"ones,":[215],"so":[216,405],"it":[217,509],"never":[219],"below":[220,318,390],"log-slope:":[222],"reading":[223,298,361],"\\(H\\)":[224],"off":[225,300],"slope":[227,297,360],"function":[231],"overestimates":[232],"exponent.":[234],"If":[235],"regularly":[238,247],"varying":[239,248,266],"\\(2H\\),":[242],"\\(1/(2H+1)\\)":[251],"converges":[256],"it,":[258],"smoothness":[261],"condition":[262],"slowly":[265],"factor,":[267],"general":[270],"still":[271],"moves":[272],"scales.":[277],"On":[278],"two-component":[280],"example":[281],"(\\(H=1/2\\)":[282],"\\(0.9\\))":[284],"agrees":[288],"numerical":[291,330,412,490],"derivative":[292],"\\(1.3\\times10^{-10}\\),":[294],"while":[295],"by":[301,382,428],"up":[302],"\\(0.038\\)":[304],"crossover;":[307],"logarithmic":[309],"correction":[310],"keeps":[311],"\\(0.008\\)":[315],"\\(0.044\\)":[317],"its":[319,411],"limit":[320],"over":[321],"eight":[322],"decades":[323],"window.":[326],"In":[327],"pre-registered":[329,493],"bench":[331],"predictions":[333],"frozen":[334],"calibration,":[337],"matched":[341],"exponents":[343],"within":[348],"three":[349],"errors":[351],"all":[353],"37":[354],"secants":[355],"four":[357],"drifts,":[358,371],"overestimated":[362],"them":[363],"every":[365,452,455,528],"secant":[366],"two":[369],"crossover":[370],"where":[373],"stops":[376],"being":[377],"jumped,":[381],"factor":[384],"\\(4.23\\)":[385],"between":[386],"levels":[388],"\\(2\\,\\%\\)":[389],"above":[392],"predicted":[394],"jump":[395],"level;":[396],"drifts":[398],"were":[399],"drawn":[400],"assumed,":[404],"verifies":[407],"mathematics":[409],"chain,":[413],"sensor.":[416],"Authorship":[417],"collaboration":[419],"All":[421],"prose":[422],"corpus":[425,460],"was":[426],"drafted":[427],"large":[429],"language":[430,536],"models":[431],"under":[432],"author's":[434],"direction":[435],"revised":[437],"author-directed":[439],"loops.":[440],"author":[442,525],"set":[443],"questions,":[445],"chose":[446],"hypotheses,":[448],"designed":[449,458],"ran":[451],"experiment,":[453],"decided":[454],"retraction,":[456],"as":[461],"system":[463],"(immutable":[464],"hub-and-spoke":[465],"record,":[466],"concept":[467],"version":[469],"identifiers,":[470],"errata":[471],"register,":[472],"hash-frozen":[473],"firmware,":[474],"pre-registration":[475],"gates).":[476],"Verification":[477],"both:":[481],"algebraic":[483],"core":[484],"machine-checked":[486],"Lean":[488],"4/Mathlib,":[489],"results":[491],"follow":[492],"protocols,":[494],"flight":[496],"logs":[497],"are":[498],"public.":[499],"Where":[500],"model-generated":[501],"reasoning":[502],"text":[503],"survives":[504],"published":[507],"document,":[508],"artifact":[512],"method":[515],"recorded":[518],"Register.":[523],"alone":[526],"approves":[527],"accountable":[532],"it;":[534],"model":[537],"author.":[540],"RETURN":[541],"TO":[542],"HUB":[543],":":[544],"https://doi.org/10.5281/zenodo.17254395":[545]},"counts_by_year":[],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2026-10-29T00:00:00"}