{"id":"https://openalex.org/W7167702005","doi":"https://doi.org/10.6084/m9.figshare.32942279","title":"Scale Graphs I: Minimum-Production Renormalization and a Strict Lyapunov Identity","display_name":"Scale Graphs I: Minimum-Production Renormalization and a Strict Lyapunov Identity","publication_year":2026,"publication_date":"2026-07-08","ids":{"openalex":"https://openalex.org/W7167702005","doi":"https://doi.org/10.6084/m9.figshare.32942279"},"language":"en","primary_location":{"id":"pmh:oai:figshare.com:article/32942279","doi":null,"is_oa":true,"landing_page_url":"https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140285709","display_name":"Andrea Caffagni","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andrea Caffagni","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5140285709"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":14.1845,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.99240794,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"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/T10591","display_name":"Theoretical and Computational Physics","score":0.364300012588501,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.364300012588501,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter 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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.3280999958515167,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.11159999668598175,"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/decimation","display_name":"decimation","score":0.7229999899864197},{"id":"https://openalex.org/keywords/renormalization-group","display_name":"renormalization group","score":0.7850000262260437},{"id":"https://openalex.org/keywords/eigenvalue-spectrum","display_name":"eigenvalue spectrum","score":0.593999981880188},{"id":"https://openalex.org/keywords/spectral-gap","display_name":"spectral gap","score":0.6259999871253967},{"id":"https://openalex.org/keywords/shadowing-effect","display_name":"shadowing","score":0.4819999933242798}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8131999969482422},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5157999992370605},{"id":"https://openalex.org/C166124518","wikidata":"https://www.wikidata.org/wiki/Q1047702","display_name":"Renormalization","level":2,"score":0.4706000089645386},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4674000144004822},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4481000006198883},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3763999938964844},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.3758000135421753},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.37049999833106995},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36880001425743103},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.3458000123500824},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3384999930858612},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3059999942779541},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2890999913215637},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.27379998564720154},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.26899999380111694},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.26669999957084656},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2529999911785126},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.25049999356269836}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:figshare.com:article/32942279","doi":null,"is_oa":true,"landing_page_url":"https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"},{"id":"doi:10.6084/m9.figshare.32942279","doi":"https://doi.org/10.6084/m9.figshare.32942279","is_oa":true,"landing_page_url":"https://doi.org/10.6084/m9.figshare.32942279","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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:10.6084/m9.figshare.32942279.v1","doi":"https://doi.org/10.6084/m9.figshare.32942279.v1","is_oa":true,"landing_page_url":"https://doi.org/10.6084/m9.figshare.32942279.v1","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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":"pmh:oai:figshare.com:article/32942279","doi":null,"is_oa":true,"landing_page_url":"https://figshare.com/articles/preprint/Scale_Graphs_I_Minimum-Production_Renormalization_and_a_Strict_Lyapunov_Identity/32942279","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"},"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/W7171288801","https://openalex.org/W7171189911","https://openalex.org/W7203581566","https://openalex.org/W7203584042","https://openalex.org/W7172426512","https://openalex.org/W7095886942","https://openalex.org/W630089378","https://openalex.org/W7221013171","https://openalex.org/W7148486598","https://openalex.org/W4313887589"],"abstract_inverted_index":{"We":[0],"isolate":[1],"the":[2,7,40,98,106,117,130,144,155,165,169,188,197,244,253,263,270],"finite-dimensional":[3],"algebraic":[4,161],"core":[5],"of":[6,13,84,109,240],"Onsager\u2013Prigogine":[8],"scale-graph":[9],"framework.":[10],"Each":[11],"edge":[12],"a":[14,19,72,102,139,219,238,249],"finite":[15],"scale":[16],"graph":[17],"carries":[18],"nonnegative":[20,183],"quadratic":[21],"production":[22,70,85,99,181],"polynomial":[23],"P_e(\u03bb)":[24,33,53],"=":[25,54,203,231,255],"a_e":[26,45,55],"\u03bb\u00b2":[27],"\u2212":[28,57,121],"2J_e":[29],"\u03bb":[30],"+":[31,59,123],"Y_e,":[32],"\u2265":[34,63],"0,":[35],"which":[36],"is":[37,80,101,116,127,217,234],"equivalent":[38],"to":[39,48,159],"Onsager":[41,170],"bound":[42],"J_e\u00b2":[43],"\u2264":[44],"Y_e":[46],"and":[47,125,138,187,233,262],"an":[49,89],"exact":[50,90],"Prigogine":[51,110],"square-completion":[52],"(\u03bb":[56],"p_e)\u00b2":[58],"\u03ba_e,":[60],"with":[61,177,209],"\u03ba_e":[62],"0.":[64],"Eliminating":[65],"intermediate":[66],"vertices":[67],"by":[68,129],"minimum":[69],"gives":[71],"canonical":[73],"dyadic":[74,136],"renormalization":[75,149],"map.":[76],"The":[77,151,214,226,259],"main":[78],"result":[79],"not":[81,235],"merely":[82],"monotonicity":[83],"under":[86],"decimation,":[87],"but":[88],"strict":[91],"Lyapunov":[92],"identity:":[93],"for":[94,143,252],"every":[95],"eliminated":[96],"vertex,":[97],"drop":[100],"computable":[103],"square":[104,119],"measuring":[105],"local":[107],"failure":[108],"closure.":[111],"In":[112,164],"current":[113],"variables":[114],"it":[115,126],"mismatch":[118],"(q\u2081":[120],"q\u2082)\u00b2/(a\u2081":[122],"a\u2082),":[124],"dominated":[128],"edgewise":[131],"alignment":[132],"defects.":[133],"This":[134],"yields":[135],"summability":[137],"Cesaro":[140],"n^(\u22121/2)":[141],"rate":[142],"mean":[145],"square-root":[146],"defect":[147],"along":[148],"orbits.":[150],"final":[152],"section":[153],"reports":[154],"numerical":[156,260],"evidence":[157],"relevant":[158],"this":[160,241],"paper":[162],"only.":[163],"archived":[166,268],"shell":[167,190,257],"ensemble,":[168],"inequality":[171],"was":[172,182,194],"tested":[173],"on":[174,184],"25,200":[175],"pairs":[176],"zero":[178],"violations,":[179],"residual":[180],"570/570":[185],"samples,":[186],"observed":[189],"RG":[191],"tangent":[192],"eigenvalue":[193,216,242],"recomputed":[195],"from":[196],"raw":[198],"RG-pair":[199],"table":[200],"as":[201,237],"\u03b4_u^shell":[202],"1.4187394,":[204],"95%":[205],"CI":[206],"[1.4002,":[207],"1.4366],":[208],"spectral-gap":[210],"ratio":[211],"approximately":[212],"3.95.":[213],"leading":[215],"real;":[218],"subleading":[220],"complex-conjugate":[221],"pair":[222],"has":[223,229],"modulus":[224],"0.2574.":[225],"shadowing":[227],"diagnostic":[228],"R\u00b2":[230],"0.0021":[232],"used":[236],"confirmation":[239],"because":[243],"shipped":[245],"v32":[246],"certificate":[247],"documents":[248],"resolution":[250],"limit":[251],"N":[254],"22":[256],"testbed.":[258],"files":[261],"Python":[264],"reproduction":[265],"script":[266],"are":[267],"in":[269],"accompanying":[271],"reproducibility":[272],"record,":[273],"DOI:":[274],"10.6084/m9.figshare.32942288.":[275]},"counts_by_year":[{"year":2026,"cited_by_count":9}],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-07-09T00:00:00"}