{"id":"https://openalex.org/W7212165456","doi":"https://doi.org/10.5281/zenodo.22682819","title":"Finiteness and Hierarchical Structure of Open Systems: A Computable Emergence Criterion via the Wuliang Method","display_name":"Finiteness and Hierarchical Structure of Open Systems: A Computable Emergence Criterion via the Wuliang Method","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212165456","doi":"https://doi.org/10.5281/zenodo.22682819"},"language":"cmn","primary_location":{"id":"doi:10.5281/zenodo.22682819","doi":"https://doi.org/10.5281/zenodo.22682819","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22682819","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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","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.22682819","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151235991","display_name":"Wuliang Wu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Wu, Liang","raw_affiliation_strings":["W3 Lab"],"raw_orcid":"https://orcid.org/0009-0006-9694-5739","affiliations":[{"raw_affiliation_string":"W3 Lab","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5151235991"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":6,"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/T14299","display_name":"Complex Systems and Dynamics","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear 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/T14299","display_name":"Complex Systems and Dynamics","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear 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/T12469","display_name":"Chaos, Complexity, and Education","score":0.0013000000035390258,"subfield":{"id":"https://openalex.org/subfields/3304","display_name":"Education"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10778","display_name":"Philosophy of Biology and Scientific Explanation","score":0.0007999999797903001,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/emergence","display_name":"emergence","score":0.5870000123977661},{"id":"https://openalex.org/keywords/open-systems","display_name":"open systems","score":0.9959999918937683},{"id":"https://openalex.org/keywords/hierarchy","display_name":"hierarchy","score":0.7570000290870667},{"id":"https://openalex.org/keywords/finiteness","display_name":"finiteness","score":0.8759999871253967},{"id":"https://openalex.org/keywords/stratification","display_name":"stratification","score":0.5059999823570251},{"id":"https://openalex.org/keywords/self-duality","display_name":"self-duality","score":0.6790000200271606},{"id":"https://openalex.org/keywords/tcp-ip-stack","display_name":"TCP/IP protocol stack","score":0.7300000190734863},{"id":"https://openalex.org/keywords/gene-regulatory-networks","display_name":"gene regulatory networks","score":1.0},{"id":"https://openalex.org/keywords/closed-systems","display_name":"closed systems","score":0.6200000047683716},{"id":"https://openalex.org/keywords/ai-architecture","display_name":"AI architecture","score":0.4869999885559082},{"id":"https://openalex.org/keywords/ai-safety","display_name":"AI safety","score":0.996999979019165}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6735000014305115},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.5608999729156494},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5273000001907349},{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.483599990606308},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.41589999198913574},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.40380001068115234},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3774000108242035},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.37229999899864197},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.367900013923645},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.3677999973297119},{"id":"https://openalex.org/C2777723229","wikidata":"https://www.wikidata.org/wiki/Q4367921","display_name":"Learnability","level":2,"score":0.34529998898506165},{"id":"https://openalex.org/C201943243","wikidata":"https://www.wikidata.org/wiki/Q7551008","display_name":"Social connectedness","level":2,"score":0.3434000015258789},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.3400000035762787},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3393999934196472},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.3370000123977661},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.328900009393692},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.314300000667572},{"id":"https://openalex.org/C2776085556","wikidata":"https://www.wikidata.org/wiki/Q183361","display_name":"Chen","level":2,"score":0.31049999594688416},{"id":"https://openalex.org/C165525559","wikidata":"https://www.wikidata.org/wiki/Q224180","display_name":"Homology (biology)","level":3,"score":0.29840001463890076},{"id":"https://openalex.org/C35234519","wikidata":"https://www.wikidata.org/wiki/Q3983973","display_name":"Homotopy lifting property","level":4,"score":0.29490000009536743},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C54486226","wikidata":"https://www.wikidata.org/wiki/Q493941","display_name":"Fibration","level":3,"score":0.2928999960422516},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2904999852180481},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.2903999984264374},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.28200000524520874},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2754000127315521},{"id":"https://openalex.org/C192943249","wikidata":"https://www.wikidata.org/wiki/Q1893382","display_name":"Stratification (seeds)","level":5,"score":0.2554999887943268},{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.2540999948978424}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.5281/zenodo.22682819","doi":"https://doi.org/10.5281/zenodo.22682819","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22682819","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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.22682820","doi":"https://doi.org/10.5281/zenodo.22682820","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22682820","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.22687384","doi":"https://doi.org/10.5281/zenodo.22687384","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22687384","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.22808897","doi":"https://doi.org/10.5281/zenodo.22808897","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22808897","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.22968897","doi":"https://doi.org/10.5281/zenodo.22968897","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22968897","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.23049182","doi":"https://doi.org/10.5281/zenodo.23049182","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23049182","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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23201809","doi":"https://doi.org/10.5281/zenodo.23201809","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23201809","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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.22682819","doi":"https://doi.org/10.5281/zenodo.22682819","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22682819","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-nc","license_id":"https://openalex.org/licenses/cc-by-nc","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/W7162403616","https://openalex.org/W2334911870","https://openalex.org/W7056836287","https://openalex.org/W7167718716","https://openalex.org/W7168083375","https://openalex.org/W7171972105","https://openalex.org/W7171002751","https://openalex.org/W7161135311","https://openalex.org/W7167724773","https://openalex.org/W7128770018"],"abstract_inverted_index":{"Open":[0],"systems,":[1,41],"systems":[2,187],"exchanging":[3],"matter,":[4],"energy,":[5],"and":[6,15,29,67,83,86,145,167,181,199,212,236],"information":[7],"with":[8],"their":[9],"environment,":[10],"span":[11],"biology,":[12],"engineering,":[13],"society,":[14],"physics,":[16],"yet":[17],"remain":[18,287],"among":[19],"the":[20,36,90,104,115,122,126,141,169,177,221,228,232,240,254,268,273,288,292],"hardest":[21],"objects":[22],"to":[23,190,256,262,298],"formalize:":[24],"far":[25],"from":[26],"equilibrium,":[27],"irreducible,":[28],"studied":[30],"only":[31],"case":[32],"by":[33,101],"case.":[34],"Emergence,":[35],"central":[37],"phenomenon":[38],"of":[39,56,108,125,234,247],"open":[40],"has":[42],"lacked":[43],"a":[44,53,192,206,248],"verifiable":[45],"mathematical":[46],"criterion":[47,158,225],"since":[48],"Anderson.":[49],"This":[50,242],"paper":[51],"advances":[52],"counter-intuitive":[54],"chain":[55],"propositions:":[57],"openness":[58,79],"enforces":[59,64],"hierarchical":[60,62],"structure;":[61],"structure":[63,124],"bounded":[65],"information;":[66],"boundedness":[68],"makes":[69],"formalization":[70],"possible.":[71],"From":[72],"this":[73,219],"we":[74],"give":[75],"three":[76],"structural":[77],"axioms,":[78],"(O1),":[80],"stratification":[81],"(L1),":[82],"finiteness":[84],"(F1),":[85],"define":[87],"emergence":[88,142,211,224],"as":[89,231,239],"irreducible":[91],"excess":[92],"over":[93],"assembly":[94,237],"prediction":[95,238],"during":[96],"level":[97],"transitions,":[98],"computed":[99],"layer":[100,102],"via":[103],"homotopy":[105,229],"cofiber":[106,230],"homology":[107],"comparison":[109,235],"maps.":[110],"The":[111,157,283],"framework":[112,255],"is":[113,121,140,149,159,220,243],"named":[114],"Wuliang":[116],"Method:":[117],"its":[118,137,146],"core":[119,138,147],"construction":[120],"three-layer":[123,165],"constructive":[127],"operator":[128],"(evidence":[129],"layer,":[130,133],"normalized":[131],"aggregation":[132],"complex":[134],"modulation":[135],"layer),":[136],"invariant":[139],"quantity":[143],"E,":[144],"property":[148],"self-duality":[150],"(forward":[151],"generation":[152,155],"constructs;":[153],"reverse":[154],"monitors).":[156],"computable":[160,223],"(polynomial":[161],"time),":[162],"constructible":[163],"(explicit":[164],"operator),":[166],"falsifiable:":[168],"same":[170],"signature":[171],"appears":[172],"in":[173],"two":[174,293],"mechanism-free":[175],"domains,":[176],"Internet":[178],"(TCP/IP":[179],"stack)":[180],"biology":[182],"(gene":[183],"regulatory":[184],"networks).":[185],"Closed":[186],"are":[188,203,296],"proven":[189],"be":[191],"degenerate":[193],"zero-flux":[194],"special":[195],"case;":[196],"static":[197],"(flux-rank)":[198],"dynamic":[200],"(behavioral-flow)":[201],"criteria":[202],"orthogonal,":[204],"yielding":[205],"four-quadrant":[207],"classification":[208],"covering":[209],"explosive":[210],"zero-emergence":[213],"perpetual":[214],"motion.":[215],"To":[216],"our":[217],"knowledge":[218],"first":[222],"that":[226],"takes":[227],"axis":[233],"baseline.":[241],"Paper":[244,251,260],"I":[245],"(foundations)":[246],"three-part":[249],"series;":[250],"II":[252],"applies":[253],"AI":[257,263],"architecture":[258],"deconstruction,":[259],"III":[261],"safety.":[264],"Revision":[265],"V4.5":[266],"(2026-10-07):":[267],"corresponding-author":[269],"block":[270],"now":[271],"includes":[272],"W3":[274],"LAB":[275],"paper-collection":[276],"website":[277],"(https://7fba2d25df5d4627a2297a066267371a.app.workbuddy.host);":[278],"no":[279],"technical":[280],"content":[281],"changes.":[282],"typeset":[284],"DOCX/PDF":[285],"files":[286,295],"V4.4/V4.3":[289],"production":[290],"versions;":[291],"Markdown":[294],"updated":[297],"V4.5.":[299]},"counts_by_year":[{"year":2026,"cited_by_count":6}],"updated_date":"2026-10-11T09:32:39.932853","created_date":"2026-09-11T00:00:00"}