{"id":"https://openalex.org/W7225756125","doi":"https://doi.org/10.5281/zenodo.24117188","title":"Left identity and complementary negative directions of the covering scale commutator are not leftover interpolating sequential type","display_name":"Left identity and complementary negative directions of the covering scale commutator are not leftover interpolating sequential type","publication_year":2026,"publication_date":"2026-11-10","ids":{"openalex":"https://openalex.org/W7225756125","doi":"https://doi.org/10.5281/zenodo.24117188"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24117188","doi":"https://doi.org/10.5281/zenodo.24117188","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24117188","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.24117188","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5146713286","display_name":"NANJIE MA","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"MA, NANJIE","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5146713286"],"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.70169997215271,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.70169997215271,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.2451000064611435,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11476","display_name":"Graph theory and applications","score":0.020400000736117363,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8671000003814697},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5619000196456909},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5180000066757202},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4059000015258789},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.3977999985218048},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3921000063419342},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.36640000343322754},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.36390000581741333},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3580999970436096},{"id":"https://openalex.org/C33222762","wikidata":"https://www.wikidata.org/wiki/Q321119","display_name":"Identity function","level":2,"score":0.35249999165534973},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.34470000863075256},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.3416000008583069},{"id":"https://openalex.org/C145620117","wikidata":"https://www.wikidata.org/wiki/Q2989763","display_name":"Commutator","level":4,"score":0.3337000012397766},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.30790001153945923},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3028999865055084},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30160000920295715},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.30000001192092896},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.2928999960422516},{"id":"https://openalex.org/C50742451","wikidata":"https://www.wikidata.org/wiki/Q1530388","display_name":"Equidistributed sequence","level":2,"score":0.28850001096725464},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.2770000100135803},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.27149999141693115},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C2776441312","wikidata":"https://www.wikidata.org/wiki/Q575710","display_name":"Gauss map","level":2,"score":0.2563000023365021},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.2526000142097473}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24117188","doi":"https://doi.org/10.5281/zenodo.24117188","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24117188","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.24117189","doi":"https://doi.org/10.5281/zenodo.24117189","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24117189","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.24117188","doi":"https://doi.org/10.5281/zenodo.24117188","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24117188","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":[],"abstract_inverted_index":{"Let":[0],"E_n":[1,19],"be":[2],"a":[3,9,96,132,150,165,252,346,353,357,390,395,421,445,456,466,528,531,562,579,590,613,636,656,723,759,763,766,824],"named":[4,10,662,739],"covering-class":[5,138,360],"embedding":[6],"and":[7,15,52,102,110,141,217,239,275,302,363,603,639,682,777,820],"T_n":[8,20],"covering-law":[11],"stepper,":[12],"both":[13],"column-stochastic,":[14],"set":[16],"Omega_n":[17,30,48],"=":[18,26,31,42,50,57,74,85,116,127,314,402,413,438],"-":[21,317],"T_{n-1}":[22],"E_{n-1}":[23],"with":[24,235,296,451,566,626,775,808],"S":[25,40,55,69,129,673],"sym(Omega_n).":[27],"Then":[28,409],"1^T":[29,39,410],"0":[32,51,104,414],"for":[33,415],"every":[34,416,586],"n":[35,84,417],">=":[36,418],"2,":[37,419],"hence":[38],"1":[41,49,56,101,266,316,325,412,619],"0.":[43,58],"The":[44,173,255,373,624],"identity":[45,426],"is":[46,53,70,131,145,148,230,247,322,326,345,436,455,527,578,654,676,690,741,752,822],"not":[47,54,149,251,352,368,389,465,495,530,559,655,786,806,814,823],"On":[59,121,429,475,498,533,612],"the":[60,65,113,124,137,142,178,184,189,198,225,260,284,310,331,349,370,424,433,479,502,518,540,551,570,597,640,645,660,666,669,683,693,706,710,715,719,737,772,797,816],"pair":[61,158],"(maximal":[62,430,522,679],"deletion,":[63,431,523,680],"port-resolved)":[64,432],"Sylvester":[66],"inertia":[67,435],"of":[68,99,205,228,232,288,299,341,348,356,377,386,394,398,423,427,506,521,544,574,582,599,601,631,665,672,686,726,733,771,788,796,800,827],"(n_-,":[71],"n_0,":[72],"n_+)":[73],"(0,1,0),":[75],"(1,0,1),":[76],"(1,0,3),":[77],"(1,0,10),":[78],"(1,0,32),":[79],"(2,0,116),":[80],"(4,0,467),":[81],"(8,0,2118)":[82],"at":[83,92,212,219,222,267,272,278,328,440,462,472,487,517,537,648,782],"2":[86,234,271,318,650],"through":[87,118,443,651],"9.":[88,120,280,474,489],"Negative":[89],"eigenvalues":[90],"appear":[91],"size":[93,119,160,220,223,268,273,279,282,329,473,488,538],"3,":[94],"form":[95,125,135,459],"four-size":[97],"plateau":[98,123,671],"index":[100],"nullity":[103],"on":[105,136,156,188,449,460,550,674,688,736],"{3,":[106],"4,":[107,214],"5,":[108,215],"6},":[109],"then":[111,469,484,594],"follow":[112],"look":[114],"n_-":[115],"2^{n-6}":[117],"that":[122,157,194,206,289,387,499,525,696,698],"Q_n(x,y)":[126],"x^T":[128],"y":[130],"Lorentzian":[133,457,670,684,711],"quadratic":[134,458],"coordinate":[139,186,311],"space,":[140],"all-ones":[143],"vector":[144,312,339],"null.":[146],"It":[147,653],"spacetime":[151,467,657,756,760],"metric.":[152],"Haynsworth":[153],"additivity":[154],"holds":[155],"from":[159,196,605],"3.":[161],"Binary":[162],"doubling":[163,197],"remains":[164],"look,":[166,529],"as":[167,420],"in":[168,622],"Companion":[169,378,616],"49":[170,379],"Theorem":[171,380,618],"5.":[172],"upward":[174],"stepper":[175],"T":[176,516],"sends":[177],"previous":[179],"negative":[180,199,262,504,542],"space":[181,187,258,774],"injectively":[182],"into":[183],"next":[185,261],"tested":[190],"sizes.":[191],"Rank":[192],"forbids":[193],"map":[195],"dimension":[200,265,270,276],"by":[201,705,709,714,718,722,728,731],"itself.":[202],"Principal":[203],"cosines":[204,287,510],"transport":[207,290],"are":[208,291,491,628,785,793,805],"0.983,":[209],"0.943,":[210],"0.916":[211],"sizes":[213,441,649],"6":[216],"0.948":[218],"7;":[221],"8":[224],"overlap":[226,297],"sum":[227],"squares":[229],"1.80":[231],"rank":[233,300],"Frobenius":[236],"residual":[237],"0.319":[238],"squared":[240],"energy":[241,303,383,547],"retention":[242,304],"about":[243,493,496],"0.90.":[244],"That":[245,343],"geography":[246,526],"injective":[248],"subspace":[249],"persistence,":[250],"parallel-transport":[253,764],"connection.":[254,532],"complementary":[256,716,773],"new":[257,825],"inside":[259],"eigenspace":[263],"has":[264,508,555,589],"7,":[269],"8,":[274],"4":[277,301,381],"At":[281],"9":[283],"four":[285],"principal":[286,509],"0.957,":[292],"0.951,":[293],"0.945,":[294],"0.941":[295],"3.598":[298],"0.900.":[305],"For":[306,561],"each":[307],"class-index":[308],"i":[309],"u^(i)":[313],"S_ii":[315],"(S":[319],"1)_i":[320],"e_i":[321],"null":[323,338],"whenever":[324],"null;":[327],"3":[330,442],"two":[332,485,632,694],"formulas":[333],"collapse":[334],"to":[335,678,692],"one":[336],"extra":[337],"independent":[340],"1.":[342],"family":[344],"rewrite":[347],"left":[350,425,707],"identity,":[351,708],"stereographic":[354,761],"projection":[355],"sphere.":[358],"Named":[359],"Hasse":[361,592],"features":[362,366],"birth-step":[364],"schedule":[365],"do":[367],"span":[369],"size-7":[371],"complement.":[372],"unique":[374,503,567,591,677,691],"paired-join":[375,552],"class":[376],"carries":[382,546],"fraction":[384,548],"0.096":[385],"complement,":[388],"claimed":[391],"22.66":[392],"percent":[393],"secondary":[396],"mode":[397,543],"S.":[399,497],"Write":[400],"F_n^(mu)":[401],"1/2":[403],"(Omega_n":[404],"M_n":[405,407],"+":[406],"Omega_n^T).":[408],"F":[411,507,545,687],"corollary":[422],"Omega.":[428],"flux":[434,667],"n_-(F)":[437],"0,1,1,1,1,1,3":[439],"9:":[444],"five-size":[446,481,724],"window":[447,482,501,685,725],"n_-=1":[448],"{4,5,6,7,8}":[450],"n_0=0,":[452],"so":[453],"Q_n^(F)":[454],"R^{W_n}":[461],"those":[463,734,746],"sizes,":[464],"metric,":[468,758],"three":[470],"negatives":[471,486],"(covering-order":[476,534],"reverse,":[477,535],"uniform-legal)":[478],"same":[480,500],"holds,":[483],"Those":[490],"statements":[492],"F,":[494,727],"direction":[505],"0.912,":[511],"0.891,":[512],"0.884,":[513],"0.893":[514],"under":[515],"successive":[519],"steps":[520],"port-resolved);":[524],"path-entropy)":[536],"7":[539],"second":[541],"0.227":[549],"class,":[553],"which":[554],"eight":[556],"linear":[557],"extensions,":[558],"one.":[560],"finite":[563,794,801],"poset":[564],"P":[565,575],"minimum":[568],"0,":[569,602],"Hasse-component":[571],"count":[572],"c_0(P)":[573,595],"minus":[576],"{0}":[577],"rooted-poset":[580],"invariant":[581],"covering-process":[583,614,633],"orders.":[584],"If":[585],"non-minimum":[587],"vertex":[588],"parent,":[593],"equals":[596],"number":[598],"covers":[600],"elements":[604],"distinct":[606],"components":[607],"have":[608],"down-set":[609],"intersection":[610],"{0}.":[611],"order,":[615],"32":[617],"forces":[620],"c_0":[621],"{1,2}.":[623],"classes":[625],"c_0=2":[627],"unordered":[629],"gluings":[630],"orders":[634],"along":[635],"fresh":[637],"minimum,":[638],"deposited":[641],"cardinalities":[642],"|W_n|":[643],"give":[644],"counts":[646],"0,1,1,3,6,18,52,183,699":[647],"10.":[652],"split.":[658],"Among":[659],"six":[661,738],"full-support":[663],"pairs":[664,695,740],"census,":[668],"{3,4,5,6}":[675],"port-resolved),":[681],"{4,...,8}":[689],"carry":[697],"window.":[699],"Occupying":[700],"leftover":[701,749],"interpolating":[702],"sequential":[703],"type":[704,799],"bilinear":[712],"form,":[713],"split,":[717],"coordinate-null":[720],"family,":[721],"c_0,":[729],"or":[730],"uniqueness":[732],"types":[735],"too":[742,753],"loose.":[743],"Remaining-never":[744],"because":[745],"objects":[747],"fail":[748],"interpolant":[750],"independence":[751],"tight.":[754],"A":[755],"Lorentz":[757],"projection,":[762],"connection,":[765],"renormalisation-group":[767],"flow,":[768],"an":[769,778],"identification":[770],"ker(E),":[776],"overlap-law":[779],"Phase-II":[780],"theorem":[781],"probability":[783],"one-third":[784],"theorems":[787],"this":[789],"note.":[790],"Covering":[791],"histories":[792],"posets":[795],"mathematical":[798],"causal":[802],"sets.":[803],"They":[804],"identified":[807],"physical":[809],"spacetime.":[810],"This":[811],"note":[812],"does":[813],"replace":[815],"parent":[817],"programme":[818],"manuscript,":[819],"it":[821],"version":[826],"Companions":[828],"13-49.":[829]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-11T00:00:00"}
