{"id":"https://openalex.org/W7225615766","doi":"https://doi.org/10.5281/zenodo.24074123","title":"Hopf Arithmetic\u2013Energy Duality B: Arithmetic\u2013Spectral Gap Correspondence on Compact Lie Group Principal Bundles\u2014Conditional Channel, Discrimination Threshold, and Boundary Declarations","display_name":"Hopf Arithmetic\u2013Energy Duality B: Arithmetic\u2013Spectral Gap Correspondence on Compact Lie Group Principal Bundles\u2014Conditional Channel, Discrimination Threshold, and Boundary Declarations","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225615766","doi":"https://doi.org/10.5281/zenodo.24074123"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24074123","doi":"https://doi.org/10.5281/zenodo.24074123","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24074123","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.24074123","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5108680601","display_name":"changzheng zhou","orcid":"https://orcid.org/0009-0002-4186-7877","observed_orcids":["https://orcid.org/0009-0002-4186-7877"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"zhou, changzheng","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0002-4186-7877","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5136242336","display_name":"ziqing zhou","orcid":"https://orcid.org/0009-0005-6883-6300","observed_orcids":["https://orcid.org/0009-0005-6883-6300"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"zhou, ziqing","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0005-6883-6300","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.5942000150680542,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.5942000150680542,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.07559999823570251,"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/T11166","display_name":"Analytic Number Theory Research","score":0.06159999966621399,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/schreier-graphs","display_name":"Schreier graphs","score":0.9769999980926514},{"id":"https://openalex.org/keywords/spectral-gap","display_name":"spectral gap","score":0.9010000228881836},{"id":"https://openalex.org/keywords/additive-energy","display_name":"additive energy","score":0.9169999957084656},{"id":"https://openalex.org/keywords/conditional-lower-bounds","display_name":"conditional lower bounds","score":0.4129999876022339},{"id":"https://openalex.org/keywords/edge-expansion","display_name":"edge expansion","score":0.34299999475479126}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8676000237464905},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.546500027179718},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.47200000286102295},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4449999928474426},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.4034000039100647},{"id":"https://openalex.org/C2781458696","wikidata":"https://www.wikidata.org/wiki/Q7307244","display_name":"Reflection group","level":4,"score":0.40139999985694885},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.39570000767707825},{"id":"https://openalex.org/C54540088","wikidata":"https://www.wikidata.org/wiki/Q3498041","display_name":"Intersection graph","level":4,"score":0.382099986076355},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3806000053882599},{"id":"https://openalex.org/C89436406","wikidata":"https://www.wikidata.org/wiki/Q1544261","display_name":"Totally disconnected space","level":3,"score":0.35830000042915344},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C36605576","wikidata":"https://www.wikidata.org/wiki/Q273188","display_name":"Cantor set","level":2,"score":0.31619998812675476},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.29840001463890076},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.2955999970436096},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.2888000011444092},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.2865999937057495},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.27480000257492065},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.2741999924182892},{"id":"https://openalex.org/C18116961","wikidata":"https://www.wikidata.org/wiki/Q734209","display_name":"Generating set of a group","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.2565000057220459}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24074123","doi":"https://doi.org/10.5281/zenodo.24074123","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24074123","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.24074124","doi":"https://doi.org/10.5281/zenodo.24074124","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24074124","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.24074123","doi":"https://doi.org/10.5281/zenodo.24074123","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24074123","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/W2178311504","https://openalex.org/W7211873979","https://openalex.org/W2008238301","https://openalex.org/W2094198986","https://openalex.org/W2963027710","https://openalex.org/W4235497163","https://openalex.org/W2137053050","https://openalex.org/W7204644362","https://openalex.org/W2117541443","https://openalex.org/W7096204711"],"abstract_inverted_index":{"This":[0,225],"paper":[1,226],"studies,":[2],"in":[3,24,187],"the":[4,12,15,25,32,43,50,56,68,71,75,89,95,116,144,152,161,163,167,170,178,188,192,197,213,217,220,231],"framework":[5,204],"of":[6,18,22,31,160,169,205,216,223],"compact":[7],"Lie":[8],"group":[9],"principal":[10],"bundles,":[11],"conditionalcorrespondence":[13],"between":[14,118],"arithmetic":[16,51],"structure":[17,106],"a":[19,148,234],"finite":[20],"set":[21],"points":[23],"base":[26],"space":[27],"andthe":[28],"spectral":[29,153,179],"regularity":[30],"Schreier":[33],"graph":[34],"it":[35],"generates.":[36],"The":[37],"two":[38,210],"routes":[39],"use":[40],"disjoint":[41],"hypothesissets:":[42],"discrimination":[44],"threshold":[45],"lemma":[46],"acts":[47,92],"only":[48,93,198],"on":[49,94,113,196,237],"input":[52],"layer":[53,98,130],"and":[54,74,99,109,140,166],"provesthat":[55,176],"product-type":[57],"sum-product":[58],"criterion":[59],"does":[60,227],"not":[61,228],"imply":[62],"an":[63,124,157],"additive":[64],"energy":[65,164],"upper":[66],"bound,with":[67],"grid":[69],"family,":[70,73],"product":[72],"root-of-unity":[76],"family":[77,201,240],"constituting":[78],"three":[79],"types":[80],"ofwitnesses":[81],"whose":[82],"roles":[83],"cannot":[84],"substitute":[85],"for":[86,151],"one":[87],"another;":[88],"group-action\u2013coordinate-embeddingincompatibility":[90],"declaration":[91],"structural":[96],"compatibility":[97,138],"shows":[100,190],"that":[101,191,230],"thegroup":[102],"action":[103],"preserves":[104],"orbit":[105],"while":[107],"addition":[108],"multiplication":[110],"are":[111],"defined":[112],"complexcoordinates,":[114],"so":[115],"channel":[117],"them":[119],"must":[120],"be":[121],"introduced":[122],"by":[123,156],"explicit":[125,158],"hypothesis.":[126],"Underthe":[127],"uniformized":[128],"hypothesis":[129],"(energy":[131],"bound":[132,150],"under":[133],"coordinate":[134],"embedding":[135],"H1\u2032,":[136],"morphismchain":[137],"H2\u2032,":[139],"spectral-gap\u2013edge-expansion":[141],"interface":[142],"H3\u2032),":[143],"conditional":[145],"maintheorem":[146,218],"gives":[147],"lower":[149],"gap":[154,180],"bounded":[155],"function":[159],"squareof":[162],"exponent":[165],"degree":[168],"generating":[171],"set.":[172],"A":[173],"circle":[174],"counterexample":[175],"unconditionally":[177],"can":[181],"tend":[182],"to":[183],"zero;":[184],"graph-by-graph":[185],"verification":[186],"toylayer":[189],"uniform":[193],"form":[194,222,236],"fails":[195],"explicitly":[199],"computable":[200],"known":[202],"withinthe":[203],"this":[206],"paper;":[207],"together":[208],"these":[209],"facts":[211],"lock":[212],"instantiation":[214],"burden":[215],"onto":[219],"family-uniform":[221],"H2\u2032.":[224],"claim":[229],"theorem":[232],"holdsin":[233],"nontrivial":[235],"any":[238],"verified":[239]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}