{"id":"https://openalex.org/W7225776013","doi":"https://doi.org/10.5281/zenodo.24143286","title":"Klein Block Studies I: Topological Necessity of Conformal Nodal Hypersurfaces on the Non-Orientable S3-Bundle over S1 and the Free-Boundary Variational Selection of Conformally Regular Solutions","display_name":"Klein Block Studies I: Topological Necessity of Conformal Nodal Hypersurfaces on the Non-Orientable S3-Bundle over S1 and the Free-Boundary Variational Selection of Conformally Regular Solutions","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225776013","doi":"https://doi.org/10.5281/zenodo.24143286"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24143286","doi":"https://doi.org/10.5281/zenodo.24143286","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143286","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.24143286","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148856208","display_name":"Canon","orcid":"https://orcid.org/0009-0001-9037-6209","observed_orcids":["https://orcid.org/0009-0001-9037-6209"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Canon","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0001-9037-6209","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5148856208"],"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.40939998626708984,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.40939998626708984,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11960","display_name":"Relativity and Gravitational Theory","score":0.3488999903202057,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"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/T10095","display_name":"Cosmology and Gravitation Theories","score":0.12759999930858612,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.59170001745224},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5156000256538391},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.46790000796318054},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.43130001425743103},{"id":"https://openalex.org/C130187892","wikidata":"https://www.wikidata.org/wiki/Q133327","display_name":"Spacetime","level":2,"score":0.40849998593330383},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3871999979019165},{"id":"https://openalex.org/C95857938","wikidata":"https://www.wikidata.org/wiki/Q658429","display_name":"Vector bundle","level":2,"score":0.3799000084400177},{"id":"https://openalex.org/C28903235","wikidata":"https://www.wikidata.org/wiki/Q746550","display_name":"Tangent bundle","level":3,"score":0.3774999976158142},{"id":"https://openalex.org/C9136319","wikidata":"https://www.wikidata.org/wiki/Q362640","display_name":"Covariant transformation","level":2,"score":0.36800000071525574},{"id":"https://openalex.org/C53216431","wikidata":"https://www.wikidata.org/wiki/Q3997736","display_name":"Weyl transformation","level":4,"score":0.366100013256073},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.365200012922287},{"id":"https://openalex.org/C125669323","wikidata":"https://www.wikidata.org/wiki/Q513175","display_name":"Conformal geometry","level":4,"score":0.34790000319480896},{"id":"https://openalex.org/C173848574","wikidata":"https://www.wikidata.org/wiki/Q3507963","display_name":"Conformal symmetry","level":3,"score":0.34119999408721924},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32589998841285706},{"id":"https://openalex.org/C5667645","wikidata":"https://www.wikidata.org/wiki/Q217255","display_name":"Lorentz transformation","level":2,"score":0.31630000472068787},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.290800005197525},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.28439998626708984},{"id":"https://openalex.org/C155765329","wikidata":"https://www.wikidata.org/wiki/Q7243128","display_name":"Primary field","level":4,"score":0.28200000524520874},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.2793999910354614},{"id":"https://openalex.org/C96403706","wikidata":"https://www.wikidata.org/wiki/Q543956","display_name":"Monodromy","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.272599995136261},{"id":"https://openalex.org/C7451433","wikidata":"https://www.wikidata.org/wiki/Q1265039","display_name":"Solenoidal vector field","level":3,"score":0.2667999863624573},{"id":"https://openalex.org/C54613284","wikidata":"https://www.wikidata.org/wiki/Q1191271","display_name":"Conformal field theory","level":3,"score":0.266400009393692},{"id":"https://openalex.org/C2780791080","wikidata":"https://www.wikidata.org/wiki/Q273711","display_name":"Einstein field equations","level":3,"score":0.2653000056743622},{"id":"https://openalex.org/C24531701","wikidata":"https://www.wikidata.org/wiki/Q5160256","display_name":"Conformal gravity","level":4,"score":0.2624000012874603},{"id":"https://openalex.org/C147452769","wikidata":"https://www.wikidata.org/wiki/Q11452","display_name":"General relativity","level":2,"score":0.2508000135421753},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24143286","doi":"https://doi.org/10.5281/zenodo.24143286","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143286","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.24143287","doi":"https://doi.org/10.5281/zenodo.24143287","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143287","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.24143286","doi":"https://doi.org/10.5281/zenodo.24143286","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24143286","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W7213221948","https://openalex.org/W7213249554","https://openalex.org/W7213348096"],"related_works":[],"abstract_inverted_index":{"Klein":[0,51,125,341,603,644,663,669,712,736],"Block":[1,126,342,604],"Studies":[2],"I:":[3],"Topological":[4,671],"Necessity":[5,640],"of":[6,21,41,71,157,175,180,213,222,236,252,311,358,366,418,529,549,552,637,641,649,659,673,700],"Conformal":[7],"Nodal":[8,720],"Hypersurfaces":[9],"on":[10,49,123,137,200,242,318,425,710,734],"the":[11,17,50,53,69,80,87,124,151,158,172,194,223,234,237,250,253,282,305,319,335,340,351,359,371,376,384,393,407,422,443,462,490,515,522,533,546,575,602,608,612,623,642,646,650,657,660,674,688,698,711,726,735],"Non-Orientable":[12,675],"S3-Bundle":[13,676],"over":[14,57,677,685],"S1":[15,58,686],"and":[16,101,161,189,245,266,270,308,363,375,387,429,454,468,489,535,571,621,656,693,696,702,723,731],"Free-Boundary":[18],"Variational":[19],"Selection":[20],"Conformally":[22],"Regular":[23],"Solutions":[24],"KLEIN":[25],"BLOCK":[26],"STUDIES":[27],"I":[28,634,744],"(BUILDS":[29],"ON":[30],"DOCUMENTS":[31],"I-III)":[32],"This":[33],"paper":[34,560],"is":[35,154,166,292,563,566,573,579,606,611],"a":[36,45,109,138,147,155,167,190,274,286,323,330,345,355,396,412,437,485,526,539,550],"rigorous":[37],"mathematical":[38],"cosmology":[39],"study":[40],"Einstein":[42],"gravity":[43],"with":[44,59,86,127,226,395,465,508,595],"relativistic":[46],"radiation":[47,313,410,593],"fluid":[48,353,388],"Block,":[52],"compact":[54],"non-orientable":[55,682],"S3-bundle":[56,684],"reflection":[60,356],"monodromy":[61],"established":[62],"in":[63,129],"Document":[64,633,665,705,743,747,751],"II.":[65],"It":[66],"asks":[67],"what":[68,562,565,572],"non-orientability":[70],"spacetime":[72],"(first":[73],"Stiefel-Whitney":[74,694],"class":[75,212],"w1":[76],"nonzero)":[77],"implies":[78],"for":[79,184,590,717],"Big":[81],"Bang":[82],"singularity":[83,394],"when":[84,182,193,344],"combined":[85],"Tzanavaris-Boyle-Turok":[88],"free-boundary":[89,259,423],"action":[90],"principle.":[91],"The":[92,257,290,457,559,586,639,668],"answer":[93],"connects":[94],"topology,":[95],"conformally":[96,326,554],"regular":[97,163,327,510,555],"(isotropic)":[98],"cosmological":[99],"singularities,":[100,263,314],"an":[102,130,177,185,210,582],"exact":[103,265],"linear":[104,413],"mode":[105,404,543],"classification":[106],"governed":[107],"by":[108,616],"Pell":[110,438,587],"equation.":[111],"KEY":[112],"RESULTS":[113],"INCLUDE:":[114],"1.":[115],"Theorem":[116,205,300,401,619],"A":[117,278,714],"(topological":[118],"necessity).":[119],"Every":[120],"continuous":[121],"coframe":[122],"values":[128],"oriented":[131],"internal":[132],"Lorentz":[133],"bundle":[134],"must":[135],"degenerate":[136],"nonempty":[139],"set":[140,165],"that":[141,607,622,679],"meets":[142],"every":[143,478],"orientation-reversing":[144],"loop.":[145],"Under":[146,329],"conformal":[148,152,195,228,297,360,377],"spatial-triad":[149],"hypothesis":[150],"factor":[153,196,378],"section":[156],"orientation":[159],"bundle,":[160],"its":[162,691],"zero":[164,199],"closed":[168,408,592],"nodal":[169,224,320],"hypersurface":[170,321],"representing":[171],"Poincare":[173],"dual":[174],"w1:":[176],"odd":[178,379],"number":[179],"three-spheres":[181],"spacelike":[183],"auxiliary":[186],"product":[187],"metric,":[188],"single":[191,397],"S3":[192],"has":[197,525],"one":[198,681],"each":[201,243],"temporal":[202],"line.":[203],"2.":[204],"B":[206],"(variational":[207],"selection).":[208],"In":[209,350,514],"off-shell":[211],"coframes":[214],"whose":[215,440],"spatial":[216],"volume":[217,276],"collapses":[218],"at":[219,477,511,556,598],"both":[220,426,512,557,599],"faces":[221,272,427],"hypersurface,":[225,614],"no":[227],"regularity":[229],"assumed,":[230],"smooth":[231,316,325,683],"variations":[232,248],"force":[233],"trace":[235],"extrinsic-curvature":[238],"momentum":[239],"to":[240,295,339,380,461,484,494],"vanish":[241],"face,":[244],"independent":[246],"face":[247],"add":[249],"vanishing":[251],"trace-free":[254],"shear":[255,284],"momentum.":[256],"resulting":[258],"condition":[260,291,348,424,588],"excludes":[261],"velocity-dominated":[262],"including":[264],"asymptotically":[267],"Kasner,":[268],"stiff-fluid,":[269],"BKL-type":[271],"under":[273,285],"stated":[275,331,580],"bound.":[277],"Hamiltonian-constraint":[279],"estimate":[280],"removes":[281],"residual":[283],"curvature":[287],"sign":[288],"condition.":[289],"shown":[293],"not":[294],"characterise":[296],"regularity.":[298],"3.":[299],"C":[301],"(conditional":[302],"descent).":[303],"Using":[304],"Newman,":[306],"Claudel-Newman":[307],"Anguige-Tod":[309,352],"theory":[310],"isotropic":[312],"any":[315,591],"three-metric":[317],"determines":[322],"local":[324],"solution.":[328],"two-ended":[332],"continuation":[333],"hypothesis,":[334],"boundary":[336,610],"geometry":[337],"descends":[338],"exactly":[343,460,680],"twisted":[346,486],"matching":[347],"holds.":[349],"gauge,":[354],"symmetry":[357],"field":[361],"equations":[362],"formal":[364],"uniqueness":[365],"their":[367],"Fuchsian":[368],"reduction":[369],"make":[370],"unphysical":[372],"metric":[373],"even":[374],"all":[381],"orders,":[382],"so":[383,455,538],"rescaled":[385],"density":[386],"velocity":[389],"extend":[390],"smoothly":[391],"through":[392,618],"time":[398],"orientation.":[399],"4.":[400],"D":[402],"(exact":[403],"classification).":[405],"On":[406],"round":[409],"cycle,":[411],"adiabatic":[414],"scalar":[415,503],"(Bardeen)":[416],"perturbation":[417],"wavenumber":[419],"k":[420,446],"satisfies":[421,687],"if":[428,431],"only":[430],"(k+1)^2":[432],"-":[433,742],"3N^2":[434],"=":[435,447,518],"1,":[436],"equation":[439],"solutions":[441,509,553],"form":[442],"sparse":[444],"tower":[445],"6,":[448],"25,":[449],"96,":[450],"361,":[451],"1350,":[452],"5041,":[453],"on.":[456],"problem":[458,578],"reduces":[459],"Poschl-Teller":[463],"operator":[464],"spectrum":[466],"N^2":[467],"Gegenbauer":[469],"eigenfunctions.":[470],"Tensor":[471],"(gravitational":[472],"wave)":[473],"modes":[474,625],"are":[475,492,505,626],"admissible":[476,624],"level":[479],"n":[480,517],">=":[481],"2":[482,519],"subject":[483],"parity":[487],"rule,":[488],"results":[491],"extended":[493],"general":[495],"isometric":[496],"monodromies.":[497],"5.":[498],"Second-order":[499],"obstruction.":[500],"Non-resonant":[501],"forced":[502,615],"channels":[504],"uniquely":[506],"solvable":[507],"faces.":[513,558],"resonant":[516],"tensor":[520,542],"channel":[521],"quadratic":[523],"source":[524],"nonzero":[527],"projection":[528],"-(20":[530],"pi/3)":[531],"times":[532],"background":[534],"amplitude":[536],"data,":[537],"pure":[540],"lowest-level":[541],"cannot":[544],"be":[545],"first-order":[547],"term":[548],"family":[551],"separates":[561],"proved,":[564],"assumed":[567],"(numbered":[568],"hypotheses":[569],"H1-H7),":[570],"conjectured;":[574],"nonlinear":[576],"return":[577],"as":[581],"explicit":[583],"open":[584],"problem.":[585],"holds":[589],"cycle":[594],"free":[596],"boundaries":[597],"ends.":[600],"What":[601],"adds":[605],"second":[609],"same":[613],"topology":[617],"A,":[620],"parity-filtered.":[627],"COMPANION":[628,740],"PAPERS":[629],"IN":[630],"THIS":[631],"ARCHITECTURE:":[632],"(WHY):":[635,745],"Shape":[636],"Reality:":[638],"Zero-Energy":[643],"BlockEstablishes":[645],"ontological":[647],"foundations":[648],"architecture,":[651],"deriving":[652],"five":[653],"Closure-Admissible":[654],"postulates":[655],"necessity":[658],"finite,":[661],"zero-energy":[662],"Block.":[664,737],"II":[666,748],"(WHAT):":[667,749],"Block:":[670,713],"Uniqueness":[672],"S1Proves":[678],"postulates,":[689],"computes":[690],"homology":[692],"classes,":[695],"proves":[697],"existence":[699],"Pin+":[701],"Pin-":[703],"structures.":[704],"III":[706,752],"(HOW):":[707,753],"Twisted":[708],"Dynamics":[709],"Conditional":[715],"Framework":[716],"Kinematic":[718],"Descent,":[719],"Defect":[721],"Structure,":[722],"Anomaly":[724],"ConstraintsFormulates":[725],"conditional":[727],"dynamical,":[728],"fermionic,":[729],"defect,":[730],"anomaly":[732],"framework":[733],"DOI":[738],"FOR":[739],"PAPERS:":[741],"10.5281/zenodo.22766109-":[746],"10.5281/zenodo.22766247-":[750],"10.5281/zenodo.22766260":[754]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}
