{"id":"https://openalex.org/W7223802402","doi":"https://doi.org/10.5281/zenodo.23690473","title":"Canonical Height Pairing, Regulator Eigenvalues, and Rank Bounds in BSD \u2014 E8 Intelligence Research","display_name":"Canonical Height Pairing, Regulator Eigenvalues, and Rank Bounds in BSD \u2014 E8 Intelligence Research","publication_year":2026,"publication_date":"2026-10-27","ids":{"openalex":"https://openalex.org/W7223802402","doi":"https://doi.org/10.5281/zenodo.23690473"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23690473","doi":"https://doi.org/10.5281/zenodo.23690473","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23690473","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","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23690473","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153285007","display_name":"Andrew Stewart Caldin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Caldin, Andrew Stewart","raw_affiliation_strings":["Independent Researcher, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher, United Kingdom","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153285007"],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9014000296592712,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9014000296592712,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.05810000002384186,"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":0.030500000342726707,"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/regulators","display_name":"regulators","score":0.6420000195503235},{"id":"https://openalex.org/keywords/birch-swinnerton-dyer-conjecture","display_name":"Birch and Swinnerton-Dyer conjecture","score":0.7139999866485596},{"id":"https://openalex.org/keywords/euclidean-lattices","display_name":"Euclidean lattices","score":0.8500000238418579},{"id":"https://openalex.org/keywords/root-lattice","display_name":"root lattice","score":0.6370000243186951},{"id":"https://openalex.org/keywords/golden-ratio","display_name":"golden ratio","score":0.5009999871253967},{"id":"https://openalex.org/keywords/j-invariant","display_name":"j-invariant","score":0.4959999918937683},{"id":"https://openalex.org/keywords/tate-shafarevich-group","display_name":"Tate-Shafarevich group","score":0.8529999852180481}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8051000237464905},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6489999890327454},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6044999957084656},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6029999852180481},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5641999840736389},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5302000045776367},{"id":"https://openalex.org/C6929976","wikidata":"https://www.wikidata.org/wiki/Q3771881","display_name":"Regulator","level":3,"score":0.5120000243186951},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4708999991416931},{"id":"https://openalex.org/C14103023","wikidata":"https://www.wikidata.org/wiki/Q11681459","display_name":"Pairing","level":3,"score":0.40389999747276306},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.3391000032424927},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.33489999175071716},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3253999948501587},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3000999987125397},{"id":"https://openalex.org/C51329190","wikidata":"https://www.wikidata.org/wiki/Q1076349","display_name":"Ising model","level":2,"score":0.2930999994277954},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.28200000524520874},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C2778778035","wikidata":"https://www.wikidata.org/wiki/Q967960","display_name":"Mathematics Subject Classification","level":2,"score":0.26170000433921814},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.2533000111579895}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23690473","doi":"https://doi.org/10.5281/zenodo.23690473","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23690473","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.23690474","doi":"https://doi.org/10.5281/zenodo.23690474","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23690474","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.23690473","doi":"https://doi.org/10.5281/zenodo.23690473","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23690473","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":0,"referenced_works":[],"related_works":["https://openalex.org/W7223554403","https://openalex.org/W7170823010","https://openalex.org/W7210288398","https://openalex.org/W7212759039","https://openalex.org/W7208822036","https://openalex.org/W7201953632","https://openalex.org/W7213957031","https://openalex.org/W7169805751","https://openalex.org/W7203738899","https://openalex.org/W7221060877"],"abstract_inverted_index":{"FINDING:":[0],"The":[1,70,116],"canonical":[2],"(N\u00e9ron\u2013Tate)":[3],"height":[4,71],"pairing":[5,72],"on":[6],"E(Q)":[7],"is":[8,93,118],"a":[9,50,76,80,94],"positive-definite":[10,60],"quadratic":[11],"form;":[12],"its":[13],"associated":[14],"matrix's":[15],"eigenvalues":[16,110],"govern":[17],"the":[18,22,29,59,107,119,122,132,154],"regulator,":[19],"which":[20],"enters":[21],"BSD":[23],"conjecture":[24],"and":[25],"bounds":[26],"rank":[27,54,99],"via":[28,131],"elliptic":[30],"curve's":[31],"arithmetic.":[32],"|":[33,68],"MATH:":[34],"\u0125(P)":[35,43],"=":[36,40,47,56,66,90,100,104,111,140],"lim_{n\u2192\u221e}":[37],"h([n]P)/n\u00b2;":[38],"\u27e8P,Q\u27e9":[39],"\u00bd(\u0125(P+Q)":[41],"\u2212":[42,44],"\u0125(Q));":[45],"Reg(E/Q)":[46],"det(\u27e8P\u1d62,P\u2c7c\u27e9)":[48],"for":[49],"basis":[51],"of":[52,58,106,153],"E(Q)/E(Q)_tors;":[53],"r":[55],"dim":[57],"lattice":[61,78,101,120],"(E(Q)/tors,":[62],"\u27e8\u00b7,\u00b7\u27e9);":[63],"BSD:":[64],"L^(r)(E,1)/r!":[65],"(\u03a9\u00b7Reg\u00b7\u220fc\u209a\u00b7#\u0428)/(#E(Q)_tors)\u00b2.":[67],"CONNECTION:":[69],"realizes":[73],"E(Q)/tors":[74],"as":[75],"Euclidean":[77],"\u2014":[79,143],"discrete":[81],"Z-module":[82],"embedded":[83],"in":[84],"R^r":[85],"with":[86,138],"Gram":[87],"matrix":[88],"G\u1d62\u2c7c":[89],"\u27e8P\u1d62,P\u2c7c\u27e9.":[91],"This":[92],"root-lattice":[95],"/":[96],"crystallographic":[97],"structure:":[98],"dimension,":[102],"regulator":[103,117],"covolume":[105],"fundamental":[108],"parallelepiped,":[109],"squared":[112],"principal":[113],"axis":[114],"lengths.":[115],"covolume;":[121],"\"geometric":[123],"harmony\"":[124],"constants":[125],"(0.618,":[126],"1.618)":[127],"appear":[128],"only":[129],"incidentally":[130],"golden-ratio":[133],"discriminant":[134],"families":[135],"(e.g.":[136],"curves":[137],"j":[139],"1728,":[141],"0)":[142],"no":[144],"Author:":[145],"Andrew":[146],"Stewart":[147],"Caldin,":[148],"Independent":[149],"Researcher,":[150],"UK.":[151],"Part":[152],"E8":[155],"Intelligence":[156],"Research":[157],"series.":[158],"Platform:":[159],"e8intelligence.com":[160]},"counts_by_year":[],"updated_date":"2026-10-29T09:56:32.521200","created_date":"2026-10-28T00:00:00"}