{"id":"https://openalex.org/W7225792312","doi":"https://doi.org/10.5281/zenodo.24145328","title":"A Restricted Jacobi Identity (Elementary Checked Boundary Case) for the Field-Level Tethered Bracket on the Divergence-Free Shear Class (Companion paper I)","display_name":"A Restricted Jacobi Identity (Elementary Checked Boundary Case) for the Field-Level Tethered Bracket on the Divergence-Free Shear Class (Companion paper I)","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225792312","doi":"https://doi.org/10.5281/zenodo.24145328"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24145328","doi":"https://doi.org/10.5281/zenodo.24145328","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24145328","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.24145328","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5147864777","display_name":"Benjamin Frohman","orcid":"https://orcid.org/0009-0006-7068-3718","observed_orcids":["https://orcid.org/0009-0006-7068-3718"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Frohman, Benjamin Stanley","raw_affiliation_strings":["Independent Researcher, Texas, United States"],"raw_orcid":"https://orcid.org/0009-0006-7068-3718","affiliations":[{"raw_affiliation_string":"Independent Researcher, Texas, United States","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5147864777"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.3037000000476837,"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"}},"topics":[{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.3037000000476837,"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"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.11169999837875366,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.10599999874830246,"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"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6899999976158142},{"id":"https://openalex.org/C133769200","wikidata":"https://www.wikidata.org/wiki/Q174565","display_name":"Bracket","level":2,"score":0.5906000137329102},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.47870001196861267},{"id":"https://openalex.org/C206159579","wikidata":"https://www.wikidata.org/wiki/Q1151539","display_name":"Jacobi identity","level":3,"score":0.460999995470047},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3982999920845032},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38989999890327454},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.3817000091075897},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.36730000376701355},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.35179999470710754},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.3237999975681305},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32359999418258667},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.323199987411499},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.32260000705718994},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.29280000925064087},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.2734000086784363},{"id":"https://openalex.org/C63316402","wikidata":"https://www.wikidata.org/wiki/Q862003","display_name":"Universal joint","level":2,"score":0.2621999979019165},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.2572000026702881},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C162269090","wikidata":"https://www.wikidata.org/wiki/Q1156047","display_name":"Rope","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24145328","doi":"https://doi.org/10.5281/zenodo.24145328","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24145328","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.24145329","doi":"https://doi.org/10.5281/zenodo.24145329","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24145329","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.24145328","doi":"https://doi.org/10.5281/zenodo.24145328","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24145328","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":1,"referenced_works":["https://openalex.org/W7225804084"],"related_works":[],"abstract_inverted_index":{"Companion":[0,151],"paper":[1,152,205,322],"I,":[2],"version":[3],"1.1":[4],"(finite-Fourier":[5],"model,":[6],"machine-checked).":[7],"In":[8],"a":[9,109,147,186,238,242,263,271,312,404],"finite-Fourier":[10,216],"model":[11,446],"of":[12,214,306,316],"vector":[13],"fields":[14],"on":[15,131,197,230,292,308,364],"T^3":[16],"we":[17],"consider":[18],"the":[19,31,61,77,94,104,113,125,129,155,169,204,219,259,285,327,349,375,384,389,407,429,435],"vector-valued":[20],"tethered":[21],"operation":[22,217,260],"{a,b}":[23],"=":[24,37,56,201,252],"[a,b]":[25],"+":[26],"T(a,b),":[27],"where":[28],"[\u00b7,\u00b7]":[29],"is":[30,136,140,280,319],"vector-field":[32],"Lie":[33,78],"bracket":[34,79,111,223,234],"and":[35,70,116,168,174,246,254,269,274,288,333,340,401,406],"T(a,b)":[36],"\u2212(\u03baw)\u27e8\u03a0_u":[38],"a,":[39],"\u03a0_u":[40,245],"b\u27e9u.":[41],"For":[42],"every":[43,97,182,420],"divergence-free":[44,69],"axis":[45,127,240],"u":[46,55],"along":[47,72],"e_x":[48],"with":[49,108,218,250],"\u27e8u,u\u27e9":[50,118],"\u2260":[51,119],"0":[52,120],"(in":[53],"particular":[54],"(sin":[57],"y,":[58,133],"0,":[59,134,202],"0))":[60],"Jacobiator":[62,130],"vanishes":[63],"whenever":[64],"all":[65],"three":[66],"arguments":[67],"are":[68,106,335],"point":[71],"e_x.":[73],"On":[74,371],"this":[75,215],"class":[76,114],"vanishes;":[80],"T":[81],"need":[82],"not":[83,146,441],"vanish,":[84],"but":[85],"it":[86],"takes":[87],"values":[88],"in":[89,93,172,227,235],"\u2102u,":[90],"which":[91,203],"lies":[92],"class,":[95],"so":[96],"nested":[98],"term":[99],"vanishes.":[100],"Lean":[101,162,183,228,300,328,342,431,437],"proves":[102],"that":[103,117,317,338,428,443],"hypotheses":[105],"inhabited":[107],"nonzero":[110],"inside":[112],"(non-vacuity)":[115],"cannot":[121,158],"be":[122,159],"dropped:":[123],"for":[124,383],"isotropic":[126,194],"e^{iy}e_x":[128],"(2cos":[132],"0)":[135],"6(\u03baw)^2":[137],"e^{iy}e_x.":[138],"This":[139],"an":[141,247],"elementary,":[142],"machine-checked":[143],"boundary":[144],"case,":[145],"new":[148],"Jacobi":[149],"theorem.":[150],"II":[153],"shows":[154,397,427],"e_x-direction":[156],"hypothesis":[157,184],"dropped.":[160],"What":[161],"checks.":[163],"shear_jacobi_holds_of_pr_ne_zero,":[164],"shearY_jacobi_holds,":[165],"paperI_nonvacuous,":[166],"shear_jacobi_fails_isotropic":[167],"supporting":[170],"lemmas":[171],"FTVerified/May/FieldJacobi.lean":[173],"FTVerified/Companion/JacobiCompanion.lean.":[175],"The":[176,193,232,321],"paper's":[177],"hypothesis-to-declaration":[178],"table":[179],"(README.md)":[180],"matches":[181],"to":[185,282,284],"stated":[187,430,436],"assumption;":[188],"no":[189,255,411],"mismatch":[190],"was":[191,378],"found.":[192],"guard":[195],"relies":[196],"Lean's":[198,275],"convention":[199],"x/0":[200],"states.":[206],"Open":[207],"gap":[208],"(not":[209],"addressed":[210],"here).":[211],"No":[212],"identification":[213],"state-dependent":[220,239,243],"geometric":[221,233,286,448],"functional":[222],"has":[224,237,262],"been":[225],"proved,":[226],"or":[229,294,415,449],"paper.":[231],"BenFrohman/frohmanian-tether-geometric":[236],"u,":[241],"projection":[244],"enstrophy":[248],"prefactor":[249],"\u03ba":[251,268],"3/2":[253],"free":[256,266],"parameter":[257],"w;":[258],"here":[261,290],"frozen":[264],"axis,":[265],"complex":[267],"w,":[270],"complex-bilinear":[272],"pairing":[273],"division":[276],"convention.":[277],"Neither":[278],"result":[279],"claimed":[281],"transport":[283],"setting,":[287],"nothing":[289],"bears":[291],"Navier\u2013Stokes":[293],"any":[295,447],"Clay":[296],"problem.":[297],"Reproducibility.":[298],"Full":[299],"source:":[301],"BenFrohman/frohmanian-tether-verified":[302],"at":[303,366],"a761aed99fba58ed4ac9bd5970577b2fc089824d":[304],"(tip":[305],"main":[307,365],"11":[309,372],"November":[310,373],"2026;":[311],"git":[313],"archive":[314],"zip":[315],"commit":[318,325,339,377],"attached).":[320],"cites":[323],"code":[324],"cf04afd;":[326],"sources,":[329],"scripts,":[330],"lakefile,":[331],"manifest":[332],"toolchain":[334],"byte-identical":[336],"between":[337],"a761aed.":[341],"leanprover/lean4:v4.30.0-rc1,":[343],"Mathlib":[344],"0c154d67103f74be3a0f2c509f72ccbf5be9f2a7.":[345],"One-command":[346],"rebuild":[347],"from":[348,381,434],"repository":[350],"root:":[351],"lake":[352,357],"exe":[353],"cache":[354],"get":[355],"&&":[356,359],"build":[358],"python3":[360],"scripts/check_jacobi_axioms.py.":[361],"GitHub":[362],"Actions":[363],"a761aed":[367],"passed":[368],"(run":[369],"42932429909).":[370],"2026":[374],"pinned":[376],"rebuilt":[379],"locally":[380],"scratch":[382],"project":[385],"modules":[386],"(attached":[387],"build.log);":[388],"attached":[390,408,421],"axiom":[391,416],"output":[392],"(axiom-output.txt:":[393],"PASS,":[394],"49":[395],"declarations)":[396],"only":[398],"propext,":[399],"Classical.choice":[400],"Quot.sound":[402],"(or":[403],"subset),":[405],"scan":[409],"found":[410],"sorry,":[412],"admit,":[413],"native_decide":[414],"declarations.":[417],"SHA256SUMS":[418],"lists":[419],"file.":[422],"Limits.":[423],"A":[424],"kernel":[425],"check":[426],"propositions":[432],"follow":[433],"definitions.":[438],"It":[439],"does":[440],"show":[442],"those":[444],"definitions":[445],"physical":[450],"object.":[451]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}
