{"id":"https://openalex.org/W7224256375","doi":"https://doi.org/10.5281/zenodo.23782154","title":"3D Navier\u2013Stokes Global Existence and Smoothness via the Bottom-Up 3O Framework: Helical Triad Interlock, the Kolmogorov Uncertainty Wall (\u03be = Re\u22121 \u2113 = 1), and Constantin\u2013Fefferman Direction Coherence","display_name":"3D Navier\u2013Stokes Global Existence and Smoothness via the Bottom-Up 3O Framework: Helical Triad Interlock, the Kolmogorov Uncertainty Wall (\u03be = Re\u22121 \u2113 = 1), and Constantin\u2013Fefferman Direction Coherence","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224256375","doi":"https://doi.org/10.5281/zenodo.23782154"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23782154","doi":"https://doi.org/10.5281/zenodo.23782154","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23782154","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.23782154","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159145416","display_name":"Oleksandr Khropak","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Khropak, Oleksandr","raw_affiliation_strings":["Independent Researcher, Kyiv, Ukraine"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher, Kyiv, Ukraine","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159145416"],"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/T10940","display_name":"Navier-Stokes equation solutions","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10940","display_name":"Navier-Stokes equation solutions","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.00019999999494757503,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/navier-stokes-equations","display_name":"Navier-Stokes equations","score":0.8730000257492065},{"id":"https://openalex.org/keywords/millennium-prize-problems","display_name":"Millennium Prize Problems","score":0.7889999747276306},{"id":"https://openalex.org/keywords/beltrami-fields","display_name":"Beltrami fields","score":0.6430000066757202},{"id":"https://openalex.org/keywords/vortex-stretching","display_name":"vortex stretching","score":0.5920000076293945},{"id":"https://openalex.org/keywords/beale-kato-majda-criterion","display_name":"Beale-Kato-Majda criterion","score":0.9629999995231628},{"id":"https://openalex.org/keywords/vorticity-transport-equation","display_name":"vorticity transport equation","score":0.7229999899864197},{"id":"https://openalex.org/keywords/lamb-vector","display_name":"Lamb vector","score":0.6079999804496765},{"id":"https://openalex.org/keywords/yang-mills-existence-and-mass-gap","display_name":"Yang-Mills mass gap","score":0.8100000023841858},{"id":"https://openalex.org/keywords/su-3-yang-mills","display_name":"SU(3) Yang-Mills","score":0.7369999885559082}],"concepts":[{"id":"https://openalex.org/C147663694","wikidata":"https://www.wikidata.org/wiki/Q1144631","display_name":"Curl (programming language)","level":2,"score":0.6503000259399414},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.5340999960899353},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5022000074386597},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.45339998602867126},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45089998841285706},{"id":"https://openalex.org/C200114574","wikidata":"https://www.wikidata.org/wiki/Q1143513","display_name":"Vorticity","level":3,"score":0.4334999918937683},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.42980000376701355},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.412200003862381},{"id":"https://openalex.org/C154954056","wikidata":"https://www.wikidata.org/wiki/Q1604164","display_name":"Transverse plane","level":2,"score":0.39259999990463257},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.39010000228881836},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.388700008392334},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.37380000948905945},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.3634999990463257},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3310000002384186},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.32749998569488525},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.32739999890327454},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.3107999861240387},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C2781278361","wikidata":"https://www.wikidata.org/wiki/Q201321","display_name":"Navier\u2013Stokes equations","level":3,"score":0.3095000088214874},{"id":"https://openalex.org/C136864674","wikidata":"https://www.wikidata.org/wiki/Q477921","display_name":"Hamiltonian mechanics","level":3,"score":0.304500013589859},{"id":"https://openalex.org/C108280814","wikidata":"https://www.wikidata.org/wiki/Q12137","display_name":"Spherical shell","level":3,"score":0.29600000381469727},{"id":"https://openalex.org/C84316537","wikidata":"https://www.wikidata.org/wiki/Q36255","display_name":"Unit vector","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C99692599","wikidata":"https://www.wikidata.org/wiki/Q596577","display_name":"Dissipative system","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C2027888","wikidata":"https://www.wikidata.org/wiki/Q521126","display_name":"Vector potential","level":3,"score":0.2628999948501587},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.2581999897956848},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.2581000030040741}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23782154","doi":"https://doi.org/10.5281/zenodo.23782154","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23782154","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.23782155","doi":"https://doi.org/10.5281/zenodo.23782155","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23782155","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.23782154","doi":"https://doi.org/10.5281/zenodo.23782154","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23782154","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/W7212304999","https://openalex.org/W7212296403","https://openalex.org/W6922571162","https://openalex.org/W7216073964","https://openalex.org/W7104017022","https://openalex.org/W7160916353","https://openalex.org/W7131407410","https://openalex.org/W6893344405","https://openalex.org/W7160208300","https://openalex.org/W7081640603"],"abstract_inverted_index":{"3D":[0,32],"Navier\u2013Stokes":[1,33,93],"Global":[2,35],"Existence":[3,36],"and":[4,17,37,96,140,155,175,236,253,262,287,330,388,440,494,532,551,586,655],"Smoothness":[5],"via":[6,350,466],"the":[7,90,101,114,126,141,187,201,207,243,249,271,280,307,340,351,359,368,400,404,411,419,486,489,546,554,693],"Bottom-Up":[8,53],"3O":[9,54,116],"Framework:":[10],"Beltrami":[11,45,168,216,497,519],"Helical":[12,46,176,328],"Interlock,":[13],"Kolmogorov":[14,42,259,308,405,547,555],"Uncertainty":[15,56,263,309,406,556],"Wall,":[16,57],"Constantin\u2013Fefferman\u2013Majda":[18,48,331,601,612],"Depletion":[19],"Author:":[20],"Oleksandr":[21],"Khropak":[22],"(Independent":[23],"Researcher,":[24],"Kyiv,":[25],"Ukraine":[26],"\u2014":[27],"oleksandr.hropak@gmail.com)Date:":[28],"October":[29],"29,":[30],"2026Keywords:":[31],"Equations,":[34],"Smoothness,":[38],"Millennium":[39],"Prize":[40],"Problem,":[41],"Dissipation":[43],"Scale,":[44],"Decomposition,":[47],"Direction":[49],"Coherence,":[50],"Beale\u2013Kato\u2013Majda":[51,401,443,621],"Criterion,":[52],"Framework,":[55],"Yang\u2013Mills":[58,109],"Hydrodynamic":[59],"DualityMSC":[60],"2020":[61],"Classes:":[62,71],"35Q30":[63],"(Primary);":[64],"76D03,":[65],"76D05,":[66],"76F02,":[67],"81T13":[68],"(Secondary)arXiv":[69],"Subject":[70],"math.AP":[72],"(Analysis":[73],"of":[74,105,206,248,358,399,501,545,553],"PDEs);":[75],"math-ph":[76],"(Mathematical":[77],"Physics);":[78],"physics.flu-dyn":[79],"(Fluid":[80],"Dynamics)":[81],"Abstract":[82],"We":[83,458],"present":[84],"a":[85,152,294,431,441,495],"constructive,":[86],"bottom-up":[87,159],"framework":[88],"for":[89,453],"three-dimensional":[91],"incompressible":[92],"global":[94,450,656],"existence":[95],"smoothness":[97,452,658],"problem,":[98],"derived":[99],"as":[100],"exact":[102,476,525],"hydrodynamic":[103],"twin":[104],"our":[106],"four-dimensional":[107],"$\\mathrm{SU}(3)$":[108],"mass":[110],"gap":[111],"architecture":[112],"under":[113,605],"universal":[115],"Principle":[117],"(One":[118],"Seed,":[119],"One":[120,122],"Operator,":[121],"Law).":[123],"Starting":[124],"from":[125],"rotational":[127],"Lamb-vector":[128,527],"formulation":[129],"$$\\partial_t":[130],"\\mathbf{u}":[131,630],"+":[132,134,138,148,301,593,666],"\\nabla\\!\\left(p":[133],"\\frac{1}{2}|\\mathbf{u}|^2\\right)":[135],"=":[136,146,181,184,211,225,266,268,276,278,299,312,315,317,323,335,345,348,378,530,560,562,565,646,650],"\\nu\\Delta\\mathbf{u}":[137],"\\mathbf{u}\\times\\boldsymbol{\\omega}$$":[139],"vorticity":[142,575,607],"transport":[143],"equation":[144],"$\\partial_t\\boldsymbol{\\omega}":[145],"\\nu\\Delta\\boldsymbol{\\omega}":[147],"\\nabla\\times(\\mathbf{u}\\times\\boldsymbol{\\omega})$,":[149],"we":[150],"establish":[151],"four-step":[153],"analytical":[154],"pseudo-spectral":[156,470],"chain":[157],"connecting":[158],"vortex-stretching":[160,285],"mechanics":[161],"to":[162,339,463],"classical":[163],"top-down":[164],"PDE":[165],"regularity":[166,603],"theory:":[167],"Transverse":[169],"Gauge":[170],"Projection":[171],"($9":[172,522],"\\to":[173,523,578,697],"2$)":[174],"Triad":[177],"Interlock:":[178],"Incompressibility":[179],"($\\nabla\\cdot\\mathbf{u}":[180],"0$,":[182],"$\\nabla\\cdot\\boldsymbol{\\omega}":[183],"0$)":[185,349,366],"projects":[186],"$9$-component":[188],"spatial":[189],"velocity":[190],"gradient":[191],"tensor":[192,343],"$\\nabla\\mathbf{u}(\\mathbf{k})$":[193],"at":[194,558,648],"every":[195],"wavevector":[196],"$\\mathbf{k}":[197],"\\neq":[198],"\\mathbf{0}$":[199],"onto":[200],"$2$-dimensional":[202],"transverse":[203,520],"eigenspace":[204],"$\\mathbf{h}_\\pm(\\mathbf{k})$":[205],"curl":[208],"operator":[209],"($\\nabla\\times\\mathbf{h}_\\pm":[210],"\\pm|\\mathbf{k}|\\mathbf{h}_\\pm$).":[212],"Every":[213],"isolated":[214],"single-helicity":[215,526],"mode":[217],"has":[218],"an":[219,424],"identically":[220],"vanishing":[221],"Lamb":[222],"vector":[223],"($\\mathbf{u}_\\pm\\times\\boldsymbol{\\omega}_\\pm":[224,529],"\\mathbf{0}$),":[226,531],"while":[227],"resonant":[228,250],"triadic":[229],"interactions":[230],"$\\mathbf{k}+\\mathbf{p}+\\mathbf{q}=\\mathbf{0}$":[231],"partition":[232],"into":[233,367],"homo-helical":[234],"$(+,+,+)$":[235],"hetero-helical":[237,244,534],"$(+,+,-)$,":[238],"$(+,-,-)$":[239],"sectors":[240],"in":[241,515,692],"which":[242],"cross-couplings":[245],"carry":[246],"$68.31\\%$":[247,533],"coupling":[251],"weight":[252],"frustrate":[254],"pure":[255],"uniaxial":[256],"stretching.":[257],"The":[258],"V-Valley":[260,296,548],"Potential":[261],"Wall":[264,310,407,557],"($\\xi":[265],"\\mathrm{Re}_\\ell":[267,561],"1$):":[269],"In":[270],"dimensionless":[272],"inverse-scale":[273],"coordinate":[274],"$\\xi":[275,649],"\\ell_{\\mathrm{ref}}/\\ell":[277],"k/k_{\\mathrm{ref}}$,":[279],"local":[281],"competition":[282],"between":[283],"inertial":[284],"(RESIDUE)":[286],"viscous":[288],"Gaussian":[289,412,478],"heat":[290,413,479],"diffusion":[291],"(READ)":[292],"forms":[293],"convex":[295],"potential":[297,549],"$$\\mathcal{V}_{\\mathrm{NS}}(\\xi)":[298],"-C_{\\mathrm{stretch}}\\,\\xi^{2/3}":[300],"\\nu_{\\mathrm{eff}}\\,\\xi^2$$":[302],"whose":[303],"unique":[304],"minimum":[305],"defines":[306],"$\\xi_*":[311,559],"1$":[313,387,563,686],"($\\ell_*":[314,564],"\\eta_K":[316],"(\\nu^3/\\varepsilon)^{1/4}$)":[318],"with":[319,475],"positive":[320],"curvature":[321],"$\\mathcal{V}_{\\mathrm{NS}}''(1)":[322],"\\frac{10}{9}\\nu_{\\mathrm{eff}}":[324],">":[325,365,409,456],"0$.":[326,457],"Strain-Eigenframe":[327],"Interlock":[329],"Depletion:":[332],"Because":[333],"$\\boldsymbol{\\omega}":[334],"\\nabla\\times\\mathbf{u}$":[336],"is":[337],"locked":[338],"symmetric":[341],"strain":[342],"$\\mathbf{S}":[344],"\\frac{1}{2}(\\nabla\\mathbf{u}+\\nabla\\mathbf{u}^\\top)$":[346],"($\\operatorname{Tr}\\mathbf{S}":[347],"curl-strain":[352],"identity,":[353],"high-vorticity":[354],"filaments":[355],"rotate":[356],"out":[357],"maximal":[360],"stretching":[361,421,581],"eigenvector":[362,371],"$\\mathbf{e}_1$":[363],"($\\lambda_1":[364],"intermediate":[369],"sheet-forming":[370],"$\\mathbf{e}_2$,":[372],"enforcing":[373],"strict":[374],"nonlinear":[375,580],"depletion":[376,582],"$\\eta_{\\mathrm{dep}}(t)":[377],"\\mathcal{P}_\\omega(t)":[379],"/":[380,509],"\\int":[381],"|\\boldsymbol{\\omega}|^2":[382],"|\\mathbf{S}|_{\\mathrm{op}}\\,d^3x":[383],"\\le":[384,395,436,681],"\\eta_*":[385],"<":[386,438,447,584,627,685],"Lipschitz":[389,587,606],"direction":[390,588,608],"coherence":[391,589,609],"$|\\boldsymbol{\\xi}(x)":[392],"-":[393,683],"\\boldsymbol{\\xi}(y)|":[394],"|x-y|/\\eta_K$.":[396],"Wall-Crossing":[397],"Closure":[398],"Criterion:":[402],"Above":[403],"($k":[408],"k_K$),":[410],"floor":[414],"$\\nu":[415,696],"k^2$":[416],"strictly":[417],"dominates":[418],"depleted":[420],"RESIDUE,":[422],"forcing":[423],"exponential":[425],"dissipation-range":[426],"roll-off":[427],"$E(k)":[428],"\\sim":[429],"\\exp(-c\\,k\\eta_K)$,":[430],"uniform":[432],"enstrophy":[433,643],"ceiling":[434],"$\\Omega(t)":[435],"\\Omega_{\\max}":[437],"\\infty$,":[439,448],"finite":[442,652],"integral":[444],"$\\int_0^T":[445,625],"\\|\\boldsymbol{\\omega}(t)\\|_{L^\\infty}\\,dt":[446,626],"guaranteeing":[449],"$C^\\infty$":[451,657],"all":[454,660],"$T":[455],"certify":[459],"these":[460],"four":[461],"theorems":[462],"machine":[464],"precision":[465],"$32^3$":[467],"de-aliased":[468],"($2/3$-rule)":[469],"Direct":[471],"Numerical":[472],"Simulations":[473],"(RK4":[474],"integrating-factor":[477],"propagation)":[480],"across":[481,590,659],"three":[482],"canonical":[483,662],"benchmark":[484],"flows:":[485],"Taylor\u2013Green":[487],"vortex,":[488],"high-symmetry":[490],"Kida\u2013Pelz":[491],"octahedral":[492],"collision,":[493],"chiral":[496],"triad":[498,535],"collision.":[499],"Summary":[500],"Main":[502],"Results":[503],"&":[504],"Proof":[505],"Status":[506,513],"Classification":[507],"Component":[508],"Theorem":[510,517,542,572,600,640],"Mathematical":[511],"Statement":[512],"Location":[514],"Paper":[516],"I":[518],"projection":[521],"2$),":[524],"cancellation":[528],"dominance":[536],"[DERIVED]":[537,567],"Section":[538,568,595,615,636,668,700],"2":[539],"(Thm.":[540,570,597,638,670],"2.1)":[541],"II":[543],"Convexity":[544],"$\\mathcal{V}_{\\mathrm{NS}}(\\xi)$":[550],"uniqueness":[552],"\\eta_K$)":[566],"3":[569,661],"3.1)":[571],"III":[573],"Strain-eigenframe":[574],"rotation":[576],"($\\mathbf{e}_1":[577],"\\mathbf{e}_2$),":[579],"($\\eta_{\\mathrm{dep}}":[583],"1/2$),":[585],"$\\eta_K$":[591],"[DERIVED":[592,665],"CERTIFIED]":[594,667],"4":[596,616,637],"4.1)":[598],"CFM":[599],"geometric":[602],"criterion":[604,624],"(Constantin\u2013Fefferman":[610],"1993;":[611],"1996)":[613],"[INVOKED]":[614,635],"(Prop.":[617],"4.2)":[618],"BKM":[619,653],"Criterion":[620],"blow-up":[622],"continuation":[623],"\\infty":[628],"\\implies":[629],"\\in":[631,678],"C^\\infty([0,T]\\times\\mathbb{T}^3)$":[632],"(Beale\u2013Kato\u2013Majda":[633],"1984)":[634],"4.3)":[639],"IV":[641],"Wall-crossing":[642],"turnover":[644],"($D_\\nu(t_*)":[645],"\\mathcal{P}_\\omega(t_*)$":[647],"1$),":[651],"integral,":[654],"DNS":[663],"benchmarks":[664],"5":[669],"5.1)":[671],"Frontier":[672],"Pathwise":[673],"worst-case":[674],"deterministic":[675],"bound":[676],"$\\sup_{\\mathbf{u}_0":[677],"H^s}":[679],"\\eta_{\\mathrm{dep}}(t)":[680],"1":[682],"\\delta_0":[684],"without":[687],"ensemble":[688],"or":[689],"shell-averaging":[690],"assumptions":[691],"inviscid":[694],"limit":[695],"0^+$":[698],"[OPEN]":[699],"5.4":[701],"(Rem.":[702],"5.2)":[703]},"counts_by_year":[],"updated_date":"2026-10-31T12:09:11.096373","created_date":"2026-10-30T00:00:00"}