The Circulation Campaign: Why Circulation-Based Estimates Cannot Close the 3D Navier–Stokes Existence and Smoothness Problem
Year: 2026
Type: preprint
Abstract: READING ORDERThe papers record an argument as it developed and are meant to be read in sequence:eight_conditions (the conditions any missing estimate must meet) -> transport_IV_events (circulation as the currency ofcollapse) -> what_must_be_conserved (circulation at every scale is the largest critical space) -> circulation_does_not_close(that currency cannot close) -> the slice budget (why: stretching is invisible to exactly thecritical functionals). Statements in the earlier papers that rely on circulation-based controlare superseded by the no-go. CONTENTS what_must_be_conserved What Must Be Conserved: Thirteen Conditions on the Missing Quantity for the Three-Dimensional Navier-Stokes Equations. The diagnosis: circulation at every scale is equivalent to the norm of the largest critical space, B^{-1}_{infinity,infinity}. circulation_does_not_close Circulation at Every Scale Does Not Close: A No-Go Statement in the Largest Critical Space for the Three-Dimensional Navier-Stokes Equations. The no-go. Part (a) is unconditional; part (b), for finite-energy data, is conditional on the Bourgain-Pavlovic norm-inflation claim for such data, and is labelled so. slice_budget Why Circulation Cannot See Stretching: an Exact Slice Budget for the Three-Dimensional Navier-Stokes Equations (+ supplement.pdf: verbatim registration records). The explanation: on a plane section, vortex stretching enters any functional int F(omega_z) only through the gradient half of the in-plane velocity, with weight wF'(w) - F(w), which vanishes exactly for the scale-critical (circulation-type) functionals. Also: a circulation-free form of the supercritical budget, with an exact growth ratio and a Fisher-information Burgers number, validated in DNS, and preregistered measurements, including one failed replication, reported as such. eight_conditions - Eight Conditions on the Missing Estimate for the Three-Dimensional Navier-Stokes Equations (synthesis note, 9 pp). transport_IV_events - Transport and the Supercriticality Gap in 3D Navier-Stokes, IV: Events (17 pp). Uses circulation as the currency; paper 2 explains why that currency cannot close. These cite further papers of the series (IV-VI, X-XII) that are not included here. code/ (for paper 3 only) dns_lean.py pseudo-spectral Navier-Stokes solver on T^3 (PyTorch, CUDA): low-storage third-order Runge-Kutta with integrating factor, 2/3 dealiasing. p145_burgers.py the slice instrument: escort weights s^q / int s^q on every z-section, effective area, Fisher information, <sigma>, self part, leak term. p145_validate.py validation: analytic Gaussian pair in imposed strain (V1); budget closure against the solver's exact time derivative (V2). p145_verdict.py registered criteria K1-K3 (P145 and the replication P145r). p146_stats.py, p146_verdict.py registered flattening statistics (P146). run_all.sh reproduces every measurement in paper 3 (about 4 x 30-60 min on an 8 GB GPU). results/ p145_verdict.json, p146_verdict.json, p145r_replication_verdict.json: the verdict files as produced for the papers. The raw per-output data (about 30 MB per run) are regenerated by run_all.sh and are not included. USE OF AIThis work was carried out with substantial help from a generative AI system (Claude, Anthropic).The author set the direction and the hypotheses; the AI searched the literature, carried outderivations and computations, wrote and ran the simulation code, and drafted the text. The AI isnot an author, and responsibility for all content rests with the author. LICENCEAs stated on the Zenodo record.
Author: Salkimbayev, Adilkhan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223230154