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Computer-assisted global regularity across nonlinear families of three-dimensional periodic Navier-Stokes flows

Jose Luis Lima de Jesus Silva

arXiv:2609.16157v1physics.flu-dyncs.LG

TL;DR

Energy decay alone cannot establish smooth evolution for three-dimensional Navier–Stokes flows. This paper uses validated finite-path comparisons and analytic bounds to prove global regularity across specified flow families, including a connected family of non-Beltrami centres. Its conclusions apply to explicit infinite-dimensional perturbation neighbourhoods and are independent of the Galerkin cutoff.

  • Problem

    Energy can decay while finer structures develop, and global finite-energy weak solutions do not establish global smoothness or uniqueness for three-dimensional Navier–Stokes flows.

    Source: §1 · p. 1

  • Method

    Finite signed reference paths, complete nonlinear residual estimates, and parameter-uniform comparisons establish regularity across centre families and smooth infinite-dimensional perturbations.

    Source: §1 · p. 3

  • Results

    The theorem establishes regularity on three explicit closed balls and a connected positive-dimensional family, with uniform estimates permitting removal of the Galerkin cutoff.

    Source: §6 · p. 35

  • Takeaways & Limitations

    The results provide a reusable family-level method for proving regularity across the stated periodic flow families.

    Source: §1 · p. 3

    An adaptive coverage theorem for changing supports remains a future step toward a general computational theory of regularity neighbourhoods.

    Source: §5 · p. 35

Abstract

from arXiv · show

Numerical simulations reveal how vortices stretch and transfer energy, but establishing smooth evolution requires bounds that remain valid beyond the simulated resolution. Here I develop a computer-assisted framework that establishes global regularity for continuous families of three-dimensional periodic Navier-Stokes flows. Its central construction combines finite reference trajectories with a common error bound that covers an interval of centre fields and infinitely many smooth perturbation modes. The method retains the complete nonlinear residual before spectral truncation and controls the evolution until viscous decay guarantees regularity for all subsequent times. Applications to cyclic-shear, Arnold-Beltrami-Childress and three-component Taylor-Green fields yield explicit perturbation radii and include initial conditions outside the direct Fourier-Wiener smallness criterion. A parameter-uniform extension covers a connected family of non-Beltrami Taylor-Green centres without repeating the proof for individual parameter values. An ensemble of 4,096 configurations, supplemented by 1,600 refinement trajectories and public turbulence data, connects the mathematical observables to spectral transfer and vortex geometry. Matched neural-operator experiments show that physics-informed training improves physical prediction, while also revealing that these gains do not necessarily improve the discovery of proof-limiting initial conditions. Together, these results provide a reusable method for establishing regularity across prescribed flow families and a quantitative setting for evaluating how learned predictions can assist rigorous computation.

1 Introduction

Three-dimensional Navier–Stokes flows can lose energy while developing finer structures, so energy decay alone does not establish smooth evolution. This work combines validated finite-path comparisons with analytic estimates to prove regularity across specified infinite-dimensional flow families.

Motivation

  • Three-dimensional Navier–Stokes theory has global finite-energy weak solutions, but weak existence and partial regularity do not establish global smoothness or uniqueness.For a specified initial family, an approximate trajectory must provide more than an energy bound to establish global smoothness.Source: §1 · p. 1

Prior regularity theory

  • Conditional criteria identify quantities whose control ensures continuation, but they do not show that every large smooth datum satisfies those criteria.Structural results include regularity criteria based on closeness to a Laplacian eigenfunction, which motivates uniform residual control here.Source: §1 · p. 2

Physical motivation

  • Vortex stretching, alignment, intermittency, and transfer toward smaller scales motivate the paper’s diagnostics, but physical observations do not replace an analytic inequality.Public DNS databases expose velocity and derivative fields for independent tests of strain alignment, intermittency, and signed production.Source: §1 · p. 2

Computer-assisted regularity

  • Prior computer-assisted methods controlled approximate Navier–Stokes solutions, while this work adds parameter-uniform comparison records for specified non-Beltrami families and perturbation balls.Earlier structured-data results used Reynolds expansions, symbolic Fourier calculations, and Sobolev a posteriori control.Source: §1 · p. 2

Neural-operator search

  • Neural operators search candidate initial fields by learning field-to-evolution maps; physics-informed training adds equation-based losses, and selected fields are recomputed with a spectral Galerkin solver.The acquisition objective targets a residual-action proxy and high-frequency tail used in numerical regularity diagnostics; the neural model is a search mechanism, not a mathematical premise.Source: §1 · p. 3

Contributions

  • The framework proves global regularity on explicit infinite-dimensional neighbourhoods and a connected coefficient family using finite signed paths, complete nonlinear residuals, and cutoff-independent estimates.Exact rational arithmetic validates finite comparison inequalities; terminal Wiener continuation establishes smoothness for subsequent times.Source: §1 · pp. 3–4

2 Mathematical methodology

The methodology establishes global smooth periodic solutions in specified neighborhoods by comparing Galerkin solutions with finite Fourier reference paths, retaining the complete nonlinear residual before spectral truncation, and carrying finite-time bounds through terminal Wiener control and continuation estimates. It also extends these bounds from individual centres to a uniform interval of non-Beltrami centre shapes, with viscosity and torus-size rescalings.

2.1 Setting and theorem

  • The cyclic-shear theorem establishes global smooth solutions in an infinite-dimensional neighborhood of a genuinely three-dimensional centre.The center uses all three spatial directions and has nontrivial quadratic evolution; the theorem covers smooth perturbations with infinitely many Fourier modes.Source: §2.1 · pp. 5–6

2.2 Centre-independent finite-path acceptance

  • A centre-independent acceptance principle turns finite-path residual and error bounds into global smooth solutions, while action estimates require additional instance-specific bounds.The principle applies uniformly over a parameter set when its hypotheses hold; an action estimate does not follow from path algebra alone.Source: §2.2 · pp. 6–7

2.3 The signed degree-three comparison path

  • The degree-three signed reference path retains every nonlinear residual output before norm estimation, and monotone majorants at a = 7/10 cover the full amplitude interval.The monotonicity is a property of the positive majorants, not an assumption that exact solutions are monotone in amplitude.Source: §2.3 · p. 8

2.4 Uniform a posteriori comparison

  • A cutoff-independent scalar error comparison is verified on 854 slabs using exact fraction arithmetic for every N ≥ 2 and every datum in the stated class.The piecewise-affine comparison bounds the Wiener error through the terminal time.Source: §2.4 · pp. 9–10

2.4.1 Exact slab construction and acceptance rule

  • Slab acceptance requires nonnegative slack, matching adjacent endpoints, the prescribed initial error, and terminal gates; independent exponential enclosures precede exact rational arithmetic.The primary and independent implementations use different exponential bounds, and the signed heat-kernel numerators are retained when forming enclosing intervals.Source: §2.4.1 · p. 10

2.5 Exceptional cutoff and terminal continuation

  • The proof handles N = 1 by energy decay and combines higher-cutoff bridge and terminal-tail estimates to obtain an explicit action bound of 418.The exceptional cutoff is treated directly because projection of the approximant would not commute with its quadratic residual; subsequent steps pass to the PDE and restore viscosity and domain size.Source: §2.5 · pp. 10–11

2.5.1 Passage from Galerkin trajectories to the PDE

  • Cutoff-independent bounds yield a strong periodic PDE solution via Aubin–Lions compactness; the terminal Wiener barrier propagates, giving global smoothness and uniqueness.Lower semicontinuity transfers the H1 and integrated H2 bounds, while the all-order Fourier estimate propagates spatial derivatives.Source: §2.5.1 · p. 12

2.5.2 Scaling in viscosity and torus size

  • The unit-viscosity results rescale to viscosity ν with initial data u0=ν(aU+h); on side-2πL tori, velocity scales as ν/L and action as ν^(1/3)/L², not to arbitrary data.These are rescalings of the proved dimensionless family.Source: §2.5.2 · p. 12

2.5 Exceptional cutoff and terminal continuation

  • For N=1, (51) handles the exceptional cutoff; for N≥2, 854 exact slabs bound pre-terminal action by 418 and the remaining action by 418/d.Together, these estimates prove the action bound for every cutoff and support terminal Wiener continuation.Source: §2.5 · p. 11, §2.5.2 · p. 12

2.6 Second centre: the equal-coefficient ABC family

  • The equal-coefficient ABC centre yields unique global smooth solutions for 0≤a≤17ν/25 under the theorem’s perturbation condition, with a zero projected centre residual.The corresponding terminal time at viscosity ν is TABC/ν.Source: §2.6 · p. 13

2.7 Third centre: the nonlinear TG3 family

  • For the TG3 centre, 0≤a≤41ν/40 and ∥h∥A2≤ν/40 yield a unique global smooth solution for every cutoff, with terminal time 873/(1024ν).Its heat path retains a nonzero quadratic residual, which is not discarded at N=1.Source: §2.7 · pp. 13–14

2.8 A uniform nonlinear coefficient family

  • For |θ|≤1/10 and 0≤a≤ν, one comparison proves global smoothness uniformly over the coefficient family, perturbations, and all cutoffs, with common terminal time 31/(32ν).Bernstein interval bounds control the complete quadratic residual before projection, and 992 common exact slabs establish the bridge bound.Source: §2.8 · pp. 14–15

3 Computational methodology

The methodology independently checks exact comparison conditions and tests a matched FNO–PINO search for proof-limiting initial conditions. The search uses PINO-proxy ranking for every proposal pool before solver rechecks.

3.1 Independent exact-arithmetic verification

  • Independent evaluation verifies 854 slab inequalities plus 13 scope and eight integration checks; four deliberately altered inputs fail their corresponding conditions.The altered inputs test residual domination, parameter coverage, terminal smallness, and cutoff compatibility.Source: §3.1 · p. 16
  • Theorem 2.5, Theorem 2.6, and Theorem 2.7 checks verify 2390, 873, and 992 inequalities, respectively, with altered inputs rejected.The checks cover ABC, TG3, and a full parameter interval, including residuals, terminal and bridge bounds, and cutoff conditions.Source: §3.1 · p. 16

3.2 Physics-informed neural-operator search

  • The neural experiment matches supervised FNO and PINO architectures, training on 127 fields and reserving 45 validation and 212 out-of-distribution fields.Both models predict three velocity components at five times; PINO adds residual, incompressibility, energy-balance, and spectral penalties to the field and initial-condition losses.Source: §3.2 · pp. 16–17
  • Each acquisition strategy receives 512 candidate evaluations and six solver rechecks per family, while random and Sobol pools are PINO-proxy-ranked before evaluation.A separate dense quadrature records the action integrand at every RK4 step, and the text distinguishes neural predictions, floating-point checks, and the exact theorem as separate evidence levels.Source: §3.2 · p. 17

4 Results and discussion

The results quantify verified regularity neighbourhoods, including a connected family of centre fields, and relate these analytic results to numerical ensemble behaviour and neural-operator predictions. They also characterize selected trajectories and public DNS through vortex geometry, eigenfield-transient scaling, a three-resolution stress test, and tensor diagnostics, with the stress-test theorem’s cutoff-uniform conclusion based on exact comparison bounds.

4.1 Position relative to standard small-data norms

  • The verified families give computable perturbation radii and terminal times, including a connected interval of centre shapes with a uniform radius.The contribution covers three independently verified instances, including two nonzero-residual centres, and extends beyond the displayed unit Wiener threshold for the cyclic-shear centre at a = 0.7, where ∥aU∥A0 = 2.1.Source: §4.1 · pp. 18–19

4.2.1 Exact geometry of the reference centre

  • For the reference centre, strain magnitude equals vorticity magnitude divided by √2, while alternating signed stretching and nonzero quadratic acceleration require control of the full signed Fourier residual.The quadratic acceleration is divergence free, so its nonzero magnitude implies a nonzero projected nonlinearity; the initial field lies on one Laplacian shell with R(0) = 0.Source: §4.2.1 · p. 19

4.2.2 Matched geometry of the three proved centres

  • Matched initial-field renderings distinguish curved cyclic-shear transport, equal-ABC vorticity coincident with velocity, and more symmetric TG3 cells, with two-sign stretching across the centres.The equal-ABC centre has zero projected nonlinearity without pointwise zero strain; TG3 is distinguished from the Beltrami ABC case by nonzero Fourier output.Source: §4.2.2 · p. 21, Figure 3 · p. 21

4.2.3 Paired-resolution configuration-space ensemble

  • Across 4,096 ensemble samples, log10 J0.5 correlated 0.916 with a and 0.538 with ρ; all 2,105 theorem-box samples and 1,257 of 1,991 exterior samples passed the numerical gates.The correlations are descriptive, not causal; loss of trust near a ≃ 1 came from the outer-layer Z gate, not numerical blowup or an analytic-theorem violation. The theorem-box median action was 1.89 × 10−3 versus 2.22 × 10−2 outside.Source: §4.2.3 · pp. 22–23
  • In the ensemble, viscous enstrophy decayed in both cohorts, while the exterior residual distribution was larger and its 95th-percentile outer-layer fraction crossed the 10−8 gate near the window’s end.The theorem-box 95th percentile stayed below 10−9, identifying the exterior crossing as the resolution boundary that motivated higher-cutoff refinement.Source: §4.2.3 · p. 24, Figure 5 · p. 24

4.2.4 High-resolution spectral-boundary refinement

  • At cutoff seven, all 124 original tail-gate failures became passes, and median moving outer-shell fractions fell from 9.29 × 10−14 at cutoff five to 6.09 × 10−22 at cutoff seven.The shell moves with the cutoff, so the trend demonstrates finite-family spectral localization rather than a cutoff-independent bound; all 64 half-step checks passed, and no cutoff-seven point approached the 10−8 gate.Source: §4.2.4 · pp. 25–26

4.2.5 Physics-informed operator search across initial-data families

  • PINO lowered all seven recorded OOD medians relative to matched FNO, including field error by 25.5%, projected Navier–Stokes residual by 68.2%, and energy-balance error by 88.3%.These fitted-model reductions are not universal across training seeds. PINO reached action-proxy Spearman correlation 0.973 versus FNO’s 0.964 and recall 0.922 versus 0.844 at 30% selection, but the composite action–tail top-decile endpoint was 0.864 for PINO versus 0.955 for FNO; none of 48 selected fields enlarged an exactly proved neighbourhood.Source: Figure 8 · p. 27, §4.2.5 · pp. 27–28

4.2.6 Objectively selected dynamic vortex geometry

  • All three selected vortex cases decrease in maximum vorticity; their initial interlocking surfaces evolve into lobes, with the rejected cutoff-four case showing no observed singular growth.The rejected case has the largest action and tail, but rejection concerns original numerical resolution; surfaces show both stretching signs, not positive stretching throughout the volume.Source: §4.2.6 · p. 28, Figure 9 · p. 29
  • For case 1088, median speed falls from 0.857 at the centre to 0.517, while median vorticity magnitude falls from 0.857 to 0.503 and median strain alignment rises from 0.608 to 0.658.At t = 0.5, median normalized stretching shifts from zero to 0.0517; the selected finite-resolution trajectory does not establish monotone alignment.Source: §4.2.6 · p. 29, Figure 10 · p. 30

4.2.7 Genuine three-dimensional eigenfield transients

  • For n = 4, 8, 16, 32, R/(ε²e−4τ) collapses toward 3τ²/4, while signed source grows as n² and critical action decreases as n−2/3.The four finite values illustrate the asymptotic scaling across a varying family, not high-frequency evolution of one fixed datum or the analytic contraction threshold.Source: §4.2.7 · p. 31

4.2.8 Three-resolution stress test and VTI time slices

  • 4.3 × 10−9 temporal errors and a 48/64 common-mode discrepancy of 9.26 × 10−7 support finite-window implementation consistency, not the theorem’s cutoff-uniform conclusion.The largest balance residual was 3.54 × 10−6, divergence norm 4.40 × 10−17, and nonlinear energy defect 1.45 × 10−15.Source: §4.2.8 · pp. 31–32
  • 1.13 × 10−5, 2.21 × 10−9, and 3.69 × 10−13 are the maximum last-two-layer enstrophy fractions at cutoffs 7, 11, and 15; the coarse tail gate fails, leaving 10 of 11 gates passed.The datum is z-invariant 2D3C, so its coherent diagonal slices are not generic three-dimensional turbulence.Source: §4.2.8 · p. 31, Figure 11 · p. 32

4.2.9 Public-DNS tensor geometry across flow classes

  • The public-DNS isotropic sample has median signed stretching 0.1965, with first and 99th percentiles −0.6256 and 0.6559; 75.79% of samples have positive production, despite mixed signs.Pooled mean absolute alignment with the intermediate strain eigenvector is 0.6613, and helicity–production correlation is −0.1178.Source: Figure 12 · p. 33, §4.2.9 · p. 33
  • Across public-DNS flows, mean signed stretching and positive fraction are 0.15665 and 0.7571 for isotropic-1024³, versus 0.04809 and 0.6221 for MHD-1024³.Isotropic-4096³ and rotating–stratified-4096³ values are 0.15661 and 0.7627, and 0.05811 and 0.6287, respectively; these are descriptive comparisons from coarse query lattices, not resolution-convergence results or theorem premises.Source: §4.2.9 · pp. 32, 34, Figure 13 · p. 34

5 Scope and outlook

The scope combines a family-level regularity theorem with configuration-space and neural-operator studies, while identifying adaptive coverage and independent validation as next steps.

5 Scope and outlook

  • The central advance is a family-level regularity theorem with explicit radii, terminal times, and bounds uniform in the Galerkin cutoff.The centre-independent acceptance theorem lets new centre-specific estimates reuse the continuum argument when they meet its acceptance conditions.Source: §5 · p. 34
  • An algorithmic coverage theorem for compact classes of finite-Fourier centres remains a next objective, replacing centre-by-centre construction with validated parameter cells.The proposed decomposition would quantify verified radius as a function of amplitude, viscosity, spectral support, and path degree.Source: §5 · p. 35
  • The 4,096-point ensemble, 512-case and 1,600-trajectory refinement, 3D diagnostics, and public-DNS samples complement the theorem; physics-informed training lowers seven OOD penalty and diagnostic medians for one fitted seed.The study assigns exact inequalities, Galerkin calculations, neural operators, and public DNS distinct evidentiary roles.Source: §5 · p. 35
  • Independent interval-arithmetic or proof-assistant validation and a wider parameter campaign would broaden the work, including tests across viscosities and changing Fourier supports.Both extensions preserve the separation between mathematical validation and physical diagnostics.Source: §5 · p. 35

6 Conclusion

Finite-reference-path comparison establishes regularity across specified families of periodic flows, with uniform bounds and explicit perturbation control. The computational programme and coefficient-family result provide reusable validation within that scope, while adaptive coverage and independent formal validation remain future directions.

6 Conclusion

  • Finite-reference-path comparison establishes regularity on three explicit closed balls and a connected positive-dimensional family around genuinely three-dimensional periodic centres.The theorem covers continuous amplitudes, centre coefficients, and smooth infinite-dimensional perturbations, with weighted Fourier–Wiener control and Galerkin-cutoff-uniform estimates.Source: §6 · p. 35
  • The computational method uses exact fraction arithmetic, independent calculations, and targeted perturbation tests to resolve nonlinear Fourier output and continuum comparison bounds.The ensemble, boundary refinement, and 3D diagnostics track residual production, spectral tails, and vortex stretching; learned operators rank candidates for solver evaluation.Source: §6 · p. 35
  • The coefficient-family theorem covers a connected continuum of non-Beltrami centre shapes with a uniform perturbation radius and an explicitly verified coefficient interval.Adaptive coverage for changing supports and independent formal validation are identified as routes toward a general computational theory.Source: §6 · p. 35

Data availability

The submission package includes numerical source data, comparison records, and a per-figure index linking figures to their data.

Data availability

  • The submission package includes numerical source data for every figure, finite comparison records, and a per-figure source-data index.It also includes neural-operator rollouts, held-out diagnostics, selected candidate coordinates, proxy recomputations, and public-DNS records with query and sampling details; no simulation or public-DNS sample premises the regularity theorems.Source: Data availability · p. 36

Code availability

A reproducibility package contains the implementations, records, solvers, verification routines, trained models, and figure-data indices.

Code availability

  • The reproducibility package contains exact-arithmetic implementations, comparison records, numerical solvers, independent verification routines, trained neural-operator definitions, and machine-readable figure-data indices.A permanent repository with a DOI is planned before journal submission and will preserve the source version used for published figures and tables.Source: Code availability · p. 36

Funding

The appendix records support for the work and supplies Fourier, projection, energy, and spectral-residual identities used in the analysis. It also characterizes when the minimized Laplacian residual vanishes and derives related spectral estimates.

Funding

  • CNPq grant No. 445344/2024-5 and Program Talentos Brasil supported the work.Source: Funding · p. 37

A Fourier, Leray, Wiener, and energy identities

  • The Fourier identities yield the Leray multiplier and Wiener bilinear estimate, while the Galerkin energy bound extends each finite-dimensional trajectory globally.The projection is non-expansive, and the energy bound excludes finite-time coefficient blowup.Source: A Fourier, Leray, Wiener, and energy… · p. 41

B Minimization of the Laplacian residual and spectral variance

  • The minimized Laplacian residual equals a spectral variance and vanishes exactly when all energetic modes lie on one Laplacian sphere.The appendix also derives a finite-spectrum variance estimate for the exceptional-cutoff argument.Source: B Minimization of the Laplacian… · p. 42
  • The appendix derives the signed nonlinear source and centred dissipation from spectral moments, then pairs the viscous equation with uN, AuN, and A2uN.Source: B Minimization of the Laplacian… · p. 42

C Enstrophy, bridge action, and terminal continuation

The section derives enstrophy and action estimates for finite Galerkin trajectories, then uses uniform bounds and cutoff removal to pass to the periodic PDE. Terminal smallness supports continuation, smoothness, and uniqueness, with scaling formulas for viscosity and domain size.

C Enstrophy, bridge action, and terminal continuation

  • The bridge and terminal estimates control action through explicit bounds, but tail and exceptional-cutoff contributions remain instance-specific.The appendix recovers bridge action coefficients below 418 without rounded decimal estimates.Source: C Enstrophy, bridge action, and… · pp. 44–45
  • After the terminal time, a Wiener-norm bound below one yields a differential inequality for enstrophy and a weighted tail estimate.The terminal bound is c = 99/100, giving d = 1 − c^2.Source: C Enstrophy, bridge action, and… · p. 44

D Cutoff removal, smoothness, uniqueness, and scaling

  • Uniform H1 and integrated H2 bounds support Galerkin compactness and passage of the quadratic term to the continuum solution.Strong convergence in L2(0, T, H1) and weak convergence in the bounded spaces provide the stated passage.Source: D Cutoff removal, smoothness,… · p. 45
  • The resulting bounds prevent finite-time continuation failure and yield positive-time smoothness; the difference-energy estimate establishes strong and weak–strong uniqueness.Source: D Cutoff removal, smoothness,… · pp. 45–46
  • Rescaling time and velocity restores viscosity, while a second rescaling incorporates torus size and gives the stated action factor ν^(1/3)/L^2.For viscosity ν, τ = νt and u(t, x) = νv(τ, x); for side length 2πL, the text gives the corresponding space-time rescaling.Source: D Cutoff removal, smoothness,… · p. 46

E Analytic field geometry and public-DNS invariants

The appendix derives geometric quantities for the analytic Taylor–Green field and clarifies how vorticity alignment is represented. It distinguishes these diagnostics from the global spectral residual and from the proof of regularity.

E Analytic field geometry and public-DNS invariants

  • For a general velocity gradient, the signed factor σ records the geometric alignment of vorticity with strain after local magnitude is removed.The normalized production Pω = ω · Sω is positive for strain that increases vorticity magnitude and negative for compression.Source: E Analytic field geometry and… · p. 47
  • The strain invariants Q and R are coordinate-invariant, and the alignment calculation contracts vorticity with ordered strain eigenvectors.Source: E Analytic field geometry and… · p. 47
  • These tensor and alignment diagnostics are not used to prove Theorem 2.2; at repeated strain eigenvalues, only projection onto the full eigenspace is basis-independent.Source: E Analytic field geometry and… · p. 47

F Dilated-eigenfield asymptotics

The appendix analyzes perturbations of a dilated eigenfield and gives exact Fourier calculations used in the comparison construction. It also verifies that the equal-coefficient ABC centre generates an exact decaying solution.

F Dilated-eigenfield asymptotics

  • At a monochromatic eigenfield, the spectral variance functional has zero first derivative, so the residual error after the expansion is cubic rather than linear.Source: F Dilated-eigenfield asymptotics · p. 48
  • Spatial dilation scales Laplacian eigenvalues by n^2 and the residual by n^4; integrating in rescaled time yields the stated asymptotic scaling.The text derives the integrated scaling using the time change dt = dτ/(νn^2).Source: F Dilated-eigenfield asymptotics · pp. 48–49

G Exact Fourier algebra and radical bounds

  • The finite Fourier calculation is determined by exact seed coefficients and convolution rules, allowing an independent implementation to reconstruct and check its norm bounds.The calculation expands bilinear terms before collecting the residual.Source: G Exact Fourier algebra and radical… · p. 50

G.1 Exact equal-ABC identities

  • For equal-coefficient ABC, the nonlinear term is a gradient annihilated by the Leray projection, so a decaying eigenfield path solves the equation exactly.Every cubical cutoff N ≥ 1 contains the field’s six-mode Fourier support.Source: §G.1 · p. 50

G.2 Exact TG3 identities

The appendices establish exact Fourier identities for the TG3 centre and its coefficient family, then use whole-interval bounds and scalar comparison to certify finite trajectories through the terminal barrier.

G.2 Exact TG3 identities

  • The TG3 centre is solenoidal and real, with all eight modes satisfying |k|^2 = 3 and every nonlinear output coefficient having magnitude 3/32.The ordered-pair calculation also supplies the θ = 0 case of the family and path algebra used in Theorem 2.6.Source: §G.2 · p. 51

G.3 Exact TG3 coefficient-family identities

  • The complete TG3 coefficient calculation gives a 12-mode residual support and parameter-uniform algebraic bounds for |θ| ≤ 1/10.The residual bound is established by exact grouping and a polynomial bound over the interval, not by sampling.Source: §G.3 · p. 52

G.4 Bernstein conversion and whole-interval bounds

  • Bernstein coefficients enclose polynomial values across the full interval, producing the rational bounds used in the Methods without a sampling assumption.For example, the degree-two coefficients for 3/4 + q + q^2/4, with q = x/10, are 3/4, 4/5, and 341/400.Source: §G.4 · p. 53

H Exact comparison induction and terminal barrier

  • The scalar comparison is justified by first-contact induction, including tangential contact, and the terminal barrier yields exponential decay once W(T*) ≤ η < ν.The comparison applies on each slab when its exact slack is nonnegative; after W ≤ η, the Wiener norm bound supplies the decay estimate.Source: H Exact comparison induction and… · p. 54

H.1 Reconstruction of the rational comparison trajectory

  • The rational trajectory construction checks accepted slabs using exact slack and endpoint conditions, and independently verifies stored fractions against whole-slab bounds and terminal smallness.All four applications stop before their maximum search times, and readers can generate or verify the trajectory with the same analytic inequalities.Source: §H.1 · p. 55

H.2 Equal-ABC comparison

The comparison induction certifies bounds for the equal-ABC and TG3 heat paths, then extends the same approach across a parameter interval of non-Beltrami Taylor-Green centres.

H.2 Equal-ABC comparison

  • For the equal-ABC heat path, comparison over 2,390 adjacent slabs establishes bounds for every 0 ≤ a ≤ 17/25 and N ≥ 1.The initial comparison value is y(0) = 1/100, and the bounds support both the Wiener barrier and the finite-time Galerkin compactness argument.Source: §H.2 · p. 56

H.3 TG3 comparison

  • For the TG3 heat path, induction over 873 slabs establishes comparison bounds for 0 ≤ a ≤ 41/40 and every N ≥ 1.The full nonlinear output is retained in the residual before projection, with no exceptional cutoff.Source: §H.3 · p. 56

H.4 Uniform coefficient-family comparison

  • One comparison problem covers the full coefficient interval |θ| ≤ 1/10 and 0 ≤ a ≤ 1, with bounds established over 992 adjacent slabs.The path modes lie in cube one; the residual occupies cube two, but its contribution is exactly projected for N = 1 and Wiener projection is contractive.Source: §H.4 · pp. 56–57

I Floating-point Galerkin update and recorded metrics

The appendix specifies how initial fields, numerical trajectories, diagnostics, and neural-operator comparisons are constructed. It also distinguishes descriptive numerical evidence from the analytic proof and records implementation choices that affect interpretation.

I Floating-point Galerkin update and recorded metrics

  • The Galerkin update constructs real, solenoidal perturbations with prescribed mode amplitudes, while the projected Fourier ODE advances non-proof simulations using fourth-order Runge–Kutta.The final projection removes roundoff components outside the divergence-free truncated space; it is not an analytic regularization.Source: I Floating-point Galerkin update and… · p. 57, §I.2 · pp. 58–59

I.1 Complete parameter maps and initial fields

  • The ensemble uses two 2,048-point Sobol strata, while the neural experiment uses 384 deterministic configurations and specifies its parameter maps and target-generation settings.The ensemble strata use seeds 20260909 and 20260910; neural targets use a 123 grid, cutoff two, unit viscosity, and RK4 step 0.005.Source: §I.1 · p. 58

I.2 Time integration and diagnostics

  • The numerical methods define trajectory, balance, spectral, and geometric diagnostics, with visual smoothing used only for display rather than for reported statistics.Saved-array vortex curves use specified interpolation, seeding, and integration procedures; those render inputs are not read by theorem checkers.Source: §I.2 · pp. 58–60

I.3 Statistical estimators and reconstruction of empirical claims

  • Ensemble summaries are descriptive functions of deterministic arrays, and Sobol points or spatially correlated DNS samples are not treated as independent replicates for p-values.Theorem membership and numerical trust are defined by explicit predicates and gates.Source: §I.3 · p. 60

J Neural-operator construction and diagnostics

  • The neural experiment compares models on 384 configurations split into 127 training, 45 validation, and 212 OOD fields, predicting velocity at five checkpoints.Its Fourier layer combines a learned spectral branch with a pointwise channel map, group normalisation, and GELU.Source: J Neural-operator construction and… · p. 61

J.1 Network operations and optimisation

  • The implemented spectral branch retains one 3 × 3 × 3 block, while the output imposes neither a Leray projection nor a hard zero-mean constraint.Each model is selected using its own training objective, so the two validation-loss ordinates are not directly comparable physical errors.Source: §J.1 · pp. 61–62

J.2 Discrete objective and measured quantities

This section specifies discrete neural-operator objectives and evaluation diagnostics, and develops supporting projection, comparison, spectral, and smooth-solution arguments, including bounds on divergence-defect effects and heat-expansion remainders with explicit objective-production coefficients.

J.2 Discrete objective and measured quantities

  • 0.955 for FNO versus 0.864 for PINO in composite top-decile recall, with paired 95% interval [−0.227, 0.045], shows no PINO improvement.The interval resamples 212 aligned OOD cases for one fitted training seed, not variation across independently trained networks.Source: §J.2 · p. 64
  • Figure 14 diagnostics show upward-biased PINO action ratios, tightly clustered dense actions across solver comparisons, and all eight cohorts above the 0.12 diversity threshold.The native-grid cutoff-sensitivity diagnostic does not enclose the complete continuous quadratic output.Source: §J.2 · pp. 63, 65, Figure 14 · p. 65

K Expanded algebraic, analytic and discrete transitions

  • Appendix K supplies intermediate derivations; pressure projection preserves zero mean and yields a finite-dimensional Galerkin ODE with unique global existence at fixed N.The continuum argument separately requires cutoff-uniform bounds.Source: K Expanded algebraic, analytic and… · p. 65, §K.1 · p. 66

K.2 Complex norms, dissipation and the scalar comparison

  • Complex coefficient-norm inequalities retain dissipation and justify scalar comparison, including at zero Fourier coefficients and when the cubic’s linear coefficient is negative.The comparison follows from a Lipschitz bound on bounded intervals and a positive-part argument.Source: §K.2 · pp. 66–67

K.3 All ordered products in the signed path

  • The signed path expansion retains all nine ordered bilinear terms before grouping, and applies the triangle inequality only after equal-mode vectors are combined.Taking absolute values before combining equal-mode vectors would produce a different bound.Source: §K.3 · pp. 67–68

K.4 Differentiating the centred spectral residual

  • The moment balances give E′ = −2νX, X′ = −2νZ + 2Π1, and Z′ = −2νW + 2Π2; the centred residual is continuous at zero in H2.Centering avoids subtracting nearly equal Z and X2/E in floating-point arithmetic.Source: §K.4 · p. 68

K.5 All-order regularity, compactness and uniqueness

  • Cutoff-uniform bounds and compactness yield a global smooth limit for each smooth datum, but not uniform high-Sobolev bounds over the entire A2 perturbation ball; the constructed smooth solution is unique.Smoothness makes each datum’s initial Hm norm finite, which is the quantifier used by the theorem.Source: §K.5 · pp. 69–70

K.6 Velocity gradients, curl and physical production

  • The vorticity equation identifies stretching as local enstrophy production, but this is not positive total production after viscosity or positivity of the spectral source.Source: §K.6 · p. 71
  • A nonzero measured gradient trace changes the second and third characteristic coefficients, so trace-free formulas require caution when interpreting discrete divergence defects.For nonzero trace dJ, the actual coefficients are (dJ² − 3dJ tr J² + 2 tr J³)/6; the trace-free definitions are QJ = −tr J²/2 and RJ = −tr J³/3.Source: §K.6 · p. 71
  • Vorticity components in a strain eigenbasis use E⊤ω, while repeated eigenvalues require the basis-independent sum of squared projections over the eigenspace.This explains the column contraction and its degeneracy qualification.Source: §K.6 · p. 71

K.7 Uniform heat expansion and nonlinear remainder

  • A contraction argument yields a uniform remainder for the heat-expansion objective, rather than only a finite-time formal series.The construction chooses |ε| ≤ (8Bsas)−1, keeping the fixed-point map in the ball with Lipschitz constant at most 1/2.Source: §K.7 · p. 72
  • The leading coefficient of DR(v) is 3ε²τ²e−4τ/2 for sufficiently small |ε|.The expansion uses AU = U, AV = 2V, ⟨U,V⟩ = 0, and a denominator bounded uniformly in τ.Source: §K.7 · p. 72
  • The integrated production coefficient in (110) is 3n²/(64ν²).In scaled variables the coefficient is 3ε²/64; integer periodic dilation yields the stated n and ν dependence.Source: §K.7 · p. 72

K.8 Discrete transforms and time integration

These appendices specify the discrete transforms, time integration, neural-operator observables, and statistical estimators used in the computations, while distinguishing numerical diagnostics from stronger convergence or exactness claims.

K.8 Discrete transforms and time integration

  • For cutoff trajectories with 4K < n, discrete means of products of four factors match their continuous means; this guarantee does not extend to unrestricted network outputs.The result follows from discrete orthogonality and the bound on total frequency.Source: §K.8 · p. 73
  • The projected finite-dimensional ODE has local error of order five for a fixed resolved ODE, but this does not establish fourth-order action-quadrature convergence or cutoff-independent error constants.The analysis expands the projected ODE stages using derivatives of its finite-dimensional vector field.Source: §K.8 · p. 73

K.9 Network indices, loss factors and acquisition

  • The spectral loss retains an extra n^-3 factor, while the energy penalty is not an exact balance for arbitrary aliased, non-solenoidal outputs.The n^-3 factor arises from averaging an already normalized Fourier array.Source: §K.9 · pp. 74–75
  • The neural generator uses P0 = I and can produce outputs that are neither solenoidal nor mean zero, unlike the analytic mean-zero Galerkin problem.The nonlinear product is formed before masking and Leray projection.Source: §K.9 · p. 75
  • Below E = 10^-20, the predicted action uses a regularized observable rather than the exact centered-grid residual quotient.The denominator is floored at 10^-20; the stated tail check gives T2 = 16/97.Source: §K.9 · p. 75
  • Acquisition optimizes 16 parameter vectors with 31 clamped Adam steps, but the piecewise derivatives do not establish global optimality.The search does not differentiate through Galerkin rechecks or exact comparison.Source: §K.9 · p. 75

K.10 Quadrature, quantiles and numerical claims

  • Refining ODE states at fixed output times does not refine the action quadrature, and R^(2/3) may lack two bounded derivatives at zeros of R.Thus a smooth-integrand order estimate cannot be assumed for that quadrature.Source: §K.10 · p. 75
  • Paired bootstrap intervals estimate median casewise differences, while recall resampling requires reranking both target and predicted scores in each draw.Keeping the original rankings changes the recall statistic.Source: §K.10 · p. 75

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