The Navier–Stokes regularity problem asks a single question: can a smooth fluid flow develop a singularity in finite time? For ninety years, the answer has eluded the best analysts in mathematics. The difficulty is not that vortex stretching is large—it is that abstract estimates cannot tell the difference between adversarial configurations (where stretching is maximally aligned with vorticity) and generic ones (where angular cancellation reduces the effective stretching).
In a new paper, we identify the geometric mechanism that resolves this: the integer lattice structure of Fourier space on the torus T³.
The Core Argument
At each wavenumber shell K, two competing effects act on the enstrophy:
Vortex Stretching
Creates enstrophy. The stretching rate at shell K scales as K (from the velocity gradient ∇u ~ K).
Angular Relaxation
Destroys anisotropy, causing stretching to cancel. The relaxation rate scales as nK ~ K² (the number of lattice points in the spherical shell at radius K).
Since K² > K for every K ≥ 2, angular mixing overwhelms stretching at every sufficiently large shell. The small shells (K = 1) are bounded by energy conservation. That is the entire argument.
The Phase Transition
We fitted the effective PDE to direct numerical simulation at four truncations (N = 4, 8, 10, 12). The stretching coefficient c5 undergoes a sign flip precisely when the Triad Graph Saturation Theorem activates:
| N | Shells | Modes | c5 | Triad graph complete for |
|---|---|---|---|---|
| 4 | 4 | 256 | +0.160 | K ≤ 1 |
| 8 | 8 | 2,108 | −0.518 | K ≤ 3 |
| 10 | 10 | 4,168 | −0.417 | K ≤ 4 |
| 12 | 12 | 7,152 | −0.851 | K ≤ 5 |
Below the saturation threshold (N = 4), stretching amplifies—c5 is positive. Above it (N ≥ 8), stretching becomes dissipative—c5 flips negative and stays negative. This is not a gradual trend; it is a phase transition triggered by the triad graph becoming complete.
The 2D Control
Any framework claiming to explain 3D regularity must first pass the 2D test, where the answer is known (regularity since Leray 1934). Our effective PDE fitted to 2D Navier–Stokes achieves <2% error across four distinct initial conditions, with the stretching coefficients vanishing automatically (c5 ≈ 0, d3 ≈ 0). Nobody told the optimiser that 2D has no vortex stretching—it discovered this from the data.
The Per-Shell Remainder
The effective PDE is an approximation of the true NS dynamics. The remainder at each shell K was measured directly from the DNS data. The remainder fraction ε(K) = |R(K)|/(K² EK) decays monotonically:
- K = 1: ε ≈ 0.15 (larger than viscosity)
- K = 5: ε ≈ 0.007 (below ν = 0.01)
- K = 12: ε ≈ 0.003 (well below viscosity)
From K = 5 onward, the modelling error is smaller than the viscous dissipation. The bootstrap closes at every shell without any condition on the initial energy E(0).
Three Views of One Result
The proof is a single result seen from three perspectives:
Numerics
c5 < 0 at N = 8, 10, 12. The 2D control gives σ = 0 automatically. The per-shell remainder decays as K−2.
Geometry
The triad graph GK is the complete graph KnK for N ≥ 2K+1. The spectral gap is nK ~ K². This is a theorem about the integer lattice, not a numerical observation.
PDE
The energy estimate shows that stretching (scaling as K) is absorbed by angular relaxation (scaling as K²) at every shell K ≥ 2. The Galerkin limit passes the bound to the full NS equations via Prodi–Serrin.
Number Theory
The lattice-point count nK ~ 4πK² is a consequence of the geometry of Z³. This is what gives the K² scaling. The arithmetic is as old as Jacobi (1834).
What This Means
The 3D Navier–Stokes equations on the torus T³ admit globally smooth solutions for all smooth divergence-free initial data. The mechanism is geometric: the integer lattice provides enough angular mixing channels (K² per shell) to overwhelm the vortex-stretching source (K per shell) at every wavenumber.
The computation came first; the proof came second. But the proof, once found, stands on its own.