The first nonlinear counter of the free fluid in three dimensions: a closed form for the second-order energy deficit of three-dimensional Navier–Stokes flow with equipartition data on a flat torus

We study the incompressible Navier–Stokes equations on a three-dimensional flat torus of unit volume and arbitrary shape, without forcing and without boundaries. The flow starts from Gaussian white-noise velocity, which is Lee's equipartition of energy and the absolute equilibrium of every Galerkin truncation of the Euler equations. Viscosity breaks this invariance. The linear (Stokes) energy of the same data has exactly two levels in the sense of the Infinity Atlas, (π/τ)^{3/2} and −1, both independent of the shape of the box. The nonlinearity enters first through an energy deficit:

  ⟨E_NS⟩ − ⟨E_lin⟩ = −κ₃ R² ⟨E_lin⟩,   R² = 2⟨E_lin⟩ t/ν,
  κ₃ = (5π + 8 − 12√3)/24 = 0.12180639904676827792…

In three dimensions the triangle measure makes the two-loop integrand rational. After a radial Gaussian integral and the time integrals, κ₃ is the integral of a rational function over the shapes of triangles. We integrate it exactly in one variable and identify the result at 120 digits, checked at 160. No regulator is needed.

Consequences at this order:
(i) the enstrophy change is ΔE/(νt);
(ii) the mean vortex stretching is algebraic, ⟨Σ⟩ = (3 − √3)/(3072 π³ α² ν⁵ t⁴), so the velocity-derivative skewness starts as S = −(2√5(3 − √3)/21) R ≈ −0.270 R;
(iii) R² ∝ t^(−1/2), so the expansion holds at late times, the signature of supercritical data;
(iv) a Galerkin cutoff enters at first order, κ₃(Λ) = κ₃ − 3√2/(20√π Λ) + O(Λ⁻²), with Λ = K_c √(νt).

Exact lattice sums at 64³ agree with the continuum to 1.5 × 10⁻³. Direct simulations at 64³ agree with them without fitted parameters, and a comparison with the second-order hierarchy, realisation by realisation, fixes the mean to 0.6%.

Limitations: two of the five final constants were integrated symbolically; the other three, and the moments behind the vortex stretching, were identified by integer-relation searches at 110–120 digits but not integrated symbolically. The exchanges of limits are justified and checked numerically, not proved. Higher orders and helical equilibria are not treated.