The first nonlinear counter of the free fluid: a closed form for the second-order energy excess of two-dimensional Navier–Stokes flow with white-noise vorticity on a flat torus

We study the incompressible Navier–Stokes equations on a flat two-dimensional torus of unit area and arbitrary shape, without forcing or boundaries, started from Gaussian white-noise vorticity. This initial law is the enstrophy Gibbs state: it is invariant for the truncated Euler dynamics, but viscosity breaks the invariance. The energy of the linear (Stokes) evolution of the same data is known exactly. In the language of the Infinity Atlas it splits into a logarithmic counter, a shape-dependent finite part (Kronecker's limit function) and a power level linear in time. We show that the nonlinearity first enters through a new level proportional to time, ⟨E_NS(t) − E_lin(t)⟩ = κ₂ R²/β + O(R⁴), R² = t/(2πβν), and we derive its counter in closed form: 4πκ₂ = 2 Li₂(1/4) − Li₂(1/2) + (95/12) log 2 − (39/8) log 3 − 1/12, κ₂ = 1.1233876401607… × 10⁻⁴. The derivation reduces a two-loop Gaussian integral to one-dimensional integrals of logarithms. It uses an exact master formula, ∫(P×Q)² e^(−a|P|²−b|Q|²−c|P+Q|²) d²P d²Q = π²/[2(ab+bc+ca)²], together with Schwinger parameters and a symmetric regulator whose infrared divergences cancel between the input and damping terms. Three consequences follow at this order:(i) the enstrophy deficit is constant in time, −κ₂/(4πβ²ν²), and the palinstrophy is unchanged;(ii) κ₂ does not depend on the shape of the torus;(iii) a Galerkin cutoff enters only through Λ = K_c√(νt), and the lattice only through a factor 1 − cτ with c ≈ 0.57. The closed form was checked in four ways: by direct evaluation at 110 digits, by an integer-relation (PSLQ) search, by symbolic integration in MATLAB, and against direct numerical simulations at 128², 256² and 512². The simulations agree with it without any fitted parameter. The value 0.90 × 10⁻⁴ measured earlier at 2.5 ≤ R ≤ 5 is identified as κ₂ plus negative higher-order corrections. Limitations: the reduction is a derivation, not yet a rigorous theorem, because the interchanges of limits are justified and checked numerically but not proved. The fourth-order coefficient κ₄ ≈ −0.031 × 10⁻⁴ is only fitted. The lattice constant c is measured, not derived.