Decaying Turbulence and the Riemann Hypothesis: The number theory behind the infinite-time singularity

Abstract

We derive a formal statistical solution of freely decaying incompressible turbulence in arbitrary dimension \(d>1\) using Navier--Stokes loop equations. The loop Fourier transform maps smooth deterministic Cauchy data in infinite space to a one-dimensional momentum-loop quantum field theory, giving a geometric origin of spontaneous stochasticity. In bounded-variation calculus the nonlinear advection term becomes a closed-loop total derivative and cancels on the compact spherical target, leaving a diffusive momentum-loop evolution. The universal attractor is the planar Euler ensemble of rational star-polygon walks. Its continuum limit splits into two parity sectors, \(\eta=N\bmod 2\). Both Euler ensembles are marginally Lyapunov-stable in the continuum limit and give dimension-independent energy scaling functions \(H(k\sqrt{\tilde\nu t})\). Their Mellin amplitudes differ only by the odd-sector prime-\(2\) Euler factor \((1-2^{-(p+17/2)})^{-1}\). Both share the Riemann-wall poles \(p=-8+i\rho_n\), generated by the non-trivial zeros \(1/2+i\rho_n\) of \(\zeta(s)\), while the odd ensemble also contains the dyadic wall \(p=-17/2+2\pi i m/\log2\), \(m\ne0\). Thus the two sectors have distinct Stokes staircases, although their spectra agree with present \(4096^3\) DNS within statistical uncertainty. Assuming the Riemann Hypothesis and simplicity of the zeros, the common Riemann-wall activations occur at \(t_n\propto\rho_n^3\) and condense into an infinite-time essential singularity.