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

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, $\eta =N2$. 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 $(1-2^{-(p+17\mathrm{/}2)}{)}^{-1}$. Both share the Riemann-wall poles $p=-8+i{\rho }_n$, generated by the non-trivial zeros $1\mathrm{/}2+i{\rho }_n$ of $\zeta (s)$, while the odd ensemble also contains the dyadic wall $p=-17\mathrm{/}2+2\pi im\mathrm{/}\log 2$, $m\mathrm{\ne }0$. 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.