arXiv2026
The statistics of a classical flow is a problem of quantum mechanics in loop space, and not by analogy: the loop functional of any flow is the loop-ordered exponential of the covariant derivative $\partial+\mathrm{i}v/ν$. We follow this representation from the Navier--Stokes equation to the Euler ensemble, the fixed point of the loop equation regularized by polygons with $N$ sides. Near the turning angle $β=π$ an exact operator identity, followed by an asymptotic continuum step and without a saddle-point approximation, reduces the ensemble to two quantum pendulums, the Mathieu equation; this supersedes our earlier WKB evaluation. A Monte Carlo simulation up to $N=10^5$ confirms the reduction. Extreme turbulence has singularities at small scales, and, as in quantum field theory, we define it by this regularization and a renormalization: the leading power of $N$ is removed, and the turbulent viscosity $\tildeν$ remains as the single physical scale. This is a proposed definition of turbulence at infinite Reynolds number; its continuum spectrum is computed under a stated convergence condition, and its identification with physical turbulence is a hypothesis. The energy spectrum is a function of $κ=k\sqrt{\tildeν(t+t_0)}$ that falls as $1/k$ and then as $k^{-2}$. At $κ=\sqrt2$ it has a weak nonanalyticity: its second derivative diverges logarithmically. At a fixed wave number the decay is therefore singular at the finite time $t_*=2/(\tildeνk^2)-t_0$, when $t_*>0$, and the vorticity correlations oscillate in space with the period $\sqrt2\,π\sqrt{\tildeν(t+t_0)}$. Of the limits of the ensemble with structure, only the pendulum limit is taken at a critical point and has a bulk dissipation that vanishes with the viscosity. A companion paper compares it with simulations and experiments.