Geometric Solution of Turbulence as Diffusion in Loop Space

arXiv:2511.02165v4 Announce Type: replace Abstract: Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics, have long constituted some of the most challenging problems in theoretical physics. This review describes a unified theoretical framework, the loop space calculus, which offers an analytical approach to these problems. The central idea is a shift in perspective from pointwise fields to integrated loop observables, a transformation that recasts the governing nonlinear equations into a universal linear diffusion equation in the space of loops. This framework, supported by recent mathematical analysis, is analytically solvable and yields an exact, parameter-free solution for decaying hydrodynamic turbulence--the Euler ensemble--which is shown to be dual to a solvable string theory. The theory's predictions include: (i) the unification of spatial and temporal scaling laws, governed by two related, infinite spectra of intermittency and decay exponents derived from the nontrivial zeros of the Riemann zeta function; (ii) a first-order phase transition in magnetohydrodynamic (MHD) turbulence; and (iii) the formation of quantized, concentric shells in passive scalar mixing. The appearance of identical mathematical structures as solutions to the turbulent regime of Yang-Mills gradient flow points to the broad applicability of this approach. The framework also yields a new type of analytic Hodge-dual matrix surface that solves the Yang-Mills fixed-point loop equation by harmonic map, opening the way for a geometric formulation of QCD string theory.