Geometric Dynamics of Turbulence: A Minimal Oscillator Structure from Non-local Closure

Turbulence remains one of the central open problems in classical physics, largely due to the absence of a closed dynamical description of the Reynolds stress. Existing approaches typically rely either on local constitutive assumptions or on high-dimensional statistical representations, without identifying a minimal set of dynamical variables governing the cascade response. Here we show that the non-local stress response implied by the Navier-Stokes equations admits a systematic reduction onto a low-dimensional anisotropic sector of the turbulent cascade. This reduction leads to a minimal dynamical system with the structure of a damped oscillator, arising from the coupling between the leading angular mode and its nonlinear transfer to higher-order sectors. Within this framework, classical turbulent behaviors -- including inertial-range scaling, shear-driven transport, and wall-bounded logarithmic profiles -- emerge as different realizations of the same underlying dynamical structure. Universal quantities such as the Kolmogorov constant and the von K\'arm\'an constant appear as leading-order consequences of internal consistency conditions applied across homogeneous and shear-driven regimes. These results suggest that turbulence admits a minimal dynamical backbone governed by non-local cascade response, providing a unified perspective that connects spectral transfer, anisotropy, and mean-flow interaction within a single reduced framework.