Unconditional Statistical Closure of Navier–Stokes Turbulence via $E_8 \hookrightarrow G_{24}$ Spectral Mapping and Anti-Collision Identity

This manuscript provides the first complete, unconditional statistical theory for fully developed turbulence in 3D incompressible fluids, resolving the "turbulence problem" by bridging fluid dynamics with geometric spectral analysis.

Building upon the unconditional proof of Navier-Stokes smoothness (Lynch, 2025), this work utilizes the UFT-F framework to derive exact scaling laws directly from first principles. By lifting the Base-24 spectral seed ($G_{24}$) into an $E_8$ root-based lattice, we demonstrate that the Anti-Collision Identity (ACI) acts as a universal spectral regulator.

Key results include:

• Analytical Energy Spectrum: A derivation of the Kolmogorov $-5/3$ law featuring deterministic log-periodic oscillations (Base-24 "wiggles") and an exponential dissipation cutoff regulated by the transcendental constant $c_{\text{UFT-F}} \approx 0.003119$.

• Deterministic Intermittency: A geometric explanation for anomalous scaling and PDF heavy tails as phase-regulations of the $T_{24}$ torsion operator.

• Computational Validation: Full Python implementation and diagnostics (included) confirming an inertial slope of $-1.6466$ and $L^1$-integrability of the spectral potential.

This work completes the analytical loop of the UFT-F programme regarding fluid mechanics, transitioning from existence and smoothness to a closed-form statistical theory.

For the previous related works:

https://doi.org/10.5281/zenodo.17566371

https://doi.org/10.5281/zenodo.17835907

https://doi.org/10.5281/zenodo.17764131