Navier–Stokes Turbulence as a Z₉ Chern–Simons Cascade: Spectral Gap, Algebraic Structure, and Convergence with Loop-Space Analysis (A Truncation Framework and Consistency Analysis)

Under Z₉ grading of the Fourier-space Navier–Stokes (NS) equations, the wavenumber inertial range decomposes into nine shells whose triadic interactions realise the fusion rule a×b≡ (a+b) mod 9, exhibiting the algebraic structure of the U (1) ₉ Chern–Simons (CS) theory of the Post-Theoretical Relativistic Holography (PTRH) programme. The rigorous results are confined to the nine-mode Z₉-graded Galerkin truncation: the standard enstrophy bound holds with viscosity; the inter-shell ratio satisfies 9^ (1/9) −1 < sin (π/9), establishing via the Bauer–Fike theorem that single-mode grading errors cannot close the spectral gap Δₘin = 2sin (π/9) ≈ 0. 684. A nine-mode numerical simulation provides a consistency check on the truncation. Gabriel's horn models finite energy at infinite scale range; the Ruelle–Pesin relation formalises the observation that present chaos accelerates approach to statistical equilibrium. Independent loop-space analysis by Migdal yields a roots-of-unity quantization of the turbulent attractor exhibiting the same algebraic architecture as the Z₉ grading — suggestive convergent evidence from an independent programme. Whether the gap persists in the full NS system (Conjecture C1) and whether Migdal's N is independently fixed to 9 (Conjecture C4) remain open. Rigorous results and conjectural claims are clearly demarcated throughout.