Structural Rigidity and Statistical Fixed Points in Three-Dimensional Turbulence: Why the Kolmogorov Constant Cannot Be Derived from Symmetry Alone

We present a structural analysis of three-dimensional turbulence. We prove the nonexistence of nontrivial quadratic spectral invariants beyond energy and show that no polynomial invariant consistent with isotropy and translational invariance exists beyond energy and helicity. We construct the Martin--Siggia--Rose functional integral for stochastic Navier--Stokes dynamics and derive the exact functional renormalization group equation. One-loop analysis yields a nontrivial fixed point reproducing the Kolmogorov scaling exponent $-5/3$. However, the amplitude remains scheme dependent. We demonstrate that the exact functional RG flow possesses a marginal amplitude direction generated by field rescaling symmetry. We then prove a general Marginal Amplitude Theorem: any local cascade system with a dissipative cutoff possesses exactly one marginal amplitude parameter. Consequently the Kolmogorov constant cannot be fixed by symmetry or RG fixed-point structure alone; it depends on dissipation-range physics. This places turbulence within the broader universality class of critical phenomena where exponents are universal but amplitudes are not.