14 / C3-L: Critical Vorticity-Moment Escape, the Active-Occupancy Dichotomy, and the Strain-Geometry Debt

C3-K compressed the gap down to the One-Frequency-Moment Gap; this round's original question is: does hypothetical blow-up really have to push the next frequency moment to infinity? The answer is unambiguously YES — proved directly via the contrapositive of Cheskidov–Dai's frequency-localized regularity criterion (Critical Vorticity-Moment Divergence, Theorem 4.1): $\nu\int\sum_{q\le Q(t)}\lambda_q^2a_q\,dt=\infty$, which can be read as the divergence of the integral of the shells' absolute vorticity — not a conjecture but the direct consequence of an already-proved theorem. Splitting this condition by threshold $\beta$ into a subthreshold part and an active part, with the subthreshold part controlled by the dissipation wavenumber ($\Lambda\in L^2$), it obtains the Critical-Moment Carrier Dichotomy (Theorem 9.1): either $\Lambda\notin L^2$ (Branch A, borne directly by frontier spikes), or $\Lambda\in L^2$ but the active moment $M_{5/2}(\beta)$ must diverge for every fixed threshold (Branch B) — combined with C3-K's already-proved $M_1(\beta)<\infty$, this yields a very clear signal: a finite first-order occupancy moment together with a divergent order-5/2 moment. The most natural candidate for raising the moment is enstrophy, whose exact identity follows directly from the vorticity equation, but integrating it immediately shows that vortex stretching $\int\omega\cdot S\omega$ must also be controlled — the Moment-Raising Geometry Debt No-Go (Proposition 20.1): raising one differential/frequency moment creates a vortex-stretching geometry debt, which is a direct logical consequence of the exact identity, not a heuristic; a scaling audit confirms the enstrophy identity itself carries no hidden scaling advantage. It then brings in the external middle-strain-eigenvalue regularity criterion (Evan Miller's theorem, together with the 2025 Guo–O endpoint Besov result): hypothetical blow-up must force $\lambda_2^+\notin L_t^2L_x^3$, and even more strongly $\lambda_2^+\notin L_t^2\dot B^{-1}_{\infty,\infty}$ — a parallel pair of necessary conditions (Spectral–Geometric Double Escape, Theorem 26.1). The document explicitly states that it has not yet been proved that spectral-moment escape implies strain-geometry escape, or the reverse implication, and this must not be smuggled in as a causal equivalence. The research frontier shifts to C3-M: can these two channels, both known to have to diverge simultaneously, be forced by genuine N–S geometry into a jointly impossible state?