25 / C3-W: Critical Pressure Rotation, Strain Active-Volume Sparseness, and the Analyticity-Scale Barrier

C3-V compressed the hypothetical singular survivor into two concentration channels: pressure rotation and strain fluctuation. This round advances both by one step. On the pressure side, the core is the Critical Pressure Mean-Forcing Bound (Theorem 4.1): the pressure forcing of the mean strain, $P_{\chi,R}=\int\chi_R\nabla^2p$, is a signed tensor, so there is no need to control $\int\chi|\nabla^2p|$; two integrations by parts (subtracting an arbitrary constant $c$) reduce it exactly to the scale-critical $L^{3/2}$ pressure oscillation $\Pi_R$. From the global bound $\int\|p\|_{3/2}^2dt\le C\|u_0\|_2^4/\nu$ (Theorem 7.1, a standard Riesz-plus-interpolation estimate), one obtains $R^2$-Weighted Pressure-Rotation Packing (Theorem 8.1) — but this is still only weighted-finite; under geometric-series scaling every generation can still have $\mathfrak R_n^P\sim1$, so the Zeno escape of pressure rotation still survives (§9). There is also Pressure-Active Core Packing (Theorems 11.1, 12.1): both the number of $b$-pressure-active cores and their time-$L_t^{4/3}$ multiplier admit a scale-independent global budget, connecting directly to Constantin's pressure-regularity theorem — a hypothetical blow-up must escape the uniform integrability of $|p|^{3/2}$ over a shrinking set; pressure-rotation escape is, in essence, a critical pressure-concentration branch. The genuinely new geometry is on the strain side: the Volume-to-One-Dimensional-Sparseness Lemma (Theorem 20.1, a purely geometric lemma) proves that a volume smaller than $\delta^3|B_r|$ already guarantees a direction along which the set is one-dimensionally $\delta$-sparse at that scale, from which follows the Strain-Intermittency-to-Sparseness Theorem (Theorem 22.1): once the effective active volume $\phi_{p,R}$ is small enough, the high-gradient region is automatically one-dimensionally, linearly sparse at scale $r_{\rm sp}\sim\phi_{p,R}^{1/3}R$ — and it is proved that $|\nabla S|\asymp|D^2u|$ pointwise (§23), which places this route on the same derivative order as the Grujić–Xu higher-derivative sparseness-regularity framework. So extreme intermittency is not a free escape: if the sparseness scale does not exceed the relevant analytic radius $\rho_{\rm an}$, a known geometric regularity mechanism may be triggered; to keep functioning as a singular survivor, the analytic scale must shrink faster than $\phi^{1/3}R$ (the Analyticity-Scale Escape Debt), or the component/sign/time interfaces must fail to match.