26 / C3-X: Joint Pressure–Strain Concentration, Finite-$k$ Gap Closure, and Analyticity-Scale Escape

C3-W already compressed the hypothetical singular survivor into two channels: pressure concentration and strain intermittency. This round asks the real question: can both escape without limit at the same time? The core is the Critical Pressure-Mass Certificate (Theorem 4.1): replacing C3-W's constant subtraction with an affine-function subtraction in the oscillation bound, the $b$-pressure-active core is thereby shown to carry a scale-independent critical pressure mass containing no $R$ at all, $\int_{B_{2R}}|p|^{3/2}\ge cb^{3/2}\nu^3$. Multi-core shrinkage (Theorem 6.1: the Small-Volume Pressure Concentration Certificate) directly yields a certificate of non-vanishing pressure mass on a small-volume set, connecting to Constantin's uniform-integrability pressure-regularity criterion — but it is explicitly flagged as merely a sufficient mechanism, not a contradiction. This round's most important no-go is in §13-32: pressure concentration does not imply pointwise overlap with strain-gradient concentration — because pressure is a nonlocal quadratic transform, a far-field source can generate local pressure in a region of low $D^2u$; the co-located case ($\Theta_{P/S}\ge\theta_0$) and the segregated case ($\Theta_{P/S}\to0$) must be tracked separately and must not be automatically merged. This round's strongest new structure is the Finite-$k$ Volume Gap-Closure Lemma (Theorem 17.1): it bridges the algebraic gap, within the Grujić–Xu higher-derivative hierarchy, between the "energy-a-priori sparseness scale" $\ell_{\rm apr}^{(k)}\sim A_k^{-1/(k+3/2)}$ and the "regularity sparseness scale" $\ell_{\rm reg}^{(k)}\sim A_k^{-1/(k+1)}$, via an explicit active-volume exponent: $\phi_k\lesssim A_k^{-\vartheta_k}$, where $\vartheta_k=3/[2(k+1)(k+3/2)]$ — in particular, at $k=2$ this gives exactly $\vartheta_2=1/7$ (Corollary 18.1), and $\vartheta_k\to0$ echoes Grujić–Xu's own direction that "the gap vanishes as $k\to\infty$." But the document repeatedly and explicitly flags: this is not an independent $k=2$ regularity theorem, only an algebraic scale bridge (§23) — genuinely applying the external theorem still requires the component/sign/threshold/analyticity conditions to all align. Even more counterintuitively, §36 shows that intermittency cannot be unboundedly strong: if $\phi_2\ll A_2^{-1/7}$, the volume collapse is in fact stronger than what is needed to close the scale gap, pushing the sparseness scale toward the regularity side — so sufficiently strong intermittency actually has a self-regularizing aspect. The genuinely surviving intersection is pressure concentration $\cap$ failure of analytic/geometric sparseness closure.