10 / C3-H: Ancestry Renormalization, the Unit-Shell Anchor, and the Critical Compactness Barrier

C3-G established a conditional causal genealogy under explicit assumptions. This round applies a viscosity-normalized critical rescaling to that genealogy, $v_n(y,s)=\nu^{-1}\lambda_n^{-1}u(x_n+y/\lambda_n,t_n+s/(\nu\lambda_n^2))$, and asks whether this directly yields a nontrivial ancient critical element to which existing backward-uniqueness/rigidity theorems can be applied. The verdict is explicit: not directly. Because of the first-crossing definition, the rescaled field retains a Persistent First-Crossing Trace (Theorem 5.1): for $s<0$ the unit-shell amplitude is strictly below the threshold, and at $s=0$ it exactly equals the threshold — a pure consequence of the scaling. Using a uniform Bernstein derivative bound plus an Arzelà–Ascoli diagonal extraction, it proves Unit-Shell Snapshot Compactness (Theorem 9.1): a nonzero, smooth, band-limited limit profile $w_\ast$ exists, satisfying the helical eigen-relation and carrying genuinely nonzero local critical mass. The backward lifespan extends to $(-\infty,0]$ because $\nu\lambda_n^2t_n\to\infty$. However, the document cites the external Seregin theorem (potential blow-up requires $\|u(t)\|_3\to\infty$, not merely $\limsup$) together with critical scale invariance to prove Renormalized global critical-norm divergence (Theorem 13.1): $\|v_n(0)\|_3\to\infty$, with the $\dot H^{1/2}$ norm likewise diverging — this is precisely the Critical Compactness Barrier (Theorem 15.1): the unit-shell anchor is compact, but the full rescaled field is unbounded in both $L^3$ and $\dot H^{1/2}$, so the Kenig–Koch or Gallagher–Koch–Planchon compactness/profile-decomposition theorems — which require a bounded critical sequence — cannot be applied directly. The document explicitly notes that amplitude normalization (dividing by the $L^3$ norm) is also not a legal N–S renormalization, since doing so would rewrite the equation itself. The remaining divergent part must be explicitly preserved as a critical background defect, not silently discarded (echoing the non-collapsing spirit of the X-Integration). A second no-go appears in the rescaling of the causal edge: the normalized time gap $\delta_n=\nu\lambda_n^2(t_n^c-t_n^p)$ is only known to satisfy $0<\delta_n\le\theta$, with no uniform positive lower bound, so $\delta_n\to0$ remains entirely possible — legality at every scale does not imply legality in the limit, and strict causality is not a property preserved under renormalization. The document closes with a Renormalization Trichotomy: Branch A (fully compact, but already proved impossible as a bounded global $L^3$ object), Branch B (background defect, requiring classification of where the divergence goes), and Branch C (causal collapse, requiring activation depth rather than physical time as the ordering parameter).