01: C1: High-Frequency Escape and the Nonlinear UV Replenishment Chain

Splits the previous integration framework's proposition $\mathrm{Blowup}(T_\ast)\Rightarrow\mathrm{XLegalUVChain}$ into three layers: C1a (high-frequency tail escape), C1b (nonlinear UV replenishment chain), C1c (persistent triadic genealogy). C1a's Theorem 4.1 proves directly: if $T_\ast$ is the maximal blow-up time, then for every fixed dyadic cutoff $J$, $\limsup_{t\uparrow T_\ast}\|P_{>J}u(t)\|_3=\infty$ — using only the Littlewood–Paley cutoff, the Bernstein inequality, and the known critical $L^3$ blow-up criterion. C1b (Theorem 11.1) is the paper's hardest result: it first proves that at a fixed time the high-frequency tail tends to zero as $J\to\infty$ (Littlewood–Paley approximation), then combines this with C1a, recursively selecting a scale–time sequence $t_n\uparrow T_\ast$, $J_n\uparrow\infty$ such that $\|P_{>J_n}u(t_{n-1})\|_3\le\varepsilon_n$ but $\|P_{>J_n}u(t_n)\|_3\ge A_n$; it uses the Duhamel identity to attribute this gap to the nonlinear term $\mathcal N_n$, then uses the contraction property of the heat semigroup on $L^3$ (linear evolution can only shrink, never amplify, the high-frequency tail) to prove $\|\mathcal N_n\|_3\ge A_n-\varepsilon_n$ — the document stresses that this "is already a causal source statement," not merely a correlational description that "the high frequencies are large," but an explicit attribution to the N–S nonlinear term $\mathbb P\nabla\cdot(u\otimes u)$ itself. It appends an X-Integration replenishment certificate ($\mathrm{XUVRepCert}_n$, 7 guards, covering time ordering, increasing scale, a single trajectory, non-amplification under heat evolution, and so on) to ensure every step has a replayable provenance, together with a Fourier parent-scale lemma: two inputs whose frequencies are far below the output scale $2^j$ cannot directly generate a shell-$j$ output; any interaction generating a high-frequency shell has at least one parent satisfying $\max(j_1,j_2)\ge j-C_0$. C1c remains OPEN: the parent-scale support rule only proves that some parent exists, which is not enough to establish a canonical parent, a quantitatively large individual triad, a single branch continuing across all $n$, cancellation that does not break the genealogy, or the existence of a nested causal branch $(j_1,t_1)\prec(j_2,t_2)\prec\cdots$. The document closes by listing two follow-up routes: Route A extracts a quantitatively persistent parent branch directly from the aggregate output (possibly requiring paraproduct grouping, square-function orthogonality, concentration compactness); Route B bypasses canonical genealogy and instead asks whether every replenishment must pay a positive, summable coercive cost — this is exactly the direction the next round, C2, takes.