48 / C5-L: Persistent Bad-Window Rigidity, Chain-Clock Defect Measures, and Root-Turnover Compression

C5-K compressed the genuine high-order spatial survivors into four categories: window-persistent sign defects, chain-clock separation, in-window root reversal, and theorem-setup defects. C5-L attacks them one by one, killing off most of the freedom in three of the four. **C5-L.1 Carrier-Relay Quotient Theorem**: for an arbitrarily chosen bad carrier $x_k(s)$ at every moment inside the window (no assumption of a continuous selection), the right-hand side of C5-I's descent bound $A_{k-1}(s)\ge\kappa_{\lambda,\delta}r_k(s)A_k(s)$ does not contain $x_k(s)$ at all — no arbitrary bad-carrier transfer can remove the descent strip; carrier identity can be quotiented out at the amplitude-chain level (though it still matters for questions like spatial compactness, causal lineage, or shared-core pressure geometry — C5-L does not invent a new "carrier-velocity defect" out of nowhere). Integrating the local $L^2$ cost over the entire window, **C5-L.2 Window Derivative-Load Strip** gives a genuinely carrier-independent global inequality, and the positive-measure sense root-load measure domination relation $\mu_{k-1}^R\ge d_k(c)\mu_k^R$ survives carrier transfer, weak time limits, and a moving bad core. **C5-L.3 Critical Harmonic Saturation Has Nonzero Descent Cost**: even when the window's best occupancy is pressed against the threshold, $\beta_k^{win}\downarrow\delta$, the descent coefficient $\kappa_{\lambda,\beta}=(1+\lambda)\beta-1$ still tends strictly to a positive value $\kappa_{\lambda,\delta}>0$ — harmonic critical saturation is not a zero-cost escape, only the safety margin shrinks, while the chain's downward descent pressure stays uniformly non-degenerate. The round's most important step directly reconnects in-window root reversal to the genuine Leray-projected Navier–Stokes equation $\partial_tu=\nu\Delta u-\mathbb P((u\cdot\nabla)u)$: after differentiating $k$ times, a genuine projected-nonlinear-source norm $\mathcal N_k^{proj}$ is defined (explicitly stating it does not smuggle in a Calderón–Zygmund estimate as a substitute), forcing **C5-L.4 Root-Turnover PDE Compression Theorem**: the variation of the log of the normalized chain root, $\operatorname{Var}_I\log\mathcal R_k$, is bounded above by the sum of the genuine viscous second-order cost and the projected-nonlinear cost — in-window root reversal is not a free temporal motif; unbounded reversal must pay either $D^{k+2}u$ viscous congestion or projected-nonlinear temporal forcing; if the cost is bounded, the normalized root path is in fact BV-compact (**C5-L.5**). On the clock side: defining the chain-clock total variation $\mathfrak V_{J,K}^{clock}=\sum|\Delta\log\tau_k|$, it is proved that a variation $\le\log4$ guarantees the whole block shares a common theorem moment (contrapositive: no common moment forces a variation $>\log4$, **C5-L.6**). The exact clock/root decomposition $\log\tau_k=g_k^{th}-2\log\mathcal R_k$ proves that chain-clock separation is not an independent time coordinate, but is the sum of derivative-root variation and theorem/factorial normalization drift, and can be compressed into a compact defect measure over order space (**C5-L.7**), with a greedy fourfold-synchronization grouping giving a clean block-partition bound: the number of synchronized groups $N_{sync}\le1+\mathfrak V^{clock}/\log4$ (**C5-L.8**). The round closes with two compression theorems: **C5-L.9 Persistent Window Compression** proves that persistent failure over an entire window must simultaneously carry a non-degenerate descent strip, a window derivative-$L^2$ load strip, plus one of a BV-compact root path, viscous congestion, or projected-nonlinear forcing — carrier transfer cannot remove the first two, nor can harmonic critical saturation remove the descent strip; **C5-L.10 Clock/Descent Block Alternative** proves that a whole block must either split into finitely many (bounded by the clock variation) synchronized clusters, with same-time inequalities legitimately applicable within each cluster, or else the clock variation itself is large and is recorded by the defect measure. At this point carrier transfer, harmonic critical saturation (treated as zero-cost), generic root reversal, and generic clock mismatch have all been removed from the independent survivor list, and only five classes of genuinely PDE-level defects remain: persistent spatial sign-window debt, viscous high-order reversal, projected-nonlinear reversal, order-clock variation/theorem-constant drift, and theorem-setup defects. The round's methodological close: once a published sufficient criterion carries an existential quantifier over an interval, the correct notion of survivor is not "timing mismatch," but persistent failure over the entire admissible interval, whose consequences should be integrated over the whole window rather than treated as an isolated pulse — formally turning "timing ambiguity" into "window-level PDE debt." Formally hands off to C5-M — Unified Defect-State Closure, Compatibility Graph Audit, and C5 Phase Boundary, the final round of the C5 series.