07 / C3-E: Viscous-Window Renewal, Phase Efficiency, and Zeno Compatibility for the Local Heterochiral Frontier

C3-D compressed the surviving core toward the local/moderately local heterochiral forward frontier (unless the nonlocal route pays an amplitude compensation or the scaling assumption breaks down). This round asks: once $k\sim p\sim q\sim\lambda$ and the nonlocal suppression vanishes, can these local triads keep sustaining consistent amplitude, phase, time window, and genealogy at ever-higher frequencies? It first establishes heat-semigroup spectral-gap decay for the high-frequency tail, then combines this with the Duhamel formula to prove the Viscous-Window Renewal Theorem (Theorem 4.1): if the high-frequency tail rises from $\varepsilon$ to $A$ over $M$ viscous windows, then at least one window's nonlinear source must itself reach the same order of magnitude — stronger than C1b's statement that "the nonlinear source must be large over some large time interval," since it pins this down precisely to a short window of order $O((\nu\lambda_J^2)^{-1})$. Combined with C1's sequence, this gives the Viscous-window UV renewal chain (Corollary 6.1). It then defines the phase efficiency $\eta_\lambda=\mathcal P_\lambda/\mathcal M_\lambda\in[0,1]$ (actual positive production over maximal amplitude capacity — a purely exact normalized diagnostic, not a turbulence closure model) and the critical local amplitude $A_q^{\rm crit}=\lambda_q^{1/2}U_q$, and uses Bernstein and Hölder to establish an upper bound on local critical production, yielding the Coherence–Amplitude Tradeoff (Theorem 13.1): $\eta_qA_q^{\rm crit}\gtrsim\nu$ — low phase efficiency must be compensated by a large critical amplitude, and vice versa. The document explicitly notes this is not a contradiction: because when $A_q^{\rm crit}\sim A$, the $L^2$ energy cost is only $\lambda_q^{-1}$, which remains summable along an exponentially growing scale sequence. The paper's most important result is a no-go (Theorem 19.1 + §20): even after proving that every generation must renew within a shrinking viscous window ($\tau_n\sim\lambda_n^{-2}$, $\lambda_n=\lambda_0r^n$), $\sum_n\tau_n<\infty$ still holds — parabolic times are geometrically summable to begin with, so "residual time compression" is not enough to rule out an infinite cascade within finite time; it only proves that if such a genealogy exists, it must exhibit Zeno-like accelerating renewal. Residence-time compression ≠ regularity proof, and this must be explicitly preserved. The document therefore points out that the genuine remaining core obstruction must incorporate the sustained legal connection of a five-dimensional space–frequency–phase–helicity–time provenance structure, formally opening the next round, C3-F: Joint Phase-Space Ancestry Obstruction, which formally folds physical-space wave-packet provenance into the ETN/X-Integration proof route.