51 / C6-B: High-Order Forcing Re-entry, Bad-Window Regeneration, and the H/F Cycle Test

C6-A's conclusion was that a projected SCC does not imply a composable PDE recurrent cycle; for the $H\leftrightarrow F$ that appears to exist on C5-M's coarse may-graph, C6-B formally tests the reverse edge $F\overset{?}{\Rightarrow}H$. The first step splits C5's forcing class into $F_{\rm visc}^{\downarrow}$ (viscous/decay-side reversal) and $F_{\rm NL}^{\pm}$ (projected nonlinear forcing that may either stabilize or amplify a chosen peak) — C6-B.1 (applying the maximum principle $\Delta f(x_\ast,t)\le0$ at the signed spatial maximum) proves $D^+A_k\le\mathcal N_k^{proj}$, so viscosity by itself cannot be the engine of forward derivative-peak regeneration, and C6-B.2 Viscous Half-Cycle Elimination therefore removes $F_{\rm visc}^{\downarrow}$ from the list of forward $H$ re-entry engines, leaving only $F_{\rm NL}^{+}$ as a possibility. But even if the Duhamel capacity $\mathfrak C_\ell^{Duh}$ of the projected nonlinear forcing is large, C6-B.3 Duhamel-Capacity No-Go (via a construction using a compactly supported abstract test field vanishing at both endpoints) proves that capacity alone cannot logically lower-bound the genuine response — an explicit Duhamel coherence coefficient $\Gamma_\ell^{Duh}=\|Z_\ell\|_\infty/\mathfrak C_\ell^{Duh}\in[0,1]$ must be kept, and it can equal $0$ even while capacity is $>0$. Even if the response peak genuinely is large, C6-B.4 Amplitude-to-Sign-Thickness No-Go (using a fixed bump function rescaled as $\phi_N(x)=\phi(Nx)$ as a witness) proves that peak amplitude alone does not determine the component/sign chain-scale thickness geometry — the same sup norm can carry arbitrarily different sparse/thick geometries. The coarse edge $F_{\rm NL}\to H$ must therefore be refined into a six-link chain: $F_{\rm NL}\to R_{\rm amp}\to R_{\rm select}\to R_{\rm sign}\to R_{\rm setup}\to R_{\rm persist}\to H$. The positive half: C6-B.5 One-Time Sign-Reentry Lemma proves that if the response itself genuinely is sign-thick on some set, the inherited heat part $\|Y_\ell\|_\infty\le\epsilon A_Z$ is small enough, and the threshold margin is strict ($\lambda_Z-\epsilon>\lambda(1+\epsilon)$), then the genuine derivative field really does inherit the same thick set at that instant; C6-B.6 Sign-Thickness Persistence Lemma proves that as long as the temporal perturbation $\Theta_\ell(s,t_\ast)$ is small enough relative to the margin $m$ ($(1+\lambda)\Theta_\ell0$, ensuring the theorem still selects the same component), theorem-setup legality, and persistence over the whole admissible window, C6-B.7 Conditional Nonlinear Re-entry Theorem proves that the genuinely valid edge is $F_{\rm NL}^{coh}\overset{C}{\to}H$, a conditional implication with six premises, not a coarse $F\to H$ — no current C5/C6 result can automatically supply these six premises from scalar forcing metadata alone. Net result: the coarse generic $H\leftrightarrow F$ cycle flagged by C6-A is formally ruled dead (not merely "still open," but genuinely REJECTED — this round's five no-gos, B-NG1 through B-NG5, break every link of the coarse implication one by one), and even the reverse $H\Rightarrow F$ is not an unconditional class-level edge either (C5-L's own trichotomy $H\Rightarrow H_{\rm compact}\vee F_{\rm visc}\vee F_{\rm NL}$ means the candidate cycle lives only inside the subtype $H_{\rm force}$). What genuinely survives is compressed into the much narrower C6-B.9 Typed H/F Cycle Reduction: $H_{\rm force}\to F_{\rm NL}^{+}\overset{\Gamma^{re}\text{ gates}}{\dashrightarrow}H_{\rm force}$, whose recurrence still needs five things proved (recurrent forward nonlinear source, $\Gamma^{re}$ staying non-degenerate, the target $H$ again being forcing-generated, time still being before $T^\ast$, and every generation evading the external Grujić–Xu gate) — none currently proved. The round's most important shift: the genuine global question is no longer "will the forcing get large," but whether $\Gamma^{Duh},\eta^{grow},m_{\rm sel},\beta_Z-\delta,\Pi_{\rm time}$ — Duhamel non-cancellation, peak-regeneration efficiency, component-selection margin, spatial sign-thickness margin, persistence reserve — can stay simultaneously non-degenerate over infinitely many re-entry generations, a genuine cycle-composition problem that is much narrower and more geometric than "can the nonlinearity stay large." Formally hands off to C6-C — Nonlinear Duhamel Coherence, Sign-Reentry Efficiency, and Cycle-Critical Saturation, with eight composition obligations (C1 Duhamel coherence dynamics through C8 H/F subcycle verdict).