31 / C4-D: Amplitude-to-Flux Branching Bridge, Local Work Cancellation, and Helical-Cancellation Rigidity

C4-C leaves the Amplitude-to-Flux Barrier: amplitude information alone cannot determine the sign of flux. This round no longer pursues the false direct implication, and instead proves a branching bridge on the genuine N–S shell evolution. The core is the Persistence-or-Fast-Crossing Dichotomy (Theorem 5.1): for any first crossing $\beta_0\to\beta_1$ of a critical shell, either the entire preceding viscous window maintains $a>\beta_0$ (which can be routed straight back to C4-A's persistence-synchronization mechanism), or the crossing is completed as a fast crossing within a single viscous time — so this round only needs to handle fast crossings. Positive Amplitude Variation Requires Positive Nonlinear Source (Theorem C4-D.2) uses a standard max-envelope argument to prove that, at the amplitude maximum, the viscous term $e\cdot\Delta f\le0$ cannot contribute a positive variation, so any positive $M'(t)$ must be supplied by the nonlinear source, $-e\cdot N\ge M'>0$ — this is an exact same-instant amplitude/source coupling. A fast crossing must pay a fixed positive-variation budget; splitting by source efficiency $\eta$ into good and bad regions, the bad region yields the Nonlinear Source-Overcapacity Impulse (Theorem 14.1), while the good region, via the Band-Limited Local Positive-Work Ball (Theorem 17.1, using Bernstein continuity to expand a pointwise source into a shell-scale positive-work ball), yields the Integrated Local Work Toll (Theorem 19.1) — a genuinely scale-invariant local nonlinear-work cost. The Local-to-Global Work Cancellation Identity (Theorem 22.1) proves that local positive work cannot simply vanish: if it does not become net shell input, it must leave a corresponding spatial negative work behind. Combining the three yields the round's core result, the Amplitude-to-Work Branching Bridge (Theorem 23.1): every fast crossing must pay at least one of a source-overcapacity impulse, positive shell nonlinear work, or spatial work cancellation — a scaling audit (§40-41) shows that both of these new branches are scale-critical $O(1)$, which explains why an ordinary finite energy budget likewise cannot shut them off, but this time it is genuinely hung on N–S structure rather than a generic pulse. If the positive-work branch is taken, it is further expanded by triad rank and helical class (rank deficiency, same-handed, radial-gap degeneration, robust opposite-handed). The round's single most important new theorem is Helical Cancellation Forces High-Mode Work Cancellation (§31, Theorem C4-D.7): it proves that the robust opposite-handed coupling ratio $\kappa_\tau=\mathcal R_\tau/(q_\tau\dot e_{q_\tau})\in[c_\ast,1]$ holds for either sign, so that if helical pair production is reversed and cancelled (the cancellation branch of C4-C), this necessarily produces, in the same instant, a negative high-mode energy work of comparable magnitude — helical cancellation is no longer an independent hidden escape channel; it is merely the critically weighted projection of energy-work cancellation, a stronger result than C4-C's own. The document assembles the complete amplitude-crossing branch tree (§34, eight named branches A through H), and announces the partial closure of the Amplitude-to-Flux Barrier (§35): the direct implication is still false, but the branching implication — that amplitude crossing implies a finite, structured branch set — is now proved. The document also confirms (§39) that most branches are already connected to the existing C3 dependency graph and are not isolated new open problems, and uses a pigeonhole argument to obtain the Recurrent Escape-Branch Reduction (Theorem 38.1): infinitely many critical crossings within a finite branch family force some branch to recur — so the next round can attack the rigidity of each branch one at a time, rather than handling every crossing-escape simultaneously.