30 / C4-C: Carrier Relay, Shared-Event Coupling, and the Amplitude-to-Flux Barrier

C4-B proved that generic turnover cost is not enough to force synchronization. This round's strategy formally pivots to Shared-Event Synchronization: rather than asking whether A and B can recur separately, it asks whether there exists some genuine N–S event whose single source, single balance, single triad algebra already forces A and B to be paid simultaneously. The core tool is helical-triad algebra (§4-9): for a single Fourier triad $k\le p\le q$, combining energy conservation with helicity conservation directly locks the derivative vector into a fixed form, splitting into four helical classes (I same-handed, II-IV opposite-handed). C4-C.2 proves that for every opposite-handed class, whenever the highest mode's energy gain $\dot e_q>0$, there is positive critical pair production $\mathcal R_\tau>0$ — Class IV is even an exact perfect coupling, $\mathcal R_{IV}=G_{IV}^q$. The coupling constants for Classes II/III degenerate to zero as the radial gap $q-p\to0$, but within the robust regime of locally comparable triads plus a non-degenerate radial gap, a genuine positive lower-bound coupling constant can be obtained (Theorem 12.1). This yields the High-Mode Energy Gain branching edge: high-mode energy gain forces same-handed carriers, or radial-gap degeneration, or net positive helicity production, or helical cancellation — the first genuine N–S helical shared-event branching edge in C4. But the round's most important guard is in §19: the C1/C3-G UV anchor is an amplitude/norm event, not an energy-gain/flux event, and the two cannot simply be equated. The Phase-Rearrangement Norm–Flux No-Go (Theorem 20.1) confirms by explicit construction that this distinction is real: fixing every mode's amplitude $|a_m|$ and changing only the phases leaves $\|u_\theta\|_2$ unchanged by Parseval's theorem, but $\|u_\theta\|_\infty$ can swing dramatically as phases align or cancel — there exists a smooth phase path along which the $L^2$-norm derivative is zero while the $L^\infty$-norm keeps increasing, proving that amplitude information alone cannot algebraically determine the sign of shell energy flux; this is the Amplitude-to-Flux Barrier. The positive, compensating result is that although UV amplitude cannot control flux, it can, via the Bernstein inequality, precisely control the critical stock at that same instant — UV Amplitude to Critical Helical Stock (Theorem 23.1) and to Strain/Vorticity Stock (Theorem 24.1) — C4's first genuinely non-asynchronous shared edge, though it is stock synchronization, not production synchronization. On the strain side, the Local Strain-Growth trichotomy edge (Theorem 28.1, a pigeonhole argument over an exact three-term sum) proves that a local strain-growth event forces pressure, or Betchov boundary flux, or positive vorticity stretching; chaining this to C3-N's exact local Betchov identity and the pointwise vorticity-stretching geometric decomposition expands it into a four-way exact edge (pressure, or Betchov, or middle-strain-weighted vorticity, or principal-stretching-aligned weighted vorticity) — currently C4's cleanest multi-step exact local edge. On the operator side, the Miller Operator-Source trichotomy edge (Theorem 37.1, a triangle inequality) proves that operator escape forces the advection term, or strain-squared, or the vorticity quadratic term to bear it simultaneously — it cannot be swapped via carrier relay for no source at all; but it also flags the Operator Cancellation Debt (§39): when the vorticity-quadratic source is large but the full Miller operator is small, the shortfall must be absorbed as cancellation by the remaining two terms. The document assembles C4's first-version shared-event closure graph (§41) and four genuinely minimal synchronization subsets (§48), and honestly lists five explicit no-gos (amplitude does not imply energy gain; energy gain does not imply net helicity production; critical stock does not imply critical production; strain growth does not imply Miller escape; a large local vorticity-quadratic source does not imply a large full operator). The next round formally locks onto the two most critical remaining gaps: the Amplitude-to-Flux Bridge and Helical-Cancellation Rigidity.