34 / C4-G: Cross-Congestion Synchronization, Operator Funnel, and UV Phase-Space Closure

C4-F left three forms of congestion: $C_{TP}$ tail/wrap, $C_{DO}$ deformation/operator, $C_{RI}$ radial-interaction. This round asks whether these three forms of congestion can be fully independent of one another. The answer is no. The document first traces each motif's origin: higher-frequency relay ($M_4$) has two sources — Relay-S (from source overcapacity) and Relay-W (from rank-deficient positive work). The Source-Impulse to Growing Deformation-Impulse Theorem (Theorem 7.1, via an annular Bernstein lower bound combined with a symmetric-gradient Korn-type lower bound) proves that source overcapacity must co-occur with a genuine $L^1_tL^2_x$ deformation-driving impulse whose lower bound grows like $\lambda_q^{1/2}$ — strong evidence of congestion, though Leray energy theory currently supplies no corresponding finite global budget for it. This gives that both Relay-S and Relay-W automatically imply deformation/operator congestion (Theorems C4-G.2, C4-G.3), while spectral-geometry degeneration ($M_6$) was, from the outset, itself a sub-case of the positive-work branch, and so likewise automatically implies it (Theorem C4-G.4). Combining the three yields the round's core result, the Cross-Congestion Funnel (§15): $M_4\subset C_{TP}\cap C_{DO}$, $M_5\subset C_{DO}$, $M_6\subset C_{RI}\cap C_{DO}$, so $M_4\vee M_5\vee M_6\Rightarrow C_{DO}$ — $C_{DO}$ is the universal forcing funnel; $C_{TP}$ and $C_{RI}$ are no longer independent exits running parallel to it, but merely phase-space coordinates attached to the same deformation/operator event. The document additionally proves that far-field relay carries an exact near-anti-parallel Fourier geometry (Theorem 18.1, via the reverse triangle inequality plus the law of cosines): the far-field parents' kinetic-energy magnitudes must be comparable and their directions must be close to anti-parallel, with the angular aperture shrinking as $O(2^{-L})$ as the relay gap $L$ grows — relay is itself a form of angular/radial interaction concentration. The round's most important achievement is the Universal Deformation-Funnel Theorem (Theorem C4-G.6): every critical UV crossing must fall into one of four synchronization channels — persistence, low strain/vorticity, positive helicity production, or deformation/operator driving — compressing the entire UV side further, from eight branches to six motifs to three congestions, down to four synchronization channels. The document then uses C3-P/Q's exact operator identities to split the fourth channel further: G-O1 Miller-operator impulse, G-O2 vorticity-quadratic impulse, G-O3 advection/sweep-deformation impulse. G-O1 in turn yields the Operator-Ratio or Higher-Derivative Impulse dichotomy (Theorem C4-G.7): either the genuine Miller-escape ratio stays elevated for a comparable proportion of time (a genuine necessary condition approaching Miller-style blow-up), or the impulse must instead be paid by $\int\|\Delta S\|_2dt$, feeding directly into the higher-derivative/derivative-chain geometry of C3-W/X/Y. The document also honestly retains the sweep warning of §35: a purely spatial-translation-style sweep can make the advection operator large without directly producing local strain growth, so C3-O's guard that an equilibrium fixed point is not the same as a dynamical fixed point must be kept, and must never be silently treated as positive strain-energy production. The UV side finally reaches a clean phase closure (§39): UV crossing no longer has a genuinely isolated recurrent escape; the frontier formally shifts from splitting UV branches to Operator-to-Gate Closure — whether this forced deformation/operator event can genuinely approach some regularity gate.