32 / C4-E: Recurrent Escape-Branch Rigidity, Transport-Free Source Routing, and UV Motif Compression

C4-D compressed critical-shell crossings into eight branches. This round's task is not to add yet more branches, but to compress the mutually-reducible branches into a finite number of recurrent structural motifs. The core tool is transport-free source routing: define $R_q^\sigma=N_q^\sigma-v_q\cdot\nabla f$ (subtracting off the low-mode pure-transport velocity field $v_q$), and prove two exact identities — Pure Transport Does Not Drive the Sup-Norm Maximum (Theorem C4-E.1: at the amplitude maximum, $\nabla|f|=0$, so the transport term vanishes exactly) and Pure Transport Does Not Drive Global Shell Energy Work (Theorem C4-E.2: because $v_q$ is divergence-free, $\int f\cdot(v_q\cdot\nabla f)=0$ holds exactly) — so amplitude sources and shell energy work are in fact both driven by the same transport-free deformation/cross-scale remainder, sharper than C4-D's direct use of the full $N_q^\sigma$. A Bony/commutator decomposition (Theorem 9.1) splits $\|R_q^\sigma\|_\infty$ into a low-mode deformation load times a comparable shell envelope, plus a high-high frequency congestion term. In the regime combining front-side safety, a fixed hysteresis ratio, and a sufficiently small threshold, the Source-Overcapacity Routing Theorem (Theorem 13.1) proves that source overcapacity must route to either the low-mode shear branch (E-SHEAR, a genuine $O(1)$ critical vorticity event at the same derivative level as BKM/Cheskidov–Dai) or the high-high-frequency branch (E-HH). The Small-Threshold Far-Relay Theorem (Theorem 17.1) further proves that under a small threshold, the nearby high-high-frequency capacity is only $O(\beta_1^2)$, not enough to pay for an $O(\beta_1)$ crossing impulse, so high-high-frequency congestion must draw on a strictly higher frequency $p\ge q+L$ — this unifies C4-D's rank deficiency with this round's source overcapacity into a single structural motif: Higher-Frequency Relay (§20), the round's first genuine motif merger. On same-handed triads, exact algebra proves $-\dot e_p=\dot e_k+\dot e_q$ (§22-23): same-handed high-mode gain is not a one-way UV transfer — the smallest mode $k$ gains simultaneously within the same event. The Homochiral Gap-or-Reverse-Co-Gain Lemma (Theorem 24.1) proves that local same-handed gain must lead either to radial-gap degeneration or to comparable-magnitude low-mode co-gain (a Bidirectional Critical Work Split) — same-handedness is therefore absorbed into spectral-geometry degeneration or critical work variation, and is no longer an independent branch. The document unifies Class II nonlocal gap collapse, Class III near-equilateral radial concentration, and same-handed gap collapse into the Spectral-Geometry Degeneration Motif (§29); and unifies spatial work cancellation, the robust helical cancellation already proved in C4-D (which forces negative high-mode work), and same-handed bidirectional splitting into the Critical Work-Variation Motif (§31) — again noting that this matches the same honesty limitation seen repeatedly across C3/C4: an ordinary energy balance controls only the net value $W^+-W^-$, not the total $W^++W^-$, and there is currently no finite unweighted budget for it. The round's core result is the UV Recurrent Motif Compression Theorem (Theorem 37.1, under explicitly flagged front-side hypotheses): every critical UV shell crossing must fall into one of six motifs — three closure-friendly motifs (persistence, UV–low-strain synchronization, UV–helicity-production synchronization, all three already synchronization successes rather than escapes) and three genuinely unresolved escape motifs (Higher-Frequency Relay, Critical Work Variation, Spectral-Geometry Degeneration). The next round formally attacks the trilemma formed by these three.