Residual Scattering Operator Theorem

TA31 converts the TA30 barrier-comparator picture into an explicit operator form. The barrier is modelled as a five-channel scattering system with Hilbert space \( H_5 = span{B_L, L, M, R, B_R} \), decomposed into a three-channel coherent triadic carrier sector \( H_M = span{L, M, R} \) and a two-channel asymmetric residual-balancing sector \( H_R = span{B_L, B_R} \). The admissible Y-gate split \( ψ_{PQ} = Π_Y ψ_{PQ} + Kψ_{PQ} \) (T96/T122), together with the three-mode helical leakage chain u_2 ↔ u_3 ↔ u_4 with no direct u_2 ↔ u_4 coupling (T127), forces a (3+2)×(3+2) block scattering structure. The barrier scattering operator takes block form S_B = [[A, C], [D, E]], where A: H_M → H_M governs coherent triadic carrier transport; D: H_M → H_R extracts phase-defective structure into residual fibres; C: H_R → H_M reinjerts circulated residual as coherent reinforcement; and E: H_R → H_R governs residual redistribution and balancing circulation. Four corollaries establish: the residual extraction channel (DM_in is the extraction term); the residual reinforcement channel (CR_in is the reinjection term); the residual circulation channel (E as the operator-level complement-sector balancing); and the coherence maintenance condition, repeated coherent transport remains stable when \( ‖DM_in‖ \) and \( ‖CR_in‖ \)remain bounded and \( ρ(E) ≤ 1 \). The spectral radius condition classifies three dynamical regimes:\( ρ(E) < 1 \) damps residual mismatch, \( ρ(E) = 1 \)gives conservative circulation, and \( ρ(E) > 1 \) produces instability and possible coherence rupture. Unitarity of S_B is not asserted. Explicit computation of block entries A, C, D, E as functions of Q5 geometry (w_H, Gray, Π_Y, K, Θ, A_L) is deferred to TA35 (geometric block-forcing definitions), TA36 (forced extraction block D), TA37 (forced circulation block E), and TA38 (forced reinjection block C). TA31 provides the structural scaffold that those theorems fill in. Physical interpretation beyond the transport formalism remains conditional.