Inversion-Comparator Residual Scattering Theorem

TA30 is a structural synthesis theorem unifying the post-T90 barrier machinery into a single operator interpretation. Earlier theorems established admissible Y-gate projection, complementary leakage sectors, antisymmetric transport survival, helical leakage propagation, lag-return phase bias, and residual phase redistribution as separate results. TA30 compresses these into one operative principle: the barrier acts as a coherence-selective inversion comparator rather than merely a separator. The full transport pipeline (T97) evolves the fibre state as \( ψ_out = D_Q M_Q Π_Y R D_N M_N ψ_full \), acting on states in \( H_{5×5} \). Every reduced transport state decomposes into an admissible crossing component \( Π_Y ψ_{PQ} \)and a complementary residual component \( Kψ_{PQ} \). From T88/T89, symmetric dressing contributions cancel under inversion conjugacy \( (S_n + S_d = 0) \), so only antisymmetric, order-sensitive transport structure survives barrier comparison. Residual transport propagates helically through the three-mode chain u_2 ↔ u_3 ↔ u_4 under the adjacency operator \( A_L = (1/√80)·[phase matrix] \) (T127), with no direct u_2 ↔ u_4 coupling. The middle mode acts as a mandatory balancing intermediary. Residual return (T94/T121) induces lag-phase observable correction \( δP = κsin(2θ)Δ_Y(t)cosφ \), with coupling strength \( κ = 1/√40 \) closing consistently with the derived transport normalization (T126). The residual scattering architecture is formalized through a block operator S_B = [[A, C], [D, E]], where A governs coherent carrier transport, D extracts phase-defective structure into residual fibres, C reinjerts circulated residual into coherent transport, and E governs residual redistribution and balancing circulation. The spectral radius of E classifies the dynamical regime: ρ(E) < 1 damps residual circulation, ρ(E) = 1 gives conservative circulation, ρ(E) > 1 produces rupture. The complement sector can function as a stabilizing coherence reservoir provided ρ(E) ≤ 1. The central structural result is that coherent transport is stabilized not by annihilating phase incompatibility but by exporting it into dynamically active residual-balancing fibres. The complement sector is not discarded leakage but an active balancing ecology. Explicit computation of block entries A, C, D, E from Q5 geometry is the primary remaining open computation, addressed in TA31–TA35. Physical interpretations (coherence-selective scattering, open-system phase stabilization, structured decoherence redistribution) remain conditional.