Forced Residual Reinjection Block

TA38 derives the reinjection block \(CB: H_R→ H_M\) of the barrier-scattering matrix from the Q5 admissibility architecture. Given the residual state circulated by TA36 and TA37, only the component of residual return lying in the \(Y\)-admissible gate subspace may re-enter the coherent carrier sector. This forces the reinjection block to take the form\[CB=Π_Y\, Uᵣₑₜᵤᵣₙ|H_R.\]The theorem also gives the exact nonvanishing condition: reinjection occurs precisely when the residual return transport has nonzero overlap with \(im(Π_Y)\). The resulting carrier output consists of direct carrier transport plus a lagged residual reinforcement term,\[ψ_Mᵒᵘᵗ(t)=ABψ_M+CBe^{tEB}DBψ_M.\]Together with TA36 and TA37, this completes the forced off-diagonal block trilogy: extraction, mediated circulation, and admissible reinjection. The result shows that the residual sector of the barrier-scattering matrix is geometrically constrained by the Q5 architecture rather than introduced as a phenomenological model. Status: solid for the forced structure of \(CB\); conditional for nontrivial reinjection pending explicit computation of the leakage generator \(A_L\).