Residual Surface Return in the Extended Q5 Leakage Sector

T94 extends the leakage-state framework of T93 by deriving the leading-order return dynamics of phase amplitude that leaves the \(Y\)-admissible crossing sector. Rather than treating leakage as a terminal loss mechanism, the theorem models leakage as a redistribution process in which rejected amplitude enters a secondary sector, circulates internally, and subsequently returns to the observable crossing sector. The extended state space is \[Hₑₓₜ=C²L,\] with dynamics governed by the coupled generator \[G={pmatrix}A & -K^{} \ & B{pmatrix},\] where \(K\) transfers amplitude into the leakage sector, \(K^{}\) returns amplitude from the leakage sector, and \(B\) governs internal circulation within \(HL\). The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state \(ψPQ(0)\), amplitude enters the leakage sector according to \[λ(t)=tKψPQ(0)+O(t²),\] and subsequently evolves through the internal circulation operator \[etB.\] The resulting leading-order return contribution to the crossing sector is \[R(t)=K^{}etBKψPQ(0).\] This return term is identified as the residual surface return of the extended Q5 architecture. A central result of the theorem is that residual return is not automatically restorative. The effect of the returned phase is determined entirely by its alignment with the gate-admissible crossing component \[ψ_Y(t)=Π_Y Ψ(t).\] The coherence response is governed by the real overlap \[ ψ_Y(t),R(t).\] Three regimes emerge: \[ ψ_Y,R > 0\] produces reinforcement of coherent crossing, \[ ψ_Y,R < 0\] produces dephasing, and \[ ψ_Y,R = 0\] produces a neutral contribution. The theorem, therefore, refines the decoherence picture introduced in T93. Decoherence is not interpreted as the disappearance of phase information but as the return of redistributed phase in a state that is no longer aligned with the admissible crossing mode. Misaligned return decreases gate-admission weight and reduces interference visibility even when the total norm remains conserved. Several corollaries follow. Generic residual return does not remain phase-locked to the crossing mode and therefore fails to restore coherence automatically. Exact recovery requires reconstruction of the full leakage-sector evolution \(etB\), making reversibility information-demanding even when the underlying dynamics remain unitary. Effective irreversibility is thus attributed to the information cost of reconstruction rather than to fundamental non-unitarity. T94 serves as the first formal return-transport theorem in the Q5 sequence. Together with T93, it provides a complete first-order picture of leakage dynamics: \[crossing sector\;{K}\;leakage sector\;{etB}\;surface circulation\;{K^{}}\;return transport.\] The theorem establishes that the coherence consequences of leakage are determined not by the existence of returned phase, but by the alignment of that returned phase with the unique \(Y\)-admissible crossing mode.