Short-Time Visibility Decay in the Extended Q5 Leakage Model

T95 derives the first quantitative prediction of the extended Q5 leakage-state framework by determining the short-time decay of interference visibility for a gate-aligned crossing state coupled to leakage modes. Building on the leakage architecture of T93 and the return-transport mechanism of T94, the theorem establishes that the initial loss of coherence is governed entirely by the leakage coupling strength and is independent of the internal leakage dynamics at leading order. The extended state space is \[Hₑₓₜ=C²L,\] with generator \[G={pmatrix}A & -K^{} \ & B{pmatrix},=iσ_y.\] An initially gate-aligned state \[Ψ(0)={pmatrix}ψ_0 \\0{pmatrix},Π_Yψ_0=ψ_0,\|ψ_0\|=1,\] begins entirely within the admissible crossing mode, with no initial leakage-sector occupation. The theorem first establishes that leakage amplitude enters the auxiliary sector at first order in time: \[λ(t)=tKψ_0+O(t^2).\] Consequently, the associated leakage weight is second order: \[\|λ(t)\|^2=t^2\|Kψ_0\|^2+O(t^3).\] Because the full evolution is norm preserving, the visible crossing-sector weight must decrease by the same amount. This leads directly to the short-time gate-admission weight \[\|Π_YΨ(t)\|^2=1-t^2\|Kψ_0\|^2+O(t^3).\] Taking the square root yields the principal result: \[V(t)=1-12\|Kψ_0\|^2t^2+O(t^3),\] where \[V(t)=\|Π_YΨ(t)\|\] is the interference visibility introduced in T93. Using the gate decoherence measure \[D_Y(t)=1-V(t)^2,\] The theorem obtains \[D_Y(t)=\|Kψ_0\|^2t^2+O(t^3).\] Thus, decoherence begins quadratically in time rather than linearly. Leakage amplitude appears immediately, but observable coherence loss emerges only after that displaced amplitude acquires measurable weight. A central consequence is that the leading decoherence coefficient depends solely on \[\|Kψ_0\|^2,\] the leakage strength of the initial state. The internal leakage-sector circulation operator \(B\) does not contribute to the leading-order visibility loss and enters only at higher orders through the return dynamics derived in T94. The theorem, therefore, separates the immediate coherence-loss mechanism from the subsequent circulation and return processes. Several corollaries follow. States satisfying \[Kψ_0=0\] experience no short-time decoherence from leakage. Stronger leakage coupling produces faster visibility decay, and the coefficient \(\|Kψ_0\|^2\) may in principle be inferred from the curvature of an interference fringe at short times. Combining the result with the reduced interference law of T91 shows that leakage suppresses the interference term through a quadratic visibility factor. The theorem provides the first explicit dynamical prediction of the extended leakage architecture. It demonstrates that phase information is not destroyed when coherence is lost. Rather, phase leaves the \(Y\)-admissible crossing mode at a rate determined by the leakage coupling, producing a universal quadratic onset of decoherence in the short-time regime.