Extended Q5 Leakage-State Decoherence

T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space \[Hₑₓₜ=C²L,\] where \(C²\) is the coherent crossing sector and \(HL\) collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight \[C(t)=\|ΠPQΨ(t)\|²,\] leakage weight \[D(t)=\|(I-ΠPQ)Ψ(t)\|²,\] and gate-admission weight \[G(t)=\|ΠYΨ(t)\|²,\] where \(Π_Y\) projects onto the \(Y\)-admissible crossing mode. A key result is that visible coherence and gate alignment are distinct quantities. A state may retain nonzero amplitude in the observable \(P/Q\) sector while failing admission into the \(Y\)-crossing mode. This leads to a geometric definition of decoherence: \[DY(t)=1-\|ΠYΨ(t)\|².\] Decoherence is therefore interpreted as loss of overlap with the admissible crossing channel rather than destruction of phase information. The theorem further establishes that interference visibility is governed by the gate-admitted component of the state: \[V(t)=\|ΠYΨ(t)\|,\] yielding the extended interference law \[(P)=12(1+V(t)sin(2t)cosφ).\] Perfect visibility corresponds to complete gate alignment, while reduced visibility reflects either leakage into non-crossing modes or misalignment with the crossing sector. An extended generator \[G={pmatrix}A & -K^{}\ & B{pmatrix}\] is introduced to couple the coherent crossing sector to the leakage sector. Under the skew-adjoint conditions \[A^{}=-A,^{}=-B,\] the evolution operator \(etG\) preserves total norm, implying \[C(t)+D(t)=1.\] Leakage therefore represents redistribution of phase weight rather than irreversible loss. The theorem provides a geometric interpretation of decoherence within the Q5 framework: phase information may remain globally conserved while becoming inaccessible to coherent crossing dynamics. In this picture, coherence is determined not merely by the presence of phase amplitude, but by its alignment with the unique \(Y\)-admissible transport mode.