Gate Projector and Leakage Complement in the Reduced Q5 Crossing Sector

T96 provides the explicit matrix realization of the gate-admission and leakage operators that were introduced abstractly in T93–T95. The theorem establishes a canonical orthogonal decomposition of the reduced Q5 crossing sector \[HPQ=C²\] into a \(Y\)-admissible crossing mode and its leakage complement. This converts the leakage architecture from an abstract projector framework into an explicitly computable model. The admissible crossing mode is defined as \[|Y={1}{√2}{pmatrix}1\{pmatrix},\] representing the quarter-turn phase relation associated with the \(Y\)-gate. The corresponding gate projector is \[Π_Y=|Y Y|=12{pmatrix}1 & -i\ & 1{pmatrix},\] and the leakage complement is \[K=I-Π_Y=12{pmatrix}1 & i\\-i & 1{pmatrix}.\] The theorem proves that both operators are orthogonal projectors satisfying \[Π_Y^2=Π_Y,^2=K,Π_YK=KΠ_Y=0,\] yielding the canonical decomposition \[HPQ=span\{|Y\}⊕(Π_Y).\] Every reduced crossing state \[ψ={pmatrix}P\{pmatrix}\] admits the unique decomposition \[ψ=Π_Yψ+Kψ,\] with norm identity \[\|ψ\|^2=\|Π_Yψ\|^2+\|Kψ\|^2.\] The theorem further derives explicit formulas for gate admission and leakage strength. The crossing-mode amplitude is \[ Y|ψ=1{√2}(P-iQ),\] and therefore the gate-admission strength is \[C_Y(ψ)=\|Π_Yψ\|^2=12|P-iQ|^2.\] For normalized states, \[D_Y(ψ)=\|Kψ\|^2=1-12|P-iQ|^2,\] so that \[C_Y(ψ)+D_Y(ψ)=1.\] Gate coherence and leakage are therefore complementary quantities. Several structural consequences follow. A state crosses cleanly if and only if \[Kψ=0,\] equivalently \[ψ=c{pmatrix}1\{pmatrix}\] for some complex scalar \(c\). Any deviation from this quarter-turn phase relation produces a nonzero leakage component. The theorem also provides the explicit realization of the abstract leakage operator appearing in T93–T95. Quantities such as the leakage norm \[\|Kψ_0\|^2\] and the short-time visibility decay derived in T95 become directly computable from the crossing amplitudes \(P\) and \(Q\). The projector formalism, therefore, acquires a concrete matrix representation suitable for explicit calculations. T96 serves as the foundational algebraic theorem of the leakage program. It identifies the admissible crossing mode, constructs the associated projector and leakage complement, establishes the orthogonal decomposition of the reduced crossing sector, and converts the abstract coherence and leakage measures of T93–T95 into explicit computable formulas.