Phase Pass-Through Condition for Reduced Q5 Transport

T83 establishes the phase pass-through condition for reduced Q5 transport. Starting from the continuous phase-flow limit established in T80,\[Tₚₕₐₛₑ^ktA=Icos t + Asin t,=kε,\]with\[A^2=-I,\]the theorem applies the normalized phase-readout functional \(L\), satisfying\[L(I)=0,(A)=1.\] It follows that\[L(etA)=L(Icos t + Asin t)=sin t.\] Thus, observable phase transmission occurs exactly when\[sin t≠ 0.\] At\[t=nπ,,\]the accumulated transport satisfies\[etA=± I,\]so the transport is diagonal/scalar in the phase readout and no phase passes through. At\[t=π/2+nπ,,\]The accumulated transport satisfies\[etA=± A,\]So the readout is purely rotational and phase pass-through is maximal. T83 is structurally important because it separates diagonal retention from observable phase transmission:\[etA=Icos t + Asin t.\]The \(Icos t\) branch retains the carrier state but is invisible to \(L\), while the \(Asin t\) branch carries the observable phase coefficient. Together with T80-T82, this completes the reduced phase-flow readout arc:\[TrawₚₕₐₛₑtA(etA)=sin t.\] Status: solid for the identity\[L(etA)=sin t\]given the T80 continuous limit and the T26-T28 normalization of \(L\); solid for the pass-through condition\[sin t≠ 0;\]conditional on the reduced phase-transport form\[Tₚₕₐₛₑ=I+ε A+O(ε^2)\]and on the T81-T82 projection-invariance bridge from raw barrier transport.