Born Readout in the Reduced Q5 Crossing Sector

T92 establishes that the squared-amplitude probability rule in the reduced Q5 crossing sector is not postulated but follows from unitarity, norm preservation, and additive refinement consistency within the reduced two-channel architecture. The theorem works in the reduced crossing sector\[HPQ = C^2,\]obtained after the \(5 → 3 → 2\) reduction. The reduced transport generator is\[A = iσ_y={pmatrix}0 & 1 \\-1 & 0{pmatrix},^ = -A,^2 = -I.\] Continuous transport is generated by\[U(t)=etA.\] The proof first establishes anti-Hermiticity:\[A^=-A.\] From this,\[U(t)^=etA^=e-tA=U(t)⁻¹,\]So the transport operator is unitary:\[U(t)^ U(t)=I.\] Unitarity then forces norm preservation:\[\|U(t)ψ\|^2=\|ψ\|^2.\] For normalized states\[ψ=P|P+Q|Q,|P|^2+|Q|^2=1,\]The theorem introduces an additive refinement consistency lemma. If a channel decomposes orthogonally,\[|P|^2=|P_1|^2+|P_2|^2,\]Then the probability must satisfy\[(P)=(P_1)+(P_2).\] Assuming probability depends only on channel norm weight,\[(P)=f(|P|^2),\]The refinement condition forces the Cauchy functional equation\[f(x+y)=f(x)+f(y),(1)=1.\] Monotonicity and nonnegativity force continuity on \([0,1]\), and the unique continuous additive solution is\[f(x)=x.\] Therefore, the unique norm-preserving probability assignment in the reduced crossing sector is\[(P)=|P|^2,(Q)=|Q|^2.\] T92 therefore upgrades the earlier reduced-sector Born interpretation from heuristic consistency to a mathematically constrained uniqueness statement within the two-channel architecture. Corollary 92.1 establishes channel resolution into the basis states\[|P=(1,0)^T,|Q=(0,1)^T,\]with probabilities determined by squared amplitudes. Corollary 92.2 shows that the Born assignment is preserved under Q5 transport because\[|P(t)|^2+|Q(t)|^2=1\]for all \(t\). Corollary 92.3 gives the architectural interpretation of measurement. Measurement is interpreted not as an external collapse force but as forced resolution into one admissible barrier-coupled channel. The barrier selects admissible channels, while the amplitude distribution determines the resolution probability:\[(channel C)=| C|ψ|^2.\] The theorem remains carefully bounded. It does not claim:- a derivation of the Born rule for arbitrary Hilbert spaces,- a complete solution to the quantum measurement problem,- or a derivation independent of the \(5 → 3 → 2\) reduction chain. The solid core is the reduced-sector statement:\[A^=-A(t)\ unitary preservation⇒(P)=|P|^2.\] T92 therefore provides the foundational probability justification underlying the interference law of T91 and strengthens the reduced Q5 transport interpretation by deriving the squared-amplitude rule from unitary reduced transport plus additive refinement consistency rather than taking it as a postulate.