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 sectorₐ = C², after the \ (5 3 2\) reduction. The reduced transport generator is = iᵧ=pmatrix0 & 1 \\-1 & 0pmatrix, ^ = -A, ² = -I. \ Continuous transport is generated by (t) =e^tA. \ The proof first establishes anti-Hermiticity: ^=-A. \ From this, (t) ^=e^tA^=e^-tA=U (t) ^-1, the transport operator is unitary: (t) ^ U (t) =I. \ Unitarity then forces norm preservation: \\|U (t) \|²=\|\|². \ For normalized states\=P|P+Q|Q, |P|²+|Q|²=1, theorem introduces an additive refinement consistency lemma. If a channel decomposes orthogonally, \|P|²=|P₁|²+|P₂|², the probability must satisfy\ (P) = (P₁) + (P₂). \ Assuming probability depends only on channel norm weight, \ (P) =f (|P|²), refinement condition forces the Cauchy functional equation (x+y) =f (x) +f (y), (1) =1. \ Monotonicity and nonnegativity force continuity on \ (0, 1\), and the unique continuous additive solution is (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, 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 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||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 (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.
Craig Edwin Holdway (Mon,) studied this question.