Conditional Uniqueness of the Amplitude-Square Form

T63 establishes a conditional uniqueness theorem for the interference and amplitude-square structures developed in T61 and T62. Starting from the reduced two-channel interference form,\[Pₜₒₜ=P_1+P_2+G(P_1,P_2,ΔΩ),\]The theorem analyzes the structural constraints imposed by symmetry, continuity, axis-vanishing behaviour, degree-1 homogeneity, and first-harmonic rotational structure generated by the reduced antisymmetric operator algebra. Under these assumptions, the interference contribution is forced into the normal form\[G(P_1,P_2,ΔΩ)=2√P_1P_2\,sin(ΔΩ),\]yielding the exact amplitude-square representation\[Pₜₒₜ=|√P_1+eiφ√P_2|^2.\]The theorem therefore shows that, within the stated structural class, the Born-compatible amplitude-square form is uniquely selected by the reduced interference geometry together with the harmonic structure induced by the rotational generator satisfying \(A^2=-I\). T63 does not derive full quantum mechanics, measurement theory, Hilbert-space axioms, or probabilistic interpretation from first principles. The result is a conditional uniqueness theorem for the algebraic normal form of two-channel interference under the stated assumptions. In particular, the homogeneity assumption is essential: without it, uniqueness does not hold. The theorem further clarifies that the complex amplitude representation is obtained from the interference structure rather than independently postulated. Status: solid for the conditional uniqueness of the amplitude-square normal form under the stated symmetry, homogeneity, and harmonic assumptions; conditional on the separability and structural assumptions defining the interference class; speculative for any claim of a complete derivation of quantum probability theory or the Born rule from Q5 geometry alone.