Emergence of Discrete Born-Type Weights in Iterated Two-Channel Exchange Systems: A Finite-Order Recurrence Law from Wave-Packet Localization Transfer, Metastable Two-State Dynamics, and Observation Selection

This preprint proves a finite-order recurrence theorem for an exchange-symmetric two-channel unitary operator shared by numerical models of wave-packet localization transfer and metastable A/B observation selection. After fixing the global phase, exchange symmetry gives U = P_s + zeta P_a and the channel amplitudes r = (1 + zeta)/2 and t = (1 - zeta)/2. Exact recurrence U^n = I selects cyclotomic eigenphases zeta = exp(-2 pi i m/n), yielding the discrete channel-projection weights |r|^2 = cos^2(pi m/n) and |t|^2 = sin^2(pi m/n). The result was identified after unexplained numerical peaks had been observed; no Born-type objective function or finite-order root was inserted into the search code. The paper does not claim a derivation of the complete Born rule or its experimental frequency interpretation. It establishes instead that finite-order closure selects a discrete sequence of Born-type transition weights in the exchange-symmetric two-channel kernel, while channel-wise normalization, weak readout, and strong-observation backreaction remain distinct downstream maps.