Trinity Theorem for Windowed Measurement\\[10pt] \large Born = Information Projection (iff),\\ Pointer = Spectral Minimum (iff),\\ Windows = Minimax Optimal

Under unified framework of de Branges--Kreĭn (DBK) canonical system, scattering--functional equation dictionary and Bregman/information geometry, this paper establishes ``Trinity Theorem'' for windowed measurement. Conclusions in three tiers: (I) Born = Information Projection (iff): Under orthogonal projection measurement (and its generalization to POVM), optimal probability vector induced by windowed readout equivalent to I-projection (minimal KL/Bregman cost) over family of linear alignment constraints if and only if it equals Born probability. (II) Pointer = Spectral Minimum (iff): For any distinguishable window family, let Ky Fan partial sum of ``windowed trace quadratic form'' minimize over all orthonormal bases, then if and only if that basis is spectral eigenbasis of measured observable (modulo degeneracy). (III) Windows = Minimax Optimal: Under constraint of bandlimited even window with normalization w(0)=1, taking Nyquist--Poisson--Euler--Maclaurin ``alias + Bernoulli layer + truncation'' non-asymptotic error upper bound as adversary, optimal window exists and (under Hilbert strongly convex proxy) unique, satisfying frequency-domain projection KKT condition. Key bridge is Birman--Kreĭn formula and Wigner--Smith delay giving phase derivative = spectral density, precisely connecting windowed readout with relative state density (LDOS).