We regard the constant-amplitude odd-harmonic sum SN (isolated peak wave) on the half-wavelength phase interval as the localized kernel of observation, and adopt two premises: (i) observation is the convolution of the kernel with phase differences and the probability is the square of the observed amplitude; (ii) phase differences finer than the band are ignored (finite-N truncation). Because SN is the truncated reproducing kernel of the odd-harmonic basis on this interval, the observed amplitude reproduces any band-limited base wave without distortion, and the observed distribution coincides exactly with |psi|² (after normalization). For a complex base wave the squaring becomes a genuine modulus |Z|² = Z conj (Z) (not Z²), reaching the real-imaginary cross structure of the Born square. The paper derives the FORM of the distribution (faithful reproduction by the reproducing-kernel property) ; the squaring rule and the probability interpretation remain postulates. It does not claim to derive the Born rule or solve the measurement problem; rather it reduces the remaining postulates to the squaring rule and the existence of randomness. The reproducing-kernel identity itself is a classical fact of Fourier analysis (Dirichlet kernel / RKHS) ; the novelty is the mapping between the physical localized wave, its reproducing kernel, and the observation model. All results are reproduced to machine precision by the accompanying verification code.
Noriaki Kihara (Sat,) studied this question.