Deriving the Form of the Born Distribution from the Reproducing-Kernel Property of a Localized Odd-Harmonic Wave: Reducing the Remaining Postulates to the Squaring Rule and Randomness

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.