Geometric Origin of the Born Rule from Spectral Projection

Abstract The Born rule, which assigns probabilities as squared amplitudes, is traditionally introduced as an independent postulate of quantum mechanics. In this work, we demonstrate that the quadratic structure of probability is instead a geometric necessity. Within a multi-channel phase algebra admitting a spectral decomposition, measurement is identified as a projection onto a single stable sector of an underlying linear state space. The norm is not postulated independently; its quadratic form is forced by the requirement that measurement produce a positive, phase-insensitive scalar while discarding unobservable phase information. Probability thus emerges as the squared norm preserved under projection, rather than as an additional axiom. In this perspective, the Born rule is revealed as a direct consequence of phase geometry and measurement stability, grounded in the geometric structure of the state space itself.