Quantum Statistics from Oscillatory Sampling

Abstract

We show that the Born rule and quantum interference emerge within an oscillatory field model analysed with standard signal processing and detection theory. This is a detection-theoretic reinterpretation—an account of why measurement probabilities are quadratic in amplitude given a specific physical model of detection—rather than an axiom-free or model-independent derivation in the sense of Gleason's theorem. Physical particles are modeled as coherent patterns in an underlying oscillatory field, with measurement formalized as finite-window demodulation followed by threshold detection in the presence of noise. The detection probability P ∝ |Ψ|² emerges as the leading-order term in a Taylor expansion, with explicit higher-order corrections O(|Ψ|⁴) that are detector-response artefacts and provide falsifiable predictions. Quantum interference arises automatically from superposition of same-frequency components; the Heisenberg uncertainty relations are equivalent to the Gabor limit from signal processing; and the Schrödinger equation emerges as the non-relativistic envelope dynamics of an oscillatory field satisfying the Klein-Gordon equation. This paper addresses single-system detection statistics only; multi-particle entanglement and Bell inequality violations require additional theoretical structure not claimed here.