The Born Rule as a Resonance Law

This upload presents The Born Rule as a Resonance Law, a concise R-Theory (Resonance Theory) account of why quantum detection probabilities scale with the squared magnitude of a local amplitude. In R-Theory, a “click” is modeled as the formation of a localized, stable resonant knot in an underlying coherent phase-field (the zero-field, 𝕆), triggered by detector–field interaction.

The central bridge to the Born rule is physical rather than axiomatic: for wave-like phenomena, transported energy inflow (intensity/power flux) is proportional to |A(x)|^2. If a detector element behaves as a noisy threshold system whose transition (click) rate is proportional to the local inflow, then normalizing click rates across the detection surface yields the standard Born probability density P(x)\propto |A(x)|^2. Interference follows immediately because amplitudes add before squaring, while classical probability addition emerges when phase coherence is randomized and cross-terms average out.

The manuscript is intended to be compatible with standard quantum predictions while offering a concrete detector-centered interpretation and a small set of empirical “handles” (threshold tuning, controlled decoherence, detector microstructure variations) that primarily affect rates/contrast rather than the normalized |A|^2 form in the single-click-per-trial regime. Limitations and open tasks are stated, including the need for a precise definition of A(x) from 𝕆 and a more rigorous universality argument across detector models.