Born Rule as Quantum Probability Core: Single-Photon Interference Confirms Squared Modulus Statistics — E8 Intelligence Research

FINDING: Born rule remains the probabilistic core of quantum mechanics; recent derivation attempts from measurement models; single-photon interference confirms wavefunction modulus-squared statistics. | MATH: Born rule: \(P(a) = |\langle a|\psi\rangle|^2\) (probability = squared modulus of amplitude). Single-photon interference: intensity \(I \propto |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\psi_1^*\psi_2)\) — cross-term encodes phase coherence. Derivation (arXiv:2408.06375) posits nondeterminism from measurement apparatus model, not from wavefunction collapse postulate alone. | CONNECTION: Interference cross-term \(2|\psi_1||\psi_2|\cos(\Delta\phi)\) — phase difference \(\Delta\phi\) maps to geometric phase (Berry phase) on projective Hilbert space. For equal amplitudes, visibility \(V = 2\sqrt{I_1I_2}/(I_1+I_2)\) reaches 1.0, but any deviation from Born rule would appear as non-quadratic power law \(P \propto |\psi|^\alpha\) with \(\alpha \neq 2\). No evidence of \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com