Principle of Action--Coherence (PAC v2.3.1)

Principle of Action–Coherence (PAC v2.3.1) This work introduces the Principle of Action–Coherence (PAC), a universal and falsifiable law that links energy, time, and coherence across all physical systems, from quanta to black holes. Unlike previous formulations where Planck’s constant $\hbar$ is treated as an external postulate, PAC shows that $\hbar$ emerges naturally as the minimal “currency of coherence” required by the wave–particle duality. Key contributions: 1D Theorem (rigorous): Derived from Fourier–Cauchy–Schwarz, showing $\sigma_E \sigma_t \ge \hbar/2$. 4D Conjecture (motivated): A covariant generalization using Quantum Fisher Information and Cramér–Rao bounds. Universal entropic index: A shape-free measure, valid even for pathological distributions. Decision rule: A unique operational test; violations at $5\sigma$ constitute a falsification. Experimental checks: Mössbauer spectroscopy, transform-limited lasers, and gravitational-wave ringdowns confirm PAC’s predictions. Supplementary Note further develops: PAC-4D under standard assumptions, the universal entropic form, robustness to multimodal and heavy-tailed distributions, a unified decision rule ensuring operational falsifiability. In summary, PAC v2.3.1 provides the first operational, falsifiable, and universal coherence law, establishing $\hbar$ not as a postulate but as the fundamental cost of maintaining wave–particle duality.