Rotational Inertia under Euclidean Variation

Abstract

We propose a mathematical mechanism for quantum fluctuations based on Euclidean self-return. A complex metric path connects Lorentzian and Euclidean sections without becoming degenerate. A Euclidean return potential and positive configuration-space inertia generate repeated metric-phase motion. After projection and mixing, this motion defines a centered process with finite Green–Kubo action \(\mathcal A_*\). Under a causal, unit-gain response assumption, a particle of mass \(m\) acquires the diffusion coefficient \[ D_{\mathrm q}(m)=\frac{\mathcal A_*}{2m}. \] Within time-symmetric stochastic mechanics, the same action scale produces the Bohm quantum-potential form and the fluctuation bound \[ \Delta x\,\Delta p_{\mathrm{fl}}\geq\frac{\mathcal A_*}{2}. \] These results motivate the conjecture \(\mathcal A_*=\hbar\). The proposed mechanism is conditional on the stated mixing and response assumptions and does not treat microscopic white noise as fundamental. **Keywords** Euclidean variation; rotational inertia; complex metrics; quantum fluctuations; stochastic mechanics; Green–Kubo relation; quantum potential; Planck constant.