We introduce a minimal, mathematically controlled modification of the classical action principle that embeds a small, divergence-free field \(f_\mu (x)\) into the Euler–Lagrange equations primarily through modifications in electrodynamics. This modification preserves locality, causality, and charge conservation while generating controlled, small deviations from standard classical trajectories, providing a unified, quantitative framework for mild classical indeterminism. The field \(f_\mu\) is Lorentz-covariant and characterized by a physically motivated spectral density, ensuring consistency across particle, scalar, and gauge systems. To leading order, we derive corrected forces, compute ensemble-averaged trajectory shifts, and identify spectral signatures accessible to high-precision experiments such as Penning traps and cathode beams. With ultraviolet-regularized spectra, the predicted deviations lie within current experimental sensitivity, establishing a direct bridge between foundational theory and empirical testability.
