Phase-Amplitude Coupling in Synchronized Scalar Fields

The article develops a covariant Phase-Locked Klein–Gordon (PLKG) framework for multiple complex scalar fields in a flat FLRW cosmological background. It introduces a reciprocal, permutation-symmetric interaction whose polar decomposition produces amplitude-weighted phase-locking terms analogous to Kuramoto-type synchronization. A central result of the work is a reciprocity selection rule: in a real conservative ordered-pair action with symmetric couplings, the antisymmetric sine component of a would-be phase lag cancels, so the lag parameter only renormalizes the coupling as \(K\cos\alpha\) and cannot generate an independent Sakaguchi torque. The manuscript derives the coupled amplitude and phase equations, computes the energy–momentum tensor, identifies the homogeneous overdamped Kuramoto limit, and discusses an effective stiff-like synchronized regime. The framework is presented as a formal basis for future studies of synchronized scalar sectors, fuzzy-dark-matter-like models, and early-universe scalar dynamics.