Amplitude and Phase Dynamics in Synchronized Scalar Fields: The Phase-Locked Klein–Gordon Model

This paper establishes a *Reciprocity Selection Theorem* for conservative quadratic theories of N complex scalar fields propagating in a flat Friedmann–Lemaître–Robertson–Walker (FLRW) background. Using a four-way decomposition of a general complex ordered coupling matrix—into symmetric-real, symmetric-imaginary, antisymmetric-real, and antisymmetric-imaginary parts—the theorem proves that real reciprocal couplings (Cᵢⱼ=Cⱼᵢ) eliminate the odd sine contribution to the polar interaction. Consequently, an apparent phase lag α can only dress the mass-squared coupling via κ≡ Kcosα and cannot generate a genuine Sakaguchi torque in a closed variational framework. The work derives four conservative coupling families: (I) the democratic Phase-Locked Klein–Gordon (PLKG) model as a rank-one exactly solvable realization; (II) general real symmetric weighted Kuramoto coupling; (III) complex symmetric coupling, which projects back to the real symmetric case; and (IV) Hermitian chiral coupling with an antisymmetric imaginary part as the minimal conservative route to displaced locking, subject to loop-closure constraints. For the democratic PLKG model, the full covariant amplitude–phase equations, stress–energy tensor, synchronization gap Msync²=κ, stability window, Fourier-space extension, and wavelet-based multiscale generalization are developed. The stiff-like equation-of-state regime wsync 1 is identified as a phase-kinetic limit requiring coherent dominance over amplitude and potential contributions. Numerical estimates for fuzzy/ultralight dark matter scales and cosmological synchronization efficiency are provided. The framework is formal and parametric; observational viability and perturbative constraints are identified as directions for future work.