Color--Phase Separation for Mixed Random Operators in Two-Speed Stochastic Klein--Gordon Systems

We study a two-component stochastic Klein--Gordon system on $\mathbb T^3$ with fixed distinct speeds and pure cross interaction $u_1u_2$. The mixed paracontrolled operators $$I_i(w<\Psi_j)\circ \Psi_k$$ are organized by color--phase separation: the pair $(j,k)$ determines the Wick or covariance contraction, while the pair $(i,j)$ determines the Duhamel--source phase gap. In the pure-cross graph, same-color contractions occur only in different-phase channels and become Fourier-diagonal Volterra multipliers; the remaining centered kernels are controlled as operator-valued second Gaussian chaoses by row/column tensor estimates. This yields a stochastic enhanced-data construction, a local paracontrolled solution map, and canonical Galerkin convergence. The result covers diagonal independent noises and Fourier-diagonal weak covariance. For $12/13<\alpha<1$ the deterministic map uses a fractional Klein--Gordon Strichartz estimate proved here, while the endpoint $\alpha=1$ uses the classical conic wave/Klein--Gordon package.