Synchronization-Controlled Harmonic Fusion: A Minimal, Simulation-Ready Phase Framework for Tokamak

We present a compact, simulation-ready phase-and-control framework to steer the (de)synchronization of modes in tokamak plasmas. The model combines (i) noisy phase oscillators with phase-lag and delay; (ii) physically anchored maps for the effective coupling Keff(Δ,Γ); (iii) per-harmonic order parameters; and (iv) control laws (gradient/MPC) that track desired synchronization while penalizing actuator power and slew. Real actuators are explicitly considered (ECRH/ECCD, ICRH, NBI, RMP, pellets/kicks) through phase-response curves and delay/lag compensation. **Contributions.** (1) A minimal, control-oriented phase model with per-harmonic order parameters; (2) physically grounded Keff(Δ,Γ) tied to real tokamak actuators; (3) gradient/MPC laws with explicit delay/lag compensation; (4) a simulation recipe enabling 1:1 Arnold tongues, r(Δ,Γ) maps, and power–error trade-offs. The article is self-contained and ready to implement with Euler–Maruyama integration and on-line estimators (lock-in and phase Kalman). It is intended as a practical bridge from diagnostics to actions for mode management in high-performance operation.