A Dynamical Bridge from Tearing-Mode Torque Balance to Forced Kuramoto Dynamics: Implications for Aperiodic Phase Structure and Mode Locking

Paper 12 in The Geometry of the Critical Line programme. Mode locking — the loss of rotation of a magnetohydrodynamic (MHD) instability relative to the laboratory frame — is a dominant precursor to major disruptions in tokamak plasmas. We show that the overdamped torque balance governing a tearing mode interacting with an error field reduces exactly to the Adler equation, the canonical first-order phase-locking model that is itself the single-oscillator limit of the forced Kuramoto network. This establishes a precise dynamical bridge between plasma mode-locking physics and the theory of coupled oscillator synchronisation. Using this bridge, we reinterpret a recent computational result: deterministic aperiodic spatial phase offsets in a forced Kuramoto network suppress synchronisation lock-in via spectral smearing. The structural identity between the Fitzpatrick and Kuramoto equations motivates the hypothesis that aperiodic perturbations to the safety-factor profile q(r) — which enter the torque balance as spatially varying phase offsets across resonant surfaces — can suppress tearing-mode locking by an analogous mechanism. We propose a spectral-flatness diagnostic for evaluating q-profile designs against locking susceptibility. We extend the result to the multi-mode case, showing that coupled tearing modes at distinct rational surfaces form a heterogeneous forced Kuramoto network in the overdamped limit. We also identify four structural correspondences between mathematical objects from an independently developed geometric framework and recognised plasma-physics invariants, graded by rigour from formal equivalence to open conjecture. The result is a mathematically exact bridge, a testable spectral-flatness diagnostic, and a compact set of open problems connecting oscillator synchronisation theory to plasma stability.