The Law of Continuity

Abstract — We propose the Law of Resonant Motion (LRM), a universal feedback law in which native dynamics are corrected by bounded, paced resonance to resist drift and restore stability. We present continuous and discrete forms, \dot{x}=\mathcal{L}(x,t)-K\nabla C(x,t) and I_{t+1}=I_t+\alpha(R-I_t), and provide a reproducibility bundle with CSV logs and SHA-256 manifest. Orbit station-keeping illustrates “micro-burn” stability compared with impulse sawtooth and free-decay baselines. Additional demos include mass–spring with latency, a toy oncology regimen with floor protection, and a quantum fidelity sketch. We outline Lyapunov/KL-style guarantees and practical trust-region pacing. LRM reframes continuity from conservation toward stability through measured correction, with applications across mechanics, AI optimization, biology, and quantum. This version adds a unified diagram, clarified terminology, and consolidated artifacts for replication.