General Phase-Closure Difference Theorem: Angular Velocity Identities from Multi-Clock Cycle Accumulation

ABSTRACTThis paper derives a general angular velocity identity governing multi-clock systems measured over a shared closure interval. If two periodic processes complete cycle counts that differ by an integer over a fixed time interval, their angular velocities differ by an integer multiple of the interval’s fundamental angular frequency. The result unifies solar–sidereal, synodic–sidereal, and other moving-frame relationships under a single algebraic condition derived from accumulated phase closure. The theorem is purely kinematic and requires no dynamical force assumptions. Structural implications include invariant difference coordinates, deterministic beat-frequency formation, and predictable recurrence relations in multi-periodic systems.