2D Pendulum-Wave Resonance Dashboard: Exact Return-Time Math

This 2D companion turns the classic pendulum-wave apparatus into a return-time instrument: instead of only watching the row swing, it exposes the arithmetic that makes the resynchronization exact. Three sliders — pendulum count N, return time T and starting oscillation count N0 — feed directly into L_n = g(T/(N0+n))^2/(4 pi^2), every pendulum is integrated independently by RK4 from the real nonlinear equation of motion, and a live coherence order parameter plus a polar phase plot make the collapse-and-recover of phase alignment measurable rather than just visible, with a per-pendulum click-to-inspect readout confirming the exact oscillation count driving each length.