Theorem: n_AR^2 - n_1 n_2 = 0 (AR geometric-mean identity)

Theorem (AR-quarter-wave-residual canonical): n_AR = sqrt(n_1 n_2) squared yields n_AR^2 = n_1 n_2 exactly. Canonical sympy pins: n1, n2 = sp.symbols('n1 n2', positive=True); n_AR = sp.sqrt(n1*n2); residual = sp.simplify(n_AR**2 - n1*n2) =…