Exact Half-Cauchy Emergence in Stormer-Verlet Integration of the Harmonic Oscillator via Kronecker-Weyl Equidistribution

We prove that the discrete trajectory generated by Stormer-Verlet (leapfrog) integration of q'' = -lambda q with constant lambda > 0 and stable step h (h sqrt(lambda) < 2) produces, under uniform sampling, an observable y_n = |u_n|/sqrt(lambda) = sqrt(1 - xi^2/4) |tan Theta_n|, where xi := h sqrt(lambda) and Theta_n is the phase angle in the canonical coordinates that diagonalise the leapfrog monodromy. The phase advances by exactly phi = 2 arcsin(xi/2) per step. By the Kronecker-Weyl equidistribution theorem, when phi/pi is irrational the angles {Theta_n} equidistribute on [0, 2 pi). Three exact corollaries follow without asymptotic approximation:(i) Var[log y] = pi^2/4 (independent of h and lambda);(ii) median(y) = sqrt(1 - xi^2/4);(iii) Q1(y) * Q3(y) = 1 - xi^2/4,where Q1, Q3 are the lower and upper quartiles. Numerical verification across xi^2 in {0.01, 0.09, 0.25, 0.49} confirms all three predictions to better than 0.3% on the median and Q1*Q3 values. This result closes Open Problem 1 of (Maino, 2026; Paper 5, DOI: 10.5281/zenodo.20031926) for the constant-coefficient autonomous case. The empirical observation reported there (that the half-Cauchy / silver-ratio statistics emerge from leapfrog integration of omega'' + Q(t) omega = 0) becomes a theorem with an explicit finite-step correction sqrt(1 - xi^2/4) that the empirical work missed. The silver-ratio quartiles sqrt(2) +/- 1 are derived directly from the half-Cauchy CDF via tan(pi/8) and tan(3 pi/8), partially closing Open Problem 3 of Paper 5; the deeper connection to the fundamental unit of Z[sqrt(2)] is observed but not derived from the integrator dynamics. The argument extends to N decoupled modes via the multi-frequency Kronecker-Weyl theorem (Proposition 4.1). Time-dependent Q(t) and coupled multi-period Calabi-Yau Picard-Fuchs systems (Open Problem 2 of Paper 5) are explicitly left open. The diagonal multi-mode case proved here gives no chi(M) correction; any topological dependence in the conjectured corrected variance must arise from genuine inter-period coupling and is not implied by the present analysis. All algebraic claims are verified by SymPy with residuals identically zero. Numerical experiments use double-precision velocity Verlet runs of 5 x 10^5 steps each.