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 yₙ = |uₙ|/sqrt (lambda) = sqrt (1 - xi²/4) |tan Thetaₙ|, where xi: = h sqrt (lambda) and Thetaₙ 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 Thetaₙ equidistribute on 0, 2 pi). Three exact corollaries follow without asymptotic approximation: (i) Var[log y = pi²/4 (independent of h and lambda) ; (ii) median (y) = sqrt (1 - xi²/4) ; (iii) Q1 (y) * Q3 (y) = 1 - xi²/4, where Q1, Q3 are the lower and upper quartiles. Numerical verification across xi² 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 - xi²/4) 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 Zsqrt (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 10⁵ steps each.
Nicholas Maino (2026) studied this question.