Sharp Safe-Time Measures for Additive Frequency Triples: Sixteen Certified Sharp Bounds and Rational Threshold Geometry

For positive integers a, b, we study the measure Fᵟ(a, b) of times t ∈ [0, 1] for which ‖at‖, ‖bt‖, and ‖(a + b)t‖ are at least δ. We certify sixteen sharp inequalities F₁⁄q(a, b) ≥ (q − 3)(q − 4)/[q(q − 1)] for integers 5 ≤ q ≤ 20, including F₁⁄₁₆(a, b) ≥ 13/20. The proof combines an exact two-torus mean of 169/256, an equispaced-grid discrepancy estimate, and exact finite certificates covering 129,568 primitive pairs over the sixteen thresholds. For every integer q ≥ 5, we additionally determine the exact measure of the extremal candidate (1, q − 2, q − 1) and give a finite-verification reduction for the corresponding global inequality. We distinguish this unconditional parametric evaluation from the conjectural all-q global minimum. For all sixteen thresholds, an exact interval-union implementation was executed and archived; a distinct midpoint-boundary implementation agrees on all q = 16 cases. We also prove a one-sided continuous-threshold lower envelope, rational-breakpoint geometry, and effective isolation of the extremal primitive pairs.