A Certified Finite Refinement for Simple and Distinct Zeros of the Riemann Zeta Function

We give a certified finite refinement of the general-window Gabor/rank–trace framework for zeros of the Riemann zeta function. Using the arbitrary-window analytic input of Alpöge and Furman, together with an existing public 17-term window and banded-Gram defect structure, we introduce a new exact rational seven-point pressure certificate and verify it globally by fail-closed interval arithmetic. The resulting certified local floor is ε0=0.0082158542.\\varepsilon_0=0.0082158542. Combined with the banded Gram estimate and a coefficient-faithful shifted-block assembly, this yields lim inf⁡T→∞N0s(T,2T)N(T,2T)>0.67342504425,\\liminf_{T\\to\\infty} \\frac{N_0^s(T,2T)}{N(T,2T)} > 0.67342504425, for the proportion of simple zeros on the critical line, and lim inf⁡T→∞Nd(T,2T)N(T,2T)>0.83671252212,\\liminf_{T\\to\\infty} \\frac{N_d(T,2T)}{N(T,2T)} > 0.83671252212, for the proportion of distinct nontrivial zeros. The finite component is independently reproducible. The window functional is enclosed by rational interval arithmetic; the six-dimensional gap inequality is certified over the full nonnegative orthant by interval subdivision together with a strong-convexity closure of the unique terminal basin; and the final block optimization is reduced to exact rational comparisons, with the unique integer optimum occurring at m=160m=160. The analytic zeta-function input is cited rather than reproved. The new contribution is the rational pair- and position-pressure coefficient system, its global certificate, the terminal-basin closure, and the resulting certified numerical improvement.