Gap distributions for arithmetic-progression semigroups: a uniform interval model and the dense transition

For numerical semigroups generated by an arithmetic progression, with a >= 2, 1 <= k <= a-1, d >= 1 and gcd(a,d)=1, we give an interval model for the distribution of all gaps at scale B=a(a/k+d). Its distribution-function error is at most 8/a uniformly in k and d. The triangular approximation has error at most 8/a+5/L, and the triangular limit holds exactly when L=a/k+d tends to infinity. We also give the explicit dense-regime limit and examples showing sharpness of the two error scales.