Exact Angular Critical-Point Counts for e^z ± 1

Abstract

For f_a(z) = e^z + a, a = ±1, we count the distinct angular critical points of log |f_a| on every zero-free circle of radius r ≥ 12. Writing N_p(r) for the number of zeros in the disk, we prove that the critical-point count N_osc(r) satisfies 0 ≤ N_osc(r) − 2N_p(r) ≤ 4, with both endpoints attained, and |N_osc(r) − 2r/π| < 4. The proof gives an exact arc decomposition: each interior arc and the last bulk arc carries one zero, while a shrinking top window carries at most two. We classify every exceptional radius for both signs, including circles containing zeros, and determine the exact angular multiplicities and radial bifurcations at all zero-free degeneracies. Non-axis degeneracies are double folds, and the unique axis degeneracy is cubic. This yields a radius-by-radius count, including the transition radii. Each function has only one maximum-modulus point on every circle, despite its linearly growing critical-point count. We also prove an unconditional argument-variation bound for these models and show that strict sign alternation fails on unbounded intervals of good radii. The upper defect bound is attained at r = 12; its lower-root certificate uses exact rational arithmetic.