Analytic Speed Limits and Extremal Rigidity for Strip-Holomorphic Monotone Profiles

We develop a sharp local-to-global transition calculus for monotone profiles whose centered holomorphic extension maps a horizontal strip into the unit disk. Schwarz-Pick contraction gives the pointwise bounds |G′| ≤ [π/(4a)](1 − G²) and 0 ≤ F′ ≤ [π/(2a)]F(1 − F), where G = 2F − 1. Integration yields optimal two-point, anchored-envelope, and threshold-duration inequalities. Equality at one real point, equality for one distinct pair, or contact with an anchored envelope at one later point forces the complete logistic profile. A speed-defect identity measures accumulated transition delay, while quantile-spacing, variance, and area bounds show that every admissible transition is at least as dispersed as the logistic extremizer. We separate the disk-contractivity hypothesis from spectral growth: compact spectral support supplies an entire function of exponential type, while strip-to-disk contractivity is the geometric condition that produces the nonlinear speed limit. The resulting framework is a portable analytic theorem; a domain realization supplies its physical clock and dynamics.