Scattering time advance under a complex growth bound: physical sharpness and optimal support cost

This mathematical-physics preprint studies scattering phase derivatives for real, compactly supported L² potentials of the Dirichlet Schrödinger operator on the half-line whose Jost functions have no zeros in the closed upper half-plane. At fixed positive momentum k₀ and prescribed scattering value S(k₀) = S₀, an independently imposed modulus bound M > 1 on an upper semicircle of radius ρ < k₀ gives the classical lower bound δ′(k₀) ≥ −2 log(M)/(πρ). The manuscript establishes sharpness within this class of local potentials while preserving both prescribed constraints. For fixed data, the infimum of the essential support endpoints required to approach this phase-slope infimum within ε has order Θ(ε⁻¹). The phase-slope infimum is not attained at finite support. The leading support coefficient remains undetermined, and no common bound on the potential norm is imposed. Appendices derive a real-axis Fourier-response bound and conditional bounds from finite noisy data with independently supplied interpolation and tail estimates. The files contain the manuscript PDF, its single-file LaTeX source, and the exact-arithmetic verification code for the synthetic-data example in Appendix C. This is a preprint.