Uniform Shannon-Edge Quantization and Normalization for Order-Zero Prolate Spheroidal Wave Functions

We study order-zero prolate spheroidal wave functions in the high-bandlimit Shannon transition, where the turning point of the angular equation reaches the regular singular endpoint. With a signed action coordinate τ = γa kept in a fixed compact interval, a real Liouville transformation reduces the endpoint problem to a Whittaker–Coulomb equation with index μ = 0. Coulomb renormalization, a Volterra endpoint construction, a parameter-uniform two-branch Kummer expansion, and first-corrected Liouville–Green propagation are combined with a continuous Sturm phase lift. Uniformly for |τₙ,ᵧ| ≤ T, this yields an explicit signed two-term quantization law with a 1/γ correction and remainder Oₜ(γ⁻⁴⁄³) through the coalescence. We also prove C¹ control of the lifted center phase. For L²(−1, 1)-normalized PSWFs, the phase derivative gives an explicit center normalization for even modes and, for both parities, the endpoint law |eₙ,ᵧ(1)|² = [πγ / {Fᵧ′(τₙ,ᵧ) · (1 + exp(πτₙ,ᵧ))}] · [1 + Oₜ(γ⁻¹)]. An exact Hellmann–Feynman identity converts this into a quantitative Shannon-edge growth law for the sinc-kernel concentration eigenvalue. The leading Gamma-phase transition is classical; the contribution here is the uniform signed subleading correction together with the point and endpoint normalization package.