Exact Strong-Field Relations and Sensitivity Hierarchy for a Four-Dimensional Nonpolynomial Regular Black Hole

This work develops an analytic characterization of the strong-field observables of a four-dimensional nonpolynomial regular black-hole geometry. Using the dimensionless parameter q = α/M², the black-hole family is reduced to a one-parameter problem, allowing exact relations for the horizon branch, extremal endpoint, photon sphere, and surface gravity to be obtained together with the ISCO condition. Controlled small-q expansions and a full numerical scan reveal a pronounced hierarchy of sensitivity. Photon-sphere, shadow, and ISCO observables remain close to their Schwarzschild values over the black-hole branch, while near-horizon quantities—particularly the surface gravity—show substantially stronger deviations as extremality is approached. The analysis complements previous numerical studies of the same geometry. The individual strong-field observables are not claimed here as first calculations; instead, the contribution is the exact one-parameter analytic organization of the model, the resulting algebraic relations, controlled perturbative expansions, and a quantitative comparison of the sensitivity of horizon-, photon-sphere-, and ISCO-scale observables. The record includes the main manuscript, supplementary material, and a reproducibility package containing the deterministic calculation and numerical scan.