Mapping parametric error profiles onto nuclear structure configurations in deformed proton radioactivity

Theoretical descriptions of proton radioactivity near the drip lines are often challenged by parametric uncertainties in nuclear potential models. To address this, a robust Bayesian uncertainty quantification (UQ) framework is established to calibrate the deformed Woods--Saxon potential coupled with the semi-classical WKB approximation for odd-$A$ proton emitters across the $Z=50$--$82$ region. Constrained by experimental half-lives, the joint and marginal posteriors and covariance topologies are extracted for the potential geometry, multipole deformations, and spectroscopic factors. Using a signed relative sensitivity index, a dynamic, mass-dependent re-ordering of parameter hierarchies is unraveled. Crucially, propagating these parametric uncertainties yields posterior predictive medians and differentiated $\sigma$ bands that tightly encompass experimental half-lives across multiple orders of magnitude without systematic bias. Furthermore, near the $Z=82$ shell closure, the structural softening of the nuclear potential manifests as a systematic expansion and volatility of the $\sigma$ bands in the deformation profiles. This correlation suggests that the macroscopic behavior of parametric error distributions can potentially be mapped onto potential energy configurations, underscoring that the Bayesian UQ methodology may serve as a sensitive probe to extract reliable nuclear structure information directly from deformed proton decay systematics.