Mapping the Shadow of Uncertainty: The Information Horizon in Quantum Spectral Inversion

Can we know the laws of physics in regions where no particle ever travels? Here, we address this epistemological question by performing a Bayesian reconstruction of a quantum Hamiltonian H solely from its discrete energy spectrum. Using an affine-invariant MCMC ensemble, we quantify the local uncertainty σ(x) of the reconstructed potential. Our results reveal a universal "Trumpet of Ignorance" structure: reconstruction error is negligible (< 1%) within the classically allowed region but diverges exponentially in the forbidden zone. We demonstrate that the recoverable information density is proportional to the local probability amplitude |ψ(x)|², implying that the structure of reality is only determinate within the "light cone" of quantum interaction.