Structural Exhaustion under Guaranteed Stability

Structural exhaustion denotes a failure mode in non-normal operator families where spectral stability is preserved while operability deteriorates. An explicit operator family is considered whose spectrum remains fixed and uniformly separated from zero. Despite this guaranteed spectral stability, the norm of the inverse grows without bound, rendering stable inversion impossible in the limit. The mechanism behind this divergence is non-normality and pseudospectral amplification, not spectral migration. The result establishes a strict separation between spectral stability (location of the spectrum) and structural operability (control of the inverse norm). Stability criteria based solely on spectral gaps are therefore insufficient in non-normal settings. The construction is elementary, finite-dimensional in nature, and independent of perturbation theory or numerical artifacts. A fully computed visual summary is provided. The core analytic inequality underlying the divergence is formally verified in Lean4.