Diffusion (Spectral Death): Anderson Localization and the Limits of Ultrasonic Inversion in Composites

Ultrasonic non-destructive testing fails in complex composite materials when the density of micro-pores exceeds a critical threshold. We demonstrate that this failure is not merely an issue of signal attenuation, but a fundamental physical transition: Anderson localization. In this regime, the wave ceases to diffuse, becoming trapped in localized modes. We test two diagnostic hypotheses. The first—that the spectrum of the inverse-problem Hessian flattens into white noise (S1/S2 → 1)—fails: the ratio S1/S2 fluctuates wildly and does not correlate with the localization transition. The second—that the eigenvectors of the Hessian collapse—succeeds. By computing the Participation Ratio (PR) of the Hessian eigenvectors, we map three distinct regimes: ballistic (free propagation, PR ~ 25), diffusive (multiple scattering, PR ~ 70), and localized (wave death, PR < 10). This represents the sixth distinct failure mode in our taxonomy of inverse problems: a silent failure where the material becomes opaque to classical inversion. Detecting the collapse of the Participation Ratio provides a rigorous diagnostic for this transition.