Density-One Spectral Blindness and Inverse Instability in Closed Thin Waveguides

We construct smooth families of pairwise noncongruent closed thin waveguides with a fixed cross-section carrying a nontrivial compact transverse symmetry. The geometry can be chosen once, before taking the thin-tube limit, so that two different forms of spectral invisibility occur simultaneously. First, the relative cosine wave trace agrees exactly over any prescribed finite observation interval. Second, below the first symmetry-active transverse threshold, an asymptotically full proportion of the ordered spectrum becomes exponentially insensitive to the global arrangement of the tube. The mechanism is based on symmetry selection and spatial separation. Local changes in the tube act trivially on the retained symmetry-invariant transverse modes, forcing the first geometry-sensitive contribution into a higher active sector. Straight guard regions then suppress communication through that sector exponentially as the tube radius decreases. A moving-energy Feshbach reduction and an exact inertia argument transfer this suppression to the physical ordered eigenvalues without requiring simple eigenvalues or analytic branch labels. The resulting instability extends beyond individual eigenvalues. Fixed subthreshold Lipschitz spectral statistics become exponentially close, density-one empirical spectral measures collapse in every Wasserstein distance while the corresponding geometries remain macroscopically separated, and no uniform inverse modulus can recover the geometry from this class of spectral data. Even predetermined exponentially sharpening smoothed spectral measurements can be made ineffective. A further near-threshold analysis identifies symmetry-protected regimes in which almost the entire pre-active spectral band remains blind to the global geometry. These results provide a quantitative obstruction to inverse spectral reconstruction in closed thin waveguides: increasingly detailed and increasingly precise subthreshold spectral information can still fail to determine globally distinct geometries.