Arbitrarily Delayed Spectral Memory in Noncongruent Closed Waveguides

We construct smooth families of closed thin waveguides that remain globally noncongruent while becoming spectrally indistinguishable over independently prescribed observational scales. For any finite wave horizon, the relative cosine wave trace can be made to vanish exactly throughout that interval by separating the geometry that distinguishes the tubes behind sufficiently long locally congruent guards. Independently, a fixed anisotropic cross-section can produce a linearly growing band of ordered eigenvalues whose splittings are exponentially small in the tube radius. The cross-section and geometry are fixed before the thin-tube limit is taken. For suitable fixed designs, at least ρL/ℓ+O(1) low eigenvalues may therefore remain pairwise O(e−A/ℓ)-close for arbitrary prescribed finite ρ and A, while the underlying tubes retain a positive geometric separation. The construction applies simultaneously to Dirichlet and Neumann boundary conditions, with corresponding extensions for compact families of physical Robin data. The resulting spectral proximity propagates to heat traces, spectral zeta functions and zeta determinants. As an inverse consequence, no reconstruction procedure based on a fixed linear-in-1/ℓ number of low eigenvalues together with any fixed exponential precision scale can uniformly recover the global geometry over this class. The results quantify how global geometric information can remain inaccessible even as the amount and precision of spectral data increase naturally in the thin-waveguide limit.