Dynamic Spectral Information in Modular Quantum Dynamics: Limits of Static Spectral Invariants

We investigate whether the full spectral trajectory arising from modular quantum dy-namics contains information that is absent from a single static spectral snapshot. Theinput data consist of a density operator and a prescribed tensor-product structure ofthe Hilbert space. From the modular generator, modular evolution of local operators isconstructed; normalized commutators are then used to define a dynamic distance ma-trix, from which a weighted graph Laplacian is obtained, whose spectrum is treated as adynamic spectral feature.We show that the spectral dynamics is nontrivial even in finite-dimensional systemswith a fixed local structure. We construct an analytic counterexample in which two gener-ators have identical spectral data at a fixed time but differ at other times. Thus, a singlestatic snapshot does not, in general, determine the full spectral trajectory. In a compu-tational test over 1044 non-isomorphic graphs with N = 7, 62 cospectrality classes withrespect to the ordinary graph Laplacian were found; for all 74 non-isomorphic cospectralpairs, the dynamic spectral feature at s = 0.1 and τ = 0.5 produced nonzero separation.At the same time, this result does not imply a universal superiority of the full trajectoryin classification accuracy: in a separate test involving local and random generators, noimprovement over the best nontrivial static snapshot was observed. Persistent extensionslikewise showed no independent diagnostic advantage in the tests considered.The results do not provide a reconstruction of spacetime and do not eliminate the needto specify the tensor-product structure and local operators externally. They establish thenarrower statement that dynamic spectral information is, in general, richer than a singlestatic spectral snapshot.