From Diagonal Decorrelation to Coherent Generator Spectroscopy

Short-time scattering rates characterize local event activity, but need not identify the slow relaxation generated by configuration-space connectivity. We prove a fixed-graph non identifiability result: when single-replacement events admit redistribution within displacement families, all diagonal activities and normalized first cumulants can remain unchanged while a bottleneck drives the spectral gap to zero and the integrated correlation time to infinity. A complementary visibility criterion determines when coherent-mode combinations recover the missing slow rate. If the limiting bottleneck coordinate lies in the measured feature space, the variational rate has an absolute error of order ε 2 ; a quantitative projection bound describes incomplete visibility. An exact 20-state example realizes both results with thirteen coherent density modes. Independent-event simulations recover the rare-crossing relative-error scaling (nε) −1/2 . Syn thetic complex-field and trajectory measurements test finite-lag bias, exposure, noise, and joint estimation, including a reported failure of nominal interval coverage. A matrix-moment construction resolves the example’s hidden dynamical directions and separates algebraic re construction from statistical stability. The contribution is the paired non-identifiability and visibility result within an established cross-wave-vector and variational framework. Its op erational scope requires phase- or coordinate-resolved access, suitable feature coverage, and sufficient rare-event sampling.