Why memory states in quantum gravity resist a clean Hilbert space

Working on the Ashtekar-Streubel phase space of radiative gravitational modes, the authors rigorously compute Dirac brackets and identify which functions on phase space qualify as proper observables — those with well-defined symplectic flows. The so-called Goldstone mode, widely used in the literature on gravitational memory, fails this test, though an infinite family of substitute 'Goldstone probes' can still measure it indirectly. Crucially, none of these probes can be built from the shear or news alone, offering a concrete explanation for persistent failures to construct a separable Hilbert space distinguishing different memory vacua. The distributional Dirac brackets between local shear and news also turn out to carry non-local corrections.