Asymptotic observable symmetry for spacetime observability of waves I: The disc

Building on the spacetime observable symmetry discovered in NWX (2025), we introduce asymptotic observable symmetry (AOS) for the wave equation. Its central object is a family of generated approximate velocities constructed from fixed profiles, carrying the same propagation-induced symmetry at different scales and asymptotically realized by exact waves. We investigate AOS on the disc through radial propagation and reflection. This gives a family of spacetime measurable observation sets satisfying the essential spacetime measurable geometric control condition for a fixed positive observation time. However, the observability inequality fails on each of these fixed sets.