Symmetry conditions for spacetime observability of wave equations on the torus

We study observability for the one-dimensional wave equation on the torus from spacetime measurable observation sets. While the Geometric Control Condition (GCC) provides a sufficient criterion in many classical settings, it is no longer sufficient in this framework. We construct explicit counterexamples showing the failure of observability despite the validity of GCC. This leads to the introduction of an additional symmetry condition on the observation set, referred to as the Observable Symmetry Condition (OSC). We prove that observability holds if and only if both GCC and OSC are satisfied. We also show that unique continuation holds if and only if both OSC and a weak form of GCC are satisfied.