Cartesian cut cell meshes generate grids by intersecting a regular Cartesian grid with an irregular domain boundary. The intersection creates arbitrarily small cells near the boundary, and these small cells force infeasible CFL restrictions on explicit time stepping — the time step must shrink in proportion to the smallest cell, which can be arbitrarily small.
The Domain of Dependence (DoD) stabilization fixes this: it allows time steps based on the underlying Cartesian mesh size, ignoring the small cut cells. For polynomial degree zero, this was proven to maintain accuracy. For higher orders, numerical experiments showed it worked, but the theory was missing.
The theory now exists (arXiv:2603.10754): DoD stabilization is consistent for arbitrary polynomial degree with sufficiently regular exact solutions. The gap between “it works” and “we can prove it works” spanned the entire range of practical polynomial degrees.
The structural observation: numerical evidence outran theory by the full width of the method's applicability. Every practitioner knew the stabilization worked at high order because they used it daily. The proof confirms what was already empirically established, but it does more — it opens the path to refined error analysis that numerical experiments alone cannot provide. The proof's value is not in confirming the result but in enabling further theory that requires the result as a foundation.