  We study the unique solvability of the discretized Helmholtz problem with
Robin boundary conditions using a conforming Galerkin $hp$-finite element
method. Well-posedness of the discrete equations is typically investigated by
applying a compact perturbation to the continuous Helmholtz problem so that a
"sufficiently rich" discretization results in a "sufficiently small"
perturbation of the continuous problem and well-posedness is inherited via
Fredholm's alternative. The qualitative notion "sufficiently rich", however,
involves unknown constants and is only of asymptotic nature.
  Our paper is focussed on a fully discrete approach by mimicking the tools for
proving well-posedness of the continuous problem directly on the discrete
level. In this way, a computable criterion is derived which certifies discrete
well-posedness without relying on an asymptotic perturbation argument. By using
this novel approach we obtain a) new stability results for the $hp$-FEM for the
Helmholtz problem b) examples for meshes such that the discretization becomes
unstable (stiffness matrix is singular), and c) a simple checking Algorithm
MOTZ "marching-of-the-zeros" which guarantees in an a posteriori way that a
given mesh is certified for a stable Helmholtz discretization.
