  Exact discrete symmetries, if non-linearly realized, can reduce the
ultraviolet sensitivity of a given theory. The scalars stemming from
spontaneous symmetry breaking are massive without breaking the discrete
symmetry, and those masses are protected from divergent quadratic corrections.
This is in contrast to non-linearly realized continuous symmetries. The
symmetry-protected masses and potentials of those discrete Goldstone bosons
offer promising physics avenues, both theoretically and in view of the blooming
experimental search for ALPs. In this text, we develop this theoretical setup
for the specific case of a triplet of $A_5$ using invariant theory, showcasing
the substantial improvements and compelling phenomenological consequences
introduced by the invariance under a discrete symmetry.
