  Cosmological relaxation of the electroweak scale is an attractive scenario
addressing the gauge hierarchy problem. Its main actor, the relaxion, is a
light spin-zero field which dynamically relaxes the Higgs mass with respect to
its natural large value. We show that the relaxion is generically stabilized at
a special position in the field space, which leads to suppression of its mass
and potentially unnatural values for the model's effective low-energy
couplings. In particular, we find that the relaxion mixing with the Higgs can
be several orders of magnitude above its naive naturalness bound. Low energy
observers may thus find the relaxion theory being fine-tuned although the
relaxion scenario itself is constructed in a technically natural way. More
generally, we identify the lower and upper bounds on the mixing angle. We
examine the experimental implications of the above observations at the
luminosity and precision frontiers. A particular attention is given to the
impressive ability of future nuclear clocks to search for rapidly oscillating
scalar ultra-light dark matter, where the future projected sensitivity is
presented.
