  We investigate radiatively stable classes of pseudo-Nambu-Goldstone boson
(pNGB) potentials for approximate spontaneously broken
$\mathrm{SO}(N+1)\to\mathrm{SO}(N)$. Using both the one-loop effective action
and symmetry, it is shown that a Gegenbauer polynomial potential is radiatively
stable, being effectively an `eigenfunction' from a radiative perspective. In
Gegenbauer pNGB models, one naturally and automatically obtains $v \propto
f/n$, where $n\in 2\mathbb{Z}$ is the order of the Gegenbauer polynomial. For a
Gegenbauer Higgs boson, this breaks the usual correlation between Higgs
coupling corrections and `$v/f$' tuning. Based on this, we argue that to
conclusively determine whether or not the Higgs is a composite pNGB in
scenarios with up to $\mathcal{O}(10\%)$ fine-tuning will require going beyond
both the Higgs coupling precision and heavy resonance mass reach of the
High-Luminosity LHC.
