  It has recently been demonstrated that magnetized black holes in composed
Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a non-minimal negative
coupling of the scalar field to the Gauss-Bonnet curvature invariant may
support spatially regular scalar hairy configurations. In particular, it has
been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the onset
of the near-horizon spontaneous scalarization phenomenon is marked by the
numerically computed dimensionless critical relation
$(BM)_{\text{crit}}\simeq0.971$, where $\{M,B\}$ are respectively the mass and
the magnetic field of the spacetime. In the present paper we prove, using
analytical techniques, that the boundary between bald Schwarzschild-Melvin
black-hole spacetimes and hairy (scalarized) black-hole solutions of the
composed Einstein-Maxwell-scalar-Gauss-Bonnet theory is characterized by the
exact dimensionless relation
$(BM)_{\text{crit}}=\sqrt{{{\sqrt{6}-2}\over{2\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}}$
for the critical magnetic strength. Intriguingly, we prove that the critical
dimensionless magnetic parameter $(BM)_{\text{crit}}$ corresponds to magnetized
black holes that support a pair of linearized non-minimally coupled thin scalar
rings that are characterized by the non-equatorial polar angular relation
$(\sin^2\theta)_{\text{scalar-ring}}={{690-72\sqrt{6}+4\sqrt{3258\sqrt{6}-7158}}\over{789}}<1$.
It is also proved that the classically allowed angular region for the
negative-coupling near-horizon spontaneous scalarization phenomenon of
magnetized Schwarzschild-Melvin spacetimes is restricted to the black-hole
poles, $\sin^2\theta_{\text{scalar}}\to0$, in the asymptotic large-strength
magnetic regime $BM\gg1$.
