  It has recently been proved numerically that spinning black holes in
Einstein-scalar theories which are characterized by a non-minimal negative
coupling of the scalar field to the Gauss-Bonnet invariant of the curved
spacetime may develop exponentially growing instabilities. Intriguingly, it has
been demonstrated that this tachyonic instability, which marks the onset of the
spontaneous scalarization phenomenon in the Einstein-Gauss-Bonnet-scalar
theory, characterizes spinning black holes whose dimensionless angular momentum
parameter ${\bar a}\equiv a/M$ is larger than some critical value ${\bar
a}_{\text{crit}}\simeq0.505$. In the present paper we prove, using {\it
analytical} techniques, that the critical rotation parameter which marks the
boundary between bald Kerr black holes and hairy (scalarized) spinning black
holes in the Einstein-Gauss-Bonnet-scalar theory is given by the exact
dimensionless relation ${\bar a}_{\text{crit}}={1\over 2}$.
