  We prove the existence of a (spectrally) stable self-similar blow-up solution
$f_0$ to the heat flow for corotational harmonic maps from $\mathbb R^3$ to the
three-sphere. In particular, our result verifies the spectral gap conjecture
stated by one of the authors and lays the groundwork for the proof of the
nonlinear stability of $f_0$. At the heart of our analysis lies a new existence
result of a monotone self-similar solution $f_0$. Although solutions of this
kind have already been constructed before, our approach reveals substantial
quantitative properties of $f_0$, leading to the stability result. A key
ingredient is the use of interval arithmetic: a rigorous computer-assisted
method for estimating functions. It is easy to verify our results by robust
numerics but the purpose of the present paper is to provide mathematically
rigorous proofs.
