  We study the critical $O(3)$ model using the numerical conformal bootstrap.
In particular, we use a recently developed cutting-surface algorithm to
efficiently map out the allowed space of CFT data from correlators involving
the leading $O(3)$ singlet $s$, vector $\phi$, and rank-2 symmetric tensor $t$.
We determine their scaling dimensions to be $(\Delta_{s}, \Delta_{\phi},
\Delta_{t}) = (0.518942(51), 1.59489(59), 1.20954(23))$, and also bound various
OPE coefficients. We additionally introduce a new "tip-finding" algorithm to
compute an upper bound on the leading rank-4 symmetric tensor $t_4$, which we
find to be relevant with $\Delta_{t_4} < 2.99056$. The conformal bootstrap thus
provides a numerical proof that systems described by the critical $O(3)$ model,
such as classical Heisenberg ferromagnets at the Curie transition, are unstable
to cubic anisotropy.
