  This paper studies the critical behavior of the 3d classical $\mathrm{O}(N)$
model with a boundary. Recently, one of us established that upon treating $N$
as a continuous variable, there exists a critical value $N_c > 2$ such that for
$2 \leq N < N_c$ the model exhibits a new extraordinary-log boundary
universality class, if the symmetry preserving interactions on the boundary are
enhanced. $N_c$ is determined by a ratio of universal amplitudes in the normal
universality class, where instead a symmetry breaking field is applied on the
boundary. We study the normal universality class using the numerical conformal
bootstrap. We find truncated solutions to the crossing equation that indicate
$N_c \approx 5$. Additionally, we use semi-definite programming to place
rigorous bounds on the boundary CFT data of interest to conclude that $N_c >
3$, under a certain positivity assumption which we check in various
perturbative limits.
