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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 boundaryuniversality class, if the symmetry preserving interactions on the boundary areenhanced. $N_c$ is determined by a ratio of universal amplitudes in the normaluniversality class, where instead a symmetry breaking field is applied on theboundary. We study the normal universality class using the numerical conformalbootstrap. We find truncated solutions to the crossing equation that indicate$N_c \approx 5$. Additionally, we use semi-definite programming to placerigorous bounds on the boundary CFT data of interest to conclude that $N_c >3$, under a certain positivity assumption which we check in variousperturbative limits.

The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap