Optimal augmentation size for BONuS under a point-mass prior

Let $\tilde m_{\textnormal{opt}}$ be the choice of augmentation size that maximizes the power of BONuS for a point-mass prior, with $m$ fixed. Optimal augmentation-size conjecture.

$$\tilde m_{\textnormal{opt}}=\Theta(\sqrt{m}).$$

The preceding expansion balances a pseudocount bias of order $1/\tilde m$ against learner degradation of order $\tilde m/m$, suggesting this scaling. Whether the exact BONuS power has this optimizer remains open.