deletions | additions       diff --git a/The_following_theorem_is_the__.tex b/The_following_theorem_is_the__.tex  index 039d6b8..ae8bf90 100644  --- a/The_following_theorem_is_the__.tex  +++ b/The_following_theorem_is_the__.tex 
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\begin{theorem}  \label{thm:ort}  Fix $\Gamma=(\Gamma_{\mathfrak{R}}$, $\Gamma_{\mathfrak{A}})$ a pair of growth spaces of rank 2 and $\mathcal{E}$ an error space of rank 2. Assume there is $\zeta: \Nats^2 \rightarrow \Reals^{>0}$ s.t. $\zeta \in \mathcal{E}$ and $\Ceil{\log $\Floor{\log  \max(-\log \zeta, 1)} 2)}  \in \Gamma_{\mathfrak{A}}$\footnote{If $\log(k+1), \log(j+1) \in \Gamma_{\mathfrak{A}}$ then this condition holds for any $\mathcal{E}$ since we can take $\zeta = 2^{-h}$ for $h$ polynomial.}. Consider $(\mu,f)$ a distributional estimation problem and $P$ an $\mathcal{E}(\Gamma)$-optimal predictor for $(\mu,f)$. Then, $P$ is also an $\mathcal{E}^{\frac{1}{2}\sharp}(\Gamma)$-optimal predictor for $(\mu,f)$. \end{theorem}