% WhiteHoleT5 / WhiteHoleMachines (math form)

\[
\begin{aligned}
&\textbf{Given:}\quad x \in \mathbb{N}^{L}\ \text{(token ids)},\quad m \in \{0,1\}^{L}\ \text{(attention mask)} \\
&\textbf{Base model (T5-like):}\quad p_{\theta}(y_t \mid y_{<t}, x, m)
=\operatorname{Softmax}\!\Big(\operatorname{Head}\big(\operatorname{Dec}_{\theta}(\operatorname{Enc}_{\theta}(x,m),\, y_{<t})\big)\Big) \\
&\textbf{White-hole wrapper:}\quad z = \operatorname{Enc}_{\theta}(x,m),\qquad \tilde{z} = H_{\psi}(z) \\
&\textbf{Forward (logits):}\quad \ell_t = \operatorname{Head}\!\left(\operatorname{Dec}_{\theta}(\tilde{z}, y_{<t})\right) \\
&\textbf{Generate:}\quad \hat{y}_{1:T} \sim \prod_{t=1}^{T} p_{\theta}(y_t \mid \hat{y}_{<t}, x, m) \\
&\textbf{Adjective list (as written):}\quad a_i = \mathrm{Parity}(3),\quad i=1,\dots,10 \\
&\textbf{Repeated acceleration loop:}\quad
\forall k \in \{1,\dots,3\cdot 10^{6}\}:\quad \hat{y}^{(k)} \leftarrow \operatorname{Generate}(x,m;\theta,\psi)
\end{aligned}
\]