\section{Formal problem}\label{sec:problem}

\paragraph{Autoregressive prediction under a computation budget.}
Let $\mathcal{V}$ be a vocabulary of size $V$ and $x_{1:T}\in\mathcal{V}^T$ a sequence. A causal language model
assigns $p_\theta(x_{t+1}\mid x_{1:t})$ and is trained to minimise the teacher-forced cross-entropy
$\Loss(\theta)=\E\big[-\log p_\theta(x_{t+1}\mid x_{1:t})\big]$. We consider models whose per-position computation
is \emph{variable}: for each position $t$ the model may execute a number $n_t\in\{0,\dots,R_{\max}\}$ of additional
\emph{actions} before emitting its distribution, each action having a known arithmetic cost. Writing $c(a)$ for the
cost of action $a$ (in units of the backbone's own operations per token), the realised cost of position $t$ is
$C_t = 1 + \sum_{r\le n_t} c(a_{t,r})$ plus the cost of deciding, and the quantities of interest are the pairs
\begin{equation}
\big(\;\text{quality}(\pi),\;\;\E_t[C_t\mid\pi]\;\big),\qquad
\big(\;\text{quality}(\pi),\;\;\operatorname{median}_t\,\text{latency}_t(\pi)\;\big),
\label{eq:frontier}
\end{equation}
traced out as the routing policy $\pi$ and its penalties vary. Quality is either the held-out cross-entropy or, for
tasks with programmatic ground truth, the exact-match accuracy of free greedy generation. The research question is
whether a policy that chooses \emph{where} to write (which of $M$ latent slots) using supervision from executed
counterfactual branches moves the frontier \eqref{eq:frontier} relative to policies that do not choose (uniform or
random slot selection, dense updates of all slots) and relative to global latent recurrence at equal cost.

\paragraph{Causal prefix constraint.}
Every quantity that influences $p_\theta(x_{t+1}\mid x_{1:t})$---including the routing decision---must be a function
of $x_{1:t}$ only. In particular the label $x_{t+1}$ is available to the \emph{training} procedure that constructs
supervision, but never to the deployed router (Section~\ref{sec:policy}).

\paragraph{What is \emph{not} studied.}
The model has no persistent memory across positions (the workspace is reset at every $t$), no retrieval, no tools,
no reinforcement learning, and no textual chain of thought. These exclusions are deliberate: they isolate the
single mechanism under test.