Section 1.1, listList of contributions; each item points to a precise statement

$\AC^0$ containment for fixed-scale quantised inference. We formalise a fixed-grid Boolean literal-threshold model and its bounded-unrolling recurrent extension. Both are simulable by $\AC^0$ circuits (Theorem 2.4). An attention-style extension is covered only if its scoring, selection and aggregation operations can each be implemented by the permitted Boolean threshold gates. This is a restriction on the computation, not a requirement that every game have the same board size. Within a play sequence we use a fixed layout of $N$ heap rows and $B$ bits per heap, retaining empty heaps as zero rows. The asymptotic statements concern families of networks for growing input lengths $NB$.

Single-frame impossibility for strong mastery. Under this model, no single-frame policy/value network can strongly master NIM in the worst case (Theorem 5.2).

Two-frame bypass via local differences and restoration. With two consecutive frames, the agent can compute $\Delta(P,P')$ in $\AC^0$ (Lemma 5.4). When $P$ has nim-sum zero, an $\AC^0$ move selector restores zero after the opponent’s move (Proposition 5.5). This is conditional restoration mastery, not an arbitrary-start verifier.

Planning analysis and the single-policy bottleneck. We analyse a restricted measurement interface inspired by rollouts, rather than a general model of MCTS. With $B$ structured policies one can recover all nim-sum bits (Proposition 5.9). Root and one-heap-modification queries to one fixed member of that family expose only one unknown linear functional (Proposition 5.10). With an explicitly bounded readout, this yields the distributional bound in Theorem 5.12.

Persistent recurrence via sequential row updates. A constant-depth recurrent update accumulates the nim-sum over $N$ heap rows and retains a summary across position separators (Proposition 5.13). With a replay of the current board or stored heap rows, this supports optimal move selection. The construction relaxes fixed total latency, not fixed-scale quantisation (Section 5.5).

Controlled empirical comparison and learnability discussion. On 20-heap, 4-bit NIM with $10^6$ supervised restoration examples, a one-frame model has low restoration accuracy while a two-frame model learns restoration (Section 7.2). In a complementary single-frame experiment, multi-head FSM-controlled shootouts achieve perfect win/loss position classification while the single-head baseline remains bottlenecked by one-bit information (Section 7.3). We also discuss why $\AC^0$ is a useful abstraction for what is robustly feasible under fixed-scale quantised inference (Section 6.3).