Section 1.2, listScope: worst-case representational statements under explicit constraints, not training results

Fixed-scale quantisation / fixed latency. The $\AC^0$ simulation (Theorem 2.4) assumes constant depth and a fixed-scale, bounded-range quantisation regime: weights/thresholds lie on a fixed grid whose range/resolution do not scale with the input length (cf. Remark 2.2). Architectures that rely on fan-in dependent re-normalisation, biases/thresholds growing with dimension, or depth growing with $n$ fall outside this abstraction. In particular, a fixed recurrent update rule need not have fixed total latency: the third bypass deliberately allows $N$ successive row updates per position.

Formal model vs. experimental networks. The formal units have nonnegative weights on Boolean literals and bounded thresholds. The experiments instead use ordinary floating-point signed linear layers, ReLU and LayerNorm, with Log-Softmax in the loss. They test the benefit of transition information under the reported training protocol, not the asymptotic lower bound or the deployment behaviour of quantised networks.

Stylised planning interface. The single-channel lower bound permits only root and one-heap-modification queries to a fixed structured policy, and requires a polynomial-size, constant-depth final readout. It is not a bound on arbitrary rollout policies, unrestricted query transformations, or an unbounded adaptive MCTS controller.

No guarantee of learnability. The multi-policy-head and recurrent constructions concern existence of suitable computations; we do not claim that self-play efficiently discovers the structured heads or an exact recurrent accumulator. Our experiments use controlled supervised restoration and policy imitation; self-play discovery is left for a separate study.