Proposition 5.14 (open item)Bridge from the recurrent strategies' game semantics to the general weak-mastery history model

Proposition 5.14 (Shallow components and complete strategies) For $N\ge1$ heaps of $B\ge1$ bits, the following constructions use polynomial-size, constant-depth Boolean components, realisable by literal-threshold networks with weights and thresholds in $\{0,1\}$.

1. (i) The two-frame selector maps a legal transition $Q\to Q'$ from $\operatorname{nim}(Q)=0$ to a legal binary-encoded response restoring zero. Retaining the response board as the next predecessor gives a winning strategy from this established invariant, with the opponent to move.

2. (ii) Each structured rollout policy and the readout from the board and its $B$ outcome bits are shallow. Executing the rollouts and then the readout gives a winning strategy from any nonzero nim-sum board, using at most $BN(2^B-1)$ simulated game moves per decision.

3. (iii) The recurrent state update is shallow. Iterating it over the replay protocol gives a winning strategy from any nonzero nim-sum board, with $O(B+\log N)$ memory and $2N+2$ updates per decision.

Here “winning strategy” quantifies over every legal opponent response. Polynomial size and constant depth refer to the indicated component, with depth independent of $N$ and $B$; they do not bound the number of sequential component evaluations.