Section 5.4, paragraph 3The controller organises which head is queried and when to stop; it adds no per-step power

Informal interpretation Motivation, interpretation or scope; not a formal claim.

1 Paper text

This viewpoint does not increase the per-step computational power of the rollout heads: each $\pi_j$ remains $\AC^0$ -bounded. Instead, the controller organises which $\AC^0$ head is iterated in a shootout and when to stop querying. Figure 3 depicts a particularly simple controller that turns the $B$ structured rollout heads from Proposition 5.9 into a sequential “bit-query” procedure for deciding whether $\sigma(P)=\operatorname{nim}(P)$ is zero (and hence whether $P$ is a $P$ -position).

Generated from paper_jmlr.tex, lines 674–676.

2 Explain this

The controller does not make any single head more powerful: each head is still a shallow policy. What it adds is organisation: which head to run next and when to stop. That is what turns separate channels into a practical procedure.

3 How this is formalised

An interpretive paragraph. The precise statements about the controller are Section 5.4, “Finite-state shootout controller (FSM over policy heads)” and Section 5.4, paragraph 4.

Coverage record
Paper anchor	`par:shootout-controller-p3` (website anchor; the manuscript has no label here)
Displayed as	Section 5.4, paragraph 3
Claim	The controller organises which head is queried and when to stop; it adds no per-step power
Coverage category	Informal interpretation
Statement source	paper_jmlr.tex lines 674–676; SHA-256 of the normalised block source `44c71a19cbbf238e…`
Depends on	Section 5.4, paragraph 4
Explanations	`explain/c-fsm-power`, `formal/c-fsm-power`
Build evidence	fresh isolated rebuild, audit report

4 Lean statement and source

No Lean statement corresponds to this claim. This is interpretation or motivation, not a formal mathematical claim.

Build evidence: fresh isolated rebuild of all 38 bundled modules with Lean (version 4.26.0, started 2026-10-10T15:21:56Z, finished 2026-10-10T15:27:00Z: 38/38 modules compiled, 1588 axiom reports, no errors. See Build for the commands and the independent audit of 2026-10-10 (dated evidence).