Section 5.4, paragraph 5Channels versus protocol: heads supply measurement channels, the FSM sequential control

Integration (channels vs. protocol). It is helpful to reinterpret Part (ii) of Theorem 5.12 in an “information-channel” language: each policy head $\pi_j$ defines a distinct rollout measurement channel which, when iterated to termination, returns a single bit $r_j=\mathrm{RO}_{\pi_j}(P)$ that depends on the hidden $B$-bit invariant $\sigma(P)$ (cf. the identities in Proposition 5.9). With a single head, repeated deterministic root rollouts return the same linear functional $\ell(\sigma)$; admissible one-heap modifications give its known affine corrections (Proposition 5.10). These measurements supply no further unknown invariant bit. With multiple heads, the agent has access to linearly independent invariant measurements and can reconstruct $\sigma(P)$ from their outcomes (Proposition 5.9). The finite-state shootout controller introduced above is the natural generalisation of this idea: it does not change what any head can compute, but organises the heads into an adaptive querying protocol—the controller decides which channel to query next (and when to stop, or when to re-query for robustness) based on previously observed rollout outcomes. In this view, multi-head provides the channels, while the FSM provides the sequential control that turns those channels into a practical evaluation procedure.