Competent Agents Reveal Exactly What a Random Walk Sees: The Complete World-Model Information in Behavior

Which properties of a partially observed environment must a competent agent’s choices reveal? Cifuentes (2026, Section 4.2) pose this identification question for goals that name only observations and actions, and leave it open. We answer it for finite communicating models that admit a policy optimal for every such goal and every hidden start. The answer is the random-walk law, the stationary distribution of observation–action words under independent uniform actions. This law determines every history-dependent controlled prediction, and it is exactly the information guaranteed by competence. An information converse shows that no quantity outside it is determined by every competent policy. A constructive recovery obtains it from sampled actions at one supplied observation, under competence averaged over the queried goals. Both directions extend to a fixed relative loss below one half on the full finite class without a common state cap, where the converse connects equal-law models through a finite common predictive realization. A known state cap changes the answer at positive loss: three deterministic-emission states preserve the characterization, while four can force one additional bit.