Example. Complementary observations defeat one-step information value [ftip-00M6]

✍️sourceAGENTDRAFTED

Let \(X,Y\) be independent uniform bits. The terminal artifact is a guess for \(X\mathbin \oplus Y\); its quality is one if correct and zero otherwise. Two observation actions reveal \(X\) and \(Y\), respectively, at cost \(c\) each, where \(0<c<1/4\). The hard budget permits both observations. A terminal guess requires no additional observation cost.

Without observations, the best expected quality is \(1/2\). Given only \(X\), the unrevealed bit \(Y\) remains uniform, so the best expected quality is still \(1/2\); the same holds with \(X,Y\) exchanged. Thus a rule that compares stopping with taking one observation and then stopping assigns either observation net value \(1/2-c<1/2\). It stops immediately.

Observing both bits determines their parity. Its expected quality is one and its quality minus observation cost is \(1-2c>1/2\). Hence neither zero one-step information value nor a myopic stopping decision certifies the full controller frontier. This is a finite calculation, not an empirical claim about a language model. The scalar cost penalty is used only to exhibit the myopic decision; the hard-budget quality frontier itself is defined in Definition [ftip-00M5].