One switch out, one switch back: who can track both?
What this is: the hidden state starts positive, flips negative, then switches back positive. The upper plot is each method's internal
belief; the lower plot is the probability that is actually scored.
Shared deterministic path: Exact Bayes and every candidate see the same two switches and common innovations.
1 · Stable: both methods predict +Watch both purple switch markers, then compare recovery.
Blue = Exact BayesOrange = Tanh clampvertical purple lines = state switches
Internal belief: blue is Exact Bayes; orange is the selected restricted update. Different curves mean different computations.
Predicted probability: the same two beliefs after decoding. Only this difference contributes to predictive error.
disagreement inside
mean logit gap
disagreement outside
mean Δp, what is scored
predictive divergence
mean KL(candidate || exact), nats
This controlled two-switch animation illustrates the mechanism; it does not prove the theorem. It is not a random typical path, and the
theorem concerns an explicit growing horizon.