Proposition: Repetition and old-function responses can diverge

The two fractions in Definitions def:oldfraction and (eq:repeatmetric) do not determine one another.

In particular:

1. an old-function strategy can make every informative post-switch response an error while never repeating its immediately preceding response;

2. a constant classifier can repeat every erroneous response without implementing the old function;

3. a memoryless classifier can have old-function response fraction one for a selected old/new pair.

Proof. Use a switch from $B$ to $A$, with alternating points in the two opposite-sign quadrants. A predictor using $B$ produces alternating labels 1 and 2. Every post-switch point is informative and every response is the old-function response, but adjacent responses differ.

On the same post-switch points, a constant response 0 is always wrong, always repeated, and is not the old-function response.

Finally, the memoryless predictor $B(X_t)$ makes the old-function response on every informative trial whenever the old function is $B$ and the new function is $A$. No stored rule ID is needed.