Theorem: Exact switch and disagreement frequencies

At a stationary rule-ID switch, the ordered pair of old and new IDs is uniform over the twelve distinct ordered pairs. The probability of a change between identical functions is

Let

For the point law in Section sec:task,

For a fresh point after a stationary ID switch, informative disagreement probability is

Conditional on a genuine label-function change, it is

Proof. Uniform stationarity and the uniform nonzero increment give probability $1/12$ to each ordered pair. Only $(1,2)$ and $(2,1)$ change IDs without changing the function.

Functions $A,B$ disagree exactly in opposite-sign quadrants, giving $1/2$. Functions $A,D$ disagree on half of each same-sign quadrant and all of each opposite-sign quadrant, giving $3/4$. Functions $B,D$ disagree on half of every quadrant, giving $1/2$. Duplication supplies the other entries.

The sum over the six unordered off-diagonal pairs is $11/4$. Therefore the mean over twelve ordered pairs is $11/24$. Divide by the genuine-switch probability $5/6$ to obtain $11/20$.