Theorem: Effective-function identification from feedback

Suppose one of $A,B,D$ is fixed for $n$ independent point trials and noiseless target feedback is received after each prediction. Retain all functions consistent with the observed point-label pairs. The probability that any incorrect distinct function remains after $n$ feedback trials is at most

The duplicate IDs 1 and 2 remain observationally identical within that fixed-rule experiment.

Proof. An incorrect function $g$ survives each fresh trial with probability $1-d(f,g)$. For all distinct effective function pairs, (eq:disagreementmatrix) gives $d(f,g)\ge 1/2$. Independence makes its survival probability at most $2^{-n}$. Union bound over the two incorrect functions.

IDs 1 and 2 prescribe identical labels for every point, so their point-label laws are identical.