Proposition: Three functions, four IDs

The functions $f_1,f_2$ are identical:

The other functions are distinct on positive-probability sets. Neither $f_0$ nor $f_1$ is a binary sign classification: both take all four label values.

Proof. The two displayed expressions differ only in the order of addition. All four sign combinations have positive probability under the continuous Gaussian mixture. Inspection of (eq:rules) distinguishes the remaining functions on open regions.