Theorem: Oai119-A finite Hellinger certificate

For every Boolean function $f:{\Omega }_2\to \{-1,1\}$, with $X$ uniform and independent symmetric noise of correlation $1/2$,

Equality holds if and only if

The gaps are exactly

In particular,

and every function that is not a signed coordinate satisfies

Proof. The representative tables and evenness of $H$ give every entry in (eq:gaptable). Lemma lem:orbits transfers them to all sixteen functions.

For parity, Lemma lem:roots gives

These bounds lie strictly between $102/1000$ and $103/1000$.

For the exceptional-vertex orbit, the same root enclosures give

and

The two numerators are, respectively, $87429$ and $87467$. Therefore

Since

this proves (eq:biasedgap).

For constants,

Thus every noncoordinate class has a strictly positive gap, and the coordinate class has gap zero. This proves the inequality, the equality classification, and (eq:uniformgap).