The two general assertions being audited

The manuscript Sharp binary-information contraction on the discrete cube, dated 24 September 2026, asserts

for every soft summary on every finite uniform cube. The inverse is the increasing branch from $[0,\ell ]$ to $[0,1]$. Soft coordinates attain equality.

For Boolean $f$ of mean $m$, its main:ck specializes this to

The refined upper curve is strictly below the universal curve when $0<|m|<1$ and $0<|\rho |\le 1$. Its separate upgrade:production asserts a stronger, mean-dependent production bound OpenAI119I.

The manuscript Hellinger contraction with arbitrary Boolean output bias, also dated 24 September 2026, asserts

for every Boolean $f$, every dimension, and every correlation. Signed coordinates attain equality. Its cor:ck derives the ordinary Courtade–Kumar bound from this assertion OpenAI119II,OpenAI119Implication.

The Courtade–Kumar formulation in bits is

For $\rho <0$, the same formula is valid because $h_2(\epsilon )=h_2(1-\epsilon )$.