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CS-127 · computer science · mathematics · Information and coding theory

Hellinger contraction with biased Boolean outputs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every Boolean f on a uniform n-dimensional discrete cube and every correlation $\rho \in [-1,1]$, is $H(Ef)-E$ $H(T_\rho f)\leq 1-H(\rho )$, where $H(a)=\surd (1-a^{2})$ and $T_\rho $ is coordinatewise noise?

Short English statement

For every Boolean f on a uniform n-dimensional discrete cube and every correlation $\rho \in [-1,1]$, is $H(Ef)-E$ $H(T_\rho f)\leq 1-H(\rho )$, where $H(a)=\surd (1-a^{2})$ and $T_\rho $ is coordinatewise noise?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-127-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-127

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 119; 2026-09-24).

Hellinger contraction with biased Boolean outputs

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 119; 2026-09-24), the asserted result is that the Hellinger loss inequality holds for every Boolean output bias and all correlations, with signed coordinates attaining the bound and an information-theoretic consequence.

Mathematical statement of the claimed result

Manuscript 1.
Let $X$ be uniform on $\{-1,1\}^n$ and independently multiply each coordinate by a sign with mean $\rho\in[-1,1]$ to obtain $Y$. Set $T_{\rho} f(x)=\mathbb E[f(Y)\mid X=x]$, $H(t)=\sqrt{1-t^2}$ and $m=\mathbb Ef$. For every $n\ge1$ and Boolean $f:\{-1,1\}^n\to\{-1,1\}$, the claimed inequality is \[ H(m)-\mathbb E H(T_{\rho} f)\le1-H(\rho). \] Both expectations use uniform measure, and $m$ need not be zero.

Outline of the argument

For manuscript 1 (family 119; 2026-09-24), asymmetric dimension induction separates compressed small coordinates from the central band. A calibrated semigroup-energy estimate controls the remaining bias range, with exact Bernstein and interval certificates validating the scalar interface.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 119; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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