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CS-128 · computer science · mathematics · Information and coding theory

Sharp binary-channel information contraction on the cube

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let X be uniform on ${-1,1}^n$, $u:{-1,1}^n\to [-1,1]$, and $Y\in {-1,1}$ have conditional mean u(X). If $T_\rho $ independently flips cube coordinates with correlation $\rho $, must $I(T_\rho u)\leq \psi (|\rho |\psi ^{-1}(I(u)))$ for all $-1\leq \rho \leq 1$, where $I(u)=h(Eu)-E$ h(u), h(a) is the binary entropy of (1+a)/2, and $\psi (a)=\log $ $2-h(a)$?

Short English statement

Let X be uniform on ${-1,1}^n$, $u:{-1,1}^n\to [-1,1]$, and $Y\in {-1,1}$ have conditional mean u(X). If $T_\rho $ independently flips cube coordinates with correlation $\rho $, must $I(T_\rho u)\leq \psi (|\rho |\psi ^{-1}(I(u)))$ for all $-1\leq \rho \leq 1$, where $I(u)=h(Eu)-E$ h(u), h(a) is the binary entropy of (1+a)/2, and $\psi (a)=\log $ $2-h(a)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-128-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-128

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).

Sharp binary-channel information contraction on the cube

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 among binary summaries with fixed information on a uniform cube, a noisy coordinate retains the most information, including the Courtade--Kumar bound and a bias-sensitive refinement.

Mathematical statement of the claimed result

Manuscript 1.
On the uniform cube $\Omega_n=\{-1,1\}^n$, $n\ge0$, take any soft binary channel $u:\Omega_n\to[-1,1]$. Define, using natural logarithms, \[ H(r)=-\tfrac{1+r}2\log\tfrac{1+r}2-\tfrac{1-r}2\log\tfrac{1-r}2, \quad\psi(r)=\log2-H(r),\quad I(u)=H(\mathbb Eu)-\mathbb EH(u). \] Under independent symmetric coordinate noise of mean $\rho\in[-1,1]$, $T_{\rho}$ is conditional averaging. The claimed sharp contraction is \[ I(T_{\rho} u)\le\psi(|\rho|\psi^{-1}(I(u))), \] where $\psi^{-1}$ is the increasing nonnegative branch from $[0,\log2]$ to $[0,1]$. For $n\ge1$, every $u(x)=ax_i$ with $|a|\le1$ attains equality. Output bias is unrestricted.

Outline of the argument

For manuscript 1 (family 119; 2026-09-24), a three-point configuration optimizes the local join. Separate entropy budgets and a reserve for thinning either common output close the dimension induction; integration of the mean-dependent entropy-production bound along the noise semigroup gives the sharp contraction.

Formal verification

For manuscript 1 (family 119; 2026-09-24), OpenAI lists Lean statements for this assertion: among binary summaries with fixed information on a uniform cube, a noisy coordinate retains the most information, including the Courtade--Kumar bound and a bias-sensitive refinement.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are CourtadeKumar.lean, SoftChannel204.lean.

Their named declarations include OAI.LeanBlast.CourtadeKumar.courtadeKumarAndAttainment, OAI.SoftChannel204.fullMain.

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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