The best bit?

The claims, in plain words

Flip n fair coins, then send them through a noisy channel that flips each one independently with probability ε. Before the noise, you may summarize the coins by a single yes-or-no answer, any Boolean function. Which summary shares the most information with the noisy observation? Kumar and Courtade conjectured in 2013 that the best is simply to report one coin, a dictator, which keeps 1 − h₂(ε) bits, at every noise level and in every dimension. Majority, for instance, keeps less.

At high noise, Ordentlich, Shayevitz and Weinstein found nearly sharp bounds for balanced functions, and Samorodnitsky proved the conjecture for all functions in a dimension-independent high-noise range; Yu extended the range for balanced functions with computer assistance. In 2017 Anantharam, Bogdanov, Chakrabarti, Jayram and Nair formulated a Hellinger conjecture that implies Courtade–Kumar, and Chen, Gohari and Nair developed a differential approach. Several other recent preprints also announce proofs of the Courtade–Kumar conjecture.

Result 119 is two OpenAI preprints dated September 24, 2026. The first (29 pages) proves a sharp contraction for any randomized binary summary: at a fixed initial information, a noisy coordinate keeps the most. For Boolean functions this gives the Courtade–Kumar inequality, and a refined bound that accounts for the function’s bias; it also proves a mean-dependent entropy-production bound. The paper notes that a production bound in an earlier preprint by Chen and coauthors, integrated from the initial information, also gives its main theorem. The second (50 pages) proves the Hellinger conjecture for every output bias and noise level: for a Boolean function of mean m, √(1 − m²) − E√(1 − (T_ρf)²) ≤ 1 − √(1 − ρ²), with ρ = 1 − 2ε and equality for signed coordinates. It implies the Courtade–Kumar bound again; its proof is computer-assisted through fixed scalar inequalities with exact rational certificates.

What it doesn’t claim

These are not the only claimed proofs of the Courtade–Kumar inequality: other groups have announced proofs by different routes, and the first paper itself notes that an earlier preprint’s production bound yields its main theorem. The Hellinger form is the distinct contribution. The refined bound for biased functions is an upper curve; the paper does not claim every biased function attains it.

Where it stands

Posted as preprints

OpenAI, 29 and 50 pages, dated September 24, 2026

Peer reviewed

Not yet

Checked in Lean

OpenAI published a Lean proof of the contraction paper, including Courtade–Kumar; not of the Hellinger paper; we have not run the check

Last checked October 19, 2026. The preprints have not been revised. OpenAI published a Lean formalization of the contraction paper: the Courtade–Kumar inequality with equality for dictators, matched against its comparator (43 files and 7,790 lines), and the sharp soft-channel contraction, bias refinement and production bound, matched against its comparator (70 files and 11,590 lines with its imports), not counting Mathlib. The Hellinger paper is not formalized. We have not run the checks. Neither paper is among those OSVerify reviewed.

How the proofs go

1. Information along the noise. As the noise grows, information drops at a rate given by an entropy production. If that production is always at least what a soft coordinate with the same information would produce, the effective amplitude shrinks at least as fast as a coordinate’s, which is the theorem. Following Chen, Gohari and Nair, the production inequality is proved by induction on dimension: split the cube along one coordinate and make the joining edges pay for the increase.
2. The joining problem. Tracking information alone fails for some pairs, so the proof keeps the mean as well, solving the local joining problem with an explicit three-point optimizer and a common entropy budget, two entropy capacities and a common-output thinning inequality.
3. The Hellinger proof. With s = √(1 − ρ²): for s ≤ 0.84, compression to increasing functions and dimension induction with an asymmetric profile and a sheared two-point comparison; for larger s, very biased functions are handled directly, and otherwise, at a first failure, an energy estimate along the noise forces a signed coordinate. The scalar inequalities, independent of the dimension, are certified with exact arithmetic.

The numbers in the video

• Every Boolean function on up to 4 bits (65,536 on 4 bits), at four noise levels: the bound held, and equality occurred exactly for the signed dictators; the refined and Hellinger bounds held too.
• At ε = 1/4, the best other function on 4 bits keeps 0.161 bits against 0.189 for a dictator; on 7 bits, majority keeps 0.134 bits.
• 3,000 random soft summaries on 5 bits never beat the sharp contraction curve.

Read the source

• Sharp binary-information contraction on the discrete cube (PDF), 29 pages.
• Hellinger contraction with arbitrary Boolean output bias (PDF), 50 pages.
• The Lean scope note.
• The collection’s catalog, where this is result 119.

The 2 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

• Sharp binary-information contraction on the discrete cube main paper
• Hellinger contraction with arbitrary Boolean output bias companion

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.