Four thirds to four?

The claims, in plain words

Map a simply connected domain W in the plane conformally onto the unit disk with a map φ. The size of |φ′| measures how much the map stretches near each point. In 1978 Brennan conjectured that the area integral of |φ′|^s over W is finite for every s between 4/3 and 4, for every such domain, with no condition on the boundary. For the inverse map f from the disk, the range becomes powers t of |f′| strictly between −2 and 2/3. The Koebe function z/(1 − z)², which maps the disk onto the plane minus a slit, shows that both endpoints fail.

The hard part is negative powers, where the derivative is very small. They are measured by integral means of |f′|^t around circles of radius r, which grow like a power of 1/(1 − r); the worst exponent over all normalized univalent maps is the integral-means spectrum B_S(t), and Brennan’s conjecture is equivalent to B_S(−2) = 1. Distortion alone gives 2. Carleson and Makarov proved B_S(t) = |t| − 1 for sufficiently negative t, without reaching −2. Bertilsson extended the area range to about 3.42, and Hedenmalm and Shimorin to about 3.75, bounding B_S(−2) by about 1.22; Sola lowered that to about 1.21.

Result 072 is two OpenAI preprints, both dated September 24, 2026. The first (31 pages) claims a bound on the inverse-square integral means with exponent 1 + ε, uniform over all normalized univalent maps; so B_S(−2) = 1, Brennan’s conjecture holds with its full range, and B_S(t) = |t| − 1 for every t ≤ −2. The second (15 pages) concerns the spectrum for maps onto bounded domains. Kraetzer’s experiments suggested t²/4 for |t| ≤ 2, which is 1/4 at t = −1; the paper proves a bound with exponent strictly below 1/4 there, uniform over all normalized univalent maps, disproving that prediction.

What they don’t claim

The exact value of the bounded spectrum at t = −1 (the gap below 1/4 is not computed), its value at t = 1, and much of the rest of the integral-means spectrum (a survey describes what is known).

Where it stands

Posted as preprints

OpenAI, both dated September 24, 2026, released October 6

Peer reviewed

Not yet

Checked in Lean

Main statements of both papers, by OpenAI (scope note); we have not run the checks

Last checked October 19, 2026. Neither preprint has been revised. OpenAI’s scope note describes Lean proofs of the main statements (Brennan and strict bound): 47 files and about 14,000 lines for the first paper and a separate 94 files and about 16,000 lines for the second, with no unfinished proofs; we have not built them. The two papers cite each other as independent companions, and neither cites papers from other results in the release. Inputs include classical distortion and compactness for univalent maps, Hopf’s tangent-turning argument and Sard’s theorem; the method is related to the conformal snowflakes of Beliaev and Smirnov. We found no public assessment by specialists.

How the proofs go

1. A dyadic operator. On the upper half-plane, with univalent maps normalized at i, changing the base point gives another map in the same compact class. A positive operator averages two such changes, at −1/2 + i/2 and 1/2 + i/2, weighted by |1/F′|²; its iterates sample points at heights 1/2, 1/4, 1/8, … above an interval. If its growth rate is at most 2, the uniform bound follows.
2. A law on maps. Otherwise the rate is 2^β with β > 1, and the paper builds a probability law on maps whose weighted changes of base point scale exactly like y^−β. Differentiating this identity fixes moments of the logarithmic derivative, which make an expected Jacobian strictly positive.
3. Saddles and maxima. For a target ξ, the potential y^k/|F − ξ| has compact superlevel sets and only maxima and saddles as critical points. An index count built on Hopf’s argument finds at least as many maxima as saddles above every level, so each saddle is paired with a maximum at least as high; integrating against the law makes the same expected Jacobian at most zero, a contradiction.
4. Strictness at t = −1. The companion runs a related argument; distinct critical values make each pairing strictly increasing, forcing a vanishing Jacobian in the borderline case, which a harmonic lower barrier rules out.

The numbers in the video

• A short Python script computes, for the Koebe function, (1 − r) times the inverse-square integral mean (22.5, 30.9, 31.89 and 31.99 at r = 0.9 to 0.9999, tending to 32), so its exponent is exactly 1 and the area integral diverges; checks the change of variables t = 2 − s and the dyadic points sampled by the operator; and evaluates Kraetzer’s predicted values.

Read the source

• Brennan’s conjecture and sharp inverse-square integral means (PDF), September 24, 2026 (31 pages).
• A strict inverse-first-power bound for univalent functions (PDF), September 24, 2026 (15 pages).
• OpenAI’s note on the scope of the Lean proofs.
• Hedenmalm and Shimorin, weighted Bergman spaces and the integral-means spectrum (arXiv).
• The collection’s catalog, where this is result 072.

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.

• Brennan's conjecture and sharp inverse-square integral means main paper
• A strict inverse-first-power bound for univalent functions 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.