---
title: "Interior $C^{1,\\alpha}$ regularity for infinity-harmonic functions"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 377; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "partial-differential-equations"
  - "unformalized"
seed_rank: 2232
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Uniform Interior $C^{1,\\alpha}$ Estimates for Infinity-Harmonic Functions"
    url: "https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Interior $C^{1,\alpha}$ regularity for infinity-harmonic functions

## One-sentence takeaway

OpenAI's result family 377 (Partial differential equations) claims: Proves uniform interior $C^{1,\alpha_d}$ regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension.

## Why it matters here

PDE results are the theory behind fluid, wave and transport simulation the engines approximate numerically. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.
- *Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions*: We prove a uniform interior $C^{1,\alpha_d}$ estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some $\alpha_d\in(0,1/3]$ depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions](https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math