---
title: "Compact counterexamples to bi-Lipschitz dimension reduction"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 098; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "convex-and-metric-geometry"
  - "lean4"
  - "formalized"
seed_rank: 1955
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 8
lineage: ai-mathematical-reasoning
cites:
  - title: "A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding"
    url: "https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Compact counterexamples to bi-Lipschitz dimension reduction

## One-sentence takeaway

OpenAI's result family 098 (Convex and metric geometry) claims: Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space.

## Why it matters here

Convex and metric geometry underlies collision, packing, embedding and spatial data-structure work. 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. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops.

## Key ideas

- Family summary (repo): The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
- *A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding*: Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
- Lean scope (lean/docs/098.md): The formalized result gives a doubling subset of real $\ell_2$, with doubling constant at most $76800$, that admits no bi-Lipschitz embedding into any finite-dimensional Euclidean space at any finite distortion. A companion gives one universal doubling constant such that every infinite-dimensional real Banach space contains a compact doubling subset with no bi-Lipschitz embedding into any finite-dimensional normed space.

## 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: formalized in part. The repo has a scope note listing 2 Comparator challenge statement(s), and that note defines which statements are covered. Do not read the Lean coverage as the full content of every manuscript in the family.
- A grep of the repo's `lean/OAI` tree found no `sorry` and no top-level `axiom` declarations. The library was not compiled for this card, and Comparator was not run.

## Links

- Manuscript: [A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding](https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf)
- Lean scope: https://github.com/openai/math/blob/main/lean/docs/098.md
- Comparator statement (Doubling Hilbert subset without finite-dimensional embedding): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DoublingHilbert.lean
- Comparator statement (Compact counterexamples in infinite-dimensional Banach spaces): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CompactBanach.lean
- 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