Brennan's Conjecture

Brennan's conjecture, part 2: B_b(-2) = 1.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement brennan_universalSpectrumBounded as open.

Recent changes upstream

2026-11-12 · Formal Conjectures: status of `brennan_universalSpectrumBounded`: research open · source

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `brennan_universalSpectrumBounded`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-brennanconjecture,
  title        = {Brennan's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/brennanconjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Brennan's conjecture, part 2: $B_{b} (- 2) = 1$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Brennanconjecture.

theorem brennan_universalSpectrumBounded :
    universalSpectrumBounded (-2) = 1

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
arXiv:2409.15074
arXiv:2512.09330
[OAI26a] OpenAI, Brennan's conjecture and sharp inverse-square integral means. OpenAI Math Release preprint (2026). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.