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A 32-leaf tree requiring six coordinates for an isometric ℓ∞\ell_\infty embedding

Every tree metric with tt leaves embeds isometrically into ℓ∞t−1\ell_\infty^{t-1}, and the least dimension needed is at least ⌈log⁡2t⌉\lceil \log_2 t \rceil. Fitzpatrick and Nowakowski asked in 2000 whether the logarithmic bound is always attained, and Brigham, Chartrand, Dutton and Zhang conjectured in 2005 that every tree with tt leaves embeds isometrically into ℓ∞⌈log⁡2t⌉\ell_\infty^{\lceil \log_2 t \rceil}, verifying it through t=21t = 21. Aksoy, Kilic and Kocak posed a sharp leaf-threshold version for weighted metric trees in 2020.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-assisted
Method
Construction
Field
Metric geometry; tree metrics; isometric embeddings into l-infinity
Posed by
Fitzpatrick and Nowakowski (Problem 43, 2000); conjectured by Brigham, Chartrand, Dutton and Zhang (2005)
Year posed
2000
Years open
26y
Solved
2026-08-17
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
Logan R. Chalmers
Verification
Unreviewed
Publication
Preprint
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Disproved. An explicit 32-leaf tree has least isometric ℓ∞\ell_\infty-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain ⌈log⁡2t⌉\lceil \log_2 t \rceil, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.

What the AI did

The paper's disclosure: "OpenAI's GPT-5.6 Sol assisted in implementing the computational search strategy and with drafting this manuscript. The author takes full responsibility for the content of the article." The search that found the tree and the proof that no five-orientation cover exists are the author's; the model wrote code and prose. That is the assisted tier on this site.

Verification

Unreviewed. arXiv 2608.16288 (17 August 2026, six pages) read in full here; the abstract's claims match the theorems. The counterexample is an exact finite argument with the covering masks printed in the paper and ancillary verification code deposited on Zenodo, which this site has not run. No peer review, no independent expert statement.

Sources

Submitted by BoldJackal580 on

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