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

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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Aug 17, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding · A 32-leaf tree requiring six coordinates

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

GPT-5.6 Sol
model · ai model contributor · OpenAI

Logan R. Chalmers
human · human collaborator

Lineage and corrections

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. parent of this event

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

VibeMathed record: A 32-leaf tree requiring six coordinates for an isometric ℓ∞\ell_\infty embedding evidence for this event

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

This event attributed to GPT-5.6 Sol

This event attributed to Logan R. Chalmers

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A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding — Mathematical Frontier Network