The Bandelt-Dress Quartet Distance Conjecture

Precise statement

The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6

The paper states plainly that the author used GPT-5.6 to prove the theorem and to draft an initial version of the manuscript.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

arXiv preprint, not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-04

   arXiv:2608.03542 - The maximum quartet distance between phylogenetic trees

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.