Kusner's Conjecture on Equilateral Sets in $\ell_p^n$

Precise statement

Kusner conjectured in 1983 that the maximum number of points in $\mathbb{R}^n$ that are pairwise at $\ell_p$-distance one is exactly $n+1$ for every $2 < p < \infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

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 Sol, Claude Fable 5

The paper's disclosure: GPT-5.6 Sol assisted in implementing the computational search strategy in code, and Claude Fable 5 was used as a tool in drafting. No mathematical step is attributed to a model by name, but the search that produced the configuration is the load-bearing computation.

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

Verification boundary

unreviewed

Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14013): the disclosure is verbatim as quoted, Kusner's 1983 attribution is in the introduction, and the data is deposited on Zenodo (10.5281/zenodo.21911503). The exact-arithmetic certificate was not re-run here. Days-old preprint, no independent review.

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

Timeline

1. 2026-08-14

   A counterexample to Kusner's conjecture on equilateral sets

   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

construction (source-reported)

Source-reported tools: construction.

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.