Kusner's conjecture is false for every $p>4$

False for every $p>4$ . This is strictly stronger than the catalog's August entry (arXiv:2608.14013, significance 30), which exhibited $n+2$ equilateral points for a single exponent and only located the infimum of failing exponents in $[4,5)$ . Together with Swanepoel for $1<p<2$ and Ge, Xu and Zhou for $2 \le p \le 4$ , Kusner's conjecture is now settled for all $1<p<\infty$ : true exactly on $[2,4]$ .

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 13, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-discovered. 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: Kusner's conjecture is false for every $p>4$ · Kusner's conjecture for $p>4$

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · OpenAI

Nathan Xiong
human · human collaborator

Evidence vector

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Named evidence

VibeMathed record: Kusner's conjecture is false for every $p>4$

specialist registry · passed

Checked here on 22 September 2026 against arXiv:2609.14794. The abstract gives the construction explicitly - for every $p>4$ an equilateral set of $8m$ points in $\mathbb R^{8m-2}$ for an $m$ depending on $p$ - and states that combining it with Swanepoel and with Ge, Xu and Zhou resolves Kusner's conjecture across the whole range $1<p<\infty$ . The attribution of the conjecture to Kusner was confirmed in the paper's introduction. The mathematics was not checked here; nine days old, no referee.