geometry-topology / Discrete geometry
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.
Temporal state
Append-only history
The Zenodo archive matched its published checksums. Its quick check covered all 1,653 point pairs, and the full FLINT-backed verifier passed exact claims C1-C6 in 80 seconds. The result is recorded as an artifact replay, not peer review.
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 polyno…
Research memory
The archived exact certificate satisfies all six stated contraction conditions for a unique exactly equilateral configuration of 58 points in $\ell_5^{56}$ inside the specified rational box; since $58>57=n+1$, this configuration is a counterexample to Kusner's conjecture.
This status is an exact replay of the authors' certificate with its pinned Python dependencies. It does not independently review every argument in the paper or constitute peer review.
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.
Replay the archived Kusner exact certificate
success · MFN independent verifier
All 1,653 pair checks completed and exact contraction claims C1-C6 passed in the full verifier.
Evidence graph
parent of · claim · counterexample
parent of · event · reproduction