Kusner's conjecture fails at a second exponent: a certificate for 58 equilateral points in ℓ₆⁵⁶
Abstract
An equilateral set in a normed space is a set of points at pairwise equal distance, and e(ℓₚⁿ) denotes the largest cardinality of one. Kusner conjectured that e(ℓₚⁿ)=n+1 for 2<p<∞. Chalmers has recently refuted this at p=5, by exhibiting 58 points in lp(5) as the unique zero, inside an explicit box, of a polynomial system with dyadic coefficients; his paper records that it is not known how far that configuration extends in the exponent, and observes that its verification scheme uses the oddness of 5. We address both points. Continued in the exponent inside the square slice that Chalmers uses to make the system square, the solution branch is a closed loop of width 0.0367 around p=5 and reaches no other integer; but the equilateral condition on 58 points in ℝ⁵⁶ is 1653 equations in 3249 unknowns, so that fold belongs to the slice and not to the problem. Continuing in the full space and refining in exact arithmetic, we obtain a Newton–Kantorovich certificate at the even exponent 6, on Chalmers' own frozen set and variable order: a centre with dyadic coordinates, a preconditioner, preconditioned residual η=4.4320 × 10⁻¹⁶ and contraction factor q=2.3563 × 10⁻⁴ at box radius 2⁻³³, so that the Kantorovich test clears by a factor 4176, and still passes at a box radius 4244 times the one used at p=5. The five arithmetic claims of the certificate, the radius window, the constants and the negative controls are checked by the Lean 4 kernel. Because the exponent is even, the sign-constancy hypothesis that the p=5 scheme needs disappears entirely. Subject to the classical reduction of the contraction hypotheses to those arithmetic claims—which is carried out here by hand and is not machine-checked, and which we flag at every use—this gives e(ℓ₆⁵⁶) ≥ 58>57: Kusner's conjecture fails at a second exponent, and at an even one, where it was open and the best bound available was e(ℓ₆⁵⁶) ≤ 169.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
927bf7fe9a7eb58c4dffd5d9471a1b3a674e7d36c11886d3f93102fa7fffa24f
Claim ledger
Stated results
KU1known data2026-08-22
(C1)-(C3): the schema, the minimum centre gap g = 215220270/2⁵0, and positivity of D on the box
KU2known data2026-08-22
(C4): the residual at the centre, ||F(c)||ᵢnf < 2⁻47 and eta = ||B F(c)||ᵢnf < 2⁻43
KU3known data2026-08-22
(C5): the contraction bound q < 2⁻14 – the expensive claim
KU4known data2026-08-22
Headline: all six claims (C1)-(C6) of the source's Proposition 4 hold for the archived certificate
KU5routine2026-08-22
Negative controls, and the finding that (C4) catches neither index error the archive warns about
KU6known2026-08-23
Lemma 3 of the source: Newton–Kantorovich existence, uniqueness, the distance bound, and invertibility, on an ℓ_∞ box in ℝ^P
KU7known2026-08-23
The interval-arithmetic form: an entrywise enclosure of the Jacobian plus one rational row-sum inequality
KU8known data2026-08-23
The archived certificate's scalars over ℝ: (C6) in the source's own form η + qρ < 2⁻¹⁰ρ < ρ, and Lemma 3 at ρ = 2⁻³³, η, q
KU9routine2026-08-23
Negative controls for Lemma 3, and its non-vacuity
KU10known data2026-08-29
(C2) per entry, Fin-quantified: at every pair k and coordinate r with a movable endpoint, w·ρ < abs Δ
KU11routine2026-08-29
The sharp box radius for per-entry sign constancy: 7104788948/2⁵0 = 54205.2 ρ, and it fails at the next integer
KU12known data2026-08-29
(C4) per row as a Finset.sum: ∀ a, abs (B·F(c))ₐ ≤ η, with η attained
KU13known2026-08-29
The bridge to ℝ: sign constancy and ‖B·F(c)‖_∞ ≤ η over ℝ — the source's Lemma 3 hypotheses at its own certificate
KU14known data2026-08-29
(C5) per row as a sparse Finset.sum: ∀ a, qRowₐ ≤ q, reproducing qNum's numeral
KU15measurement2026-08-29
MEASUREMENT: Finset.sum versus Array fold under native_decide on this certificate, the shared-box timing variance, and the verdict on strategy experiment A1
This ledger entry is reported in prose and is not bound to a Lean theorem.KU16routine2026-08-30
The certificate's box-radius window is exactly [61, 2548288865]/2⁵⁰, sharp at both ends — uniqueness in a box 19441.9× wider than the source's
KU17routine2026-08-30
The certificate's unused margin: ρ(1−q(ρ)) > 10⁷·η at ρ† = 1274000000/2⁵⁰, plus ‖B‖_∞, max abs Δ and the Jacobian row bound
KU18routine2026-08-30
The 58-point configuration, and every configuration in the source's box, has trivial isometry group
KU19prose2026-08-30
PROSE: the source's Corollary 2 made quantitative — e(ℓₚ⁵⁶) ≥ 58 for every real p with abs(p−5) ≤ 4×10⁻¹²
This ledger entry is reported in prose and is not bound to a Lean theorem.KU20candidate2026-08-30
A Newton–Kantorovich certificate at exponent 6: 58 equilateral points in ℓ₆⁵⁶, on the source's own frozen set
KU21routine2026-08-30
The exponent-6 radius window: the Kantorovich test passes at R = 556278737/2⁵⁰ and fails one unit above; its constants, and the negative controls
KU22measurement2026-08-30
MEASUREMENT: the source's square slice folds into a closed loop of width 0.0367, while the full-space branch runs to p = 7 and breaks only at p ≈ 4.9267
This ledger entry is reported in prose and is not bound to a Lean theorem.KU23prose2026-08-30
PROSE: Kusner's conjecture fails at p = 6 — e(ℓ₆⁵⁶) ≥ 58 > 57
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. L. R. Chalmers, *A counterexample to Kusner's conjecture on equilateral sets*, arXiv:2608.14013 (math.MG, v1, 14 Aug 2026). Certificate data: doi:10.5281/zenodo.21911503.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7