Five positive point charges in ℝ³ whose potential has at least 24 nondegenerate critical points — exceeding the conjectured bound of 16. Maxwell conjectured that n point charges produce at most (n−1)² critical points of their potential.
| field | as recorded at source |
|---|---|
| the claim | Five positive point charges in ℝ³ whose potential has at least 24 nondegenerate critical points — exceeding the conjectured bound of 16. |
| the problem | Maxwell conjectured that n point charges produce at most (n−1)² critical points of their potential. |
| credited AI system | OpenAI GPT-5.6 Sol (idea); written up by Arathoon, Ball and Kvalheim |
| verification at source | Three-author verification; Mathematica and Maple checks — floating point, no code artifact, no certificate, and no concrete ε. |
| our verdict | CONFIRMED at ε = 1/6 only |
At the fixed rational ε = 1/6 with qε = 859/248832 exactly, 24 pairwise-disjoint boxes, each certifying existence and local uniqueness of a critical point by the Krawczyk operator and nondegeneracy by an interval Hessian determinant that excludes zero. 24 > 16 is then exact integer arithmetic.
| [1] Exact configuration data (paper §2 + eq. (1)) and sqrt soundness | |
|---|---|
| 1 | q_eps = (3/4)(1/6)^3 - (5/32)(1/6)^5 = 859/248832 (paper eq. (1) at eps = 1/6) |
| 2 | q_eps > 0 — all five charges positive (hypothesis of the construction) |
| 3 | 24 > (n-1)^2 = 16 at n = 5 — the arithmetic of the refutation (exact integers) |
| 4 | eps and q_eps enclosed in proved intervals of width 8.67e-19 |
| 5 | sqrt(3) enclosure proved by exact rational squaring: lo^2 < 3 < hi^2 |
| 6 | sqrtIv = exp(log/2) containment verified on 7 rationals by exact squaring |
| 7 | float grad/Hess agree with finite differences of V (worst dev 5.2e-10) |
| [2] Candidates (float Newton from paper Lemma 2 seeds — no evidentiary weight) | |
|---|---|
| 1 | 24 distinct candidates found (got 24) |
| 2 | orbit shells |x| = 4.60e-14, 1.58e-3, 8.88e-3, 1.18e-2, 1.60e-2, 9.49e-2, 2.17e-1 with sizes [1,3,2,6,6,3,3] (origin; 3 planar-in; 2 axial; 6+6 off-planar; 3 planar-out; 3 edge) |
| 3 | theta=pi ray carries 3 planar equilibria at x = -1.580e-3, -9.490e-2, -2.172e-1 (paper Fig. 1d shows 3) |
| [3] Certificates: Krawczyk existence+uniqueness box and interval Hessian per point | |
|---|---|
| 1 | Krawczyk certifies existence + local uniqueness in all 24 boxes (got 24) |
| 2 | interval det Hess excludes 0 on all 24 certified boxes (got 24) — nondegeneracy |
| 3 | harmonicity witness: 0 in trace Hess interval on all 24 boxes (got 24) |
| 4 | all 276 box pairs disjoint; min coordinate gap 1.37e-3 |
| 5 | CLAIM GATE: 24 disjoint certified nondegenerate critical points => 24 > 16 = (5-1)^2 |
| [4] Structure against the paper (det signs and Morse indices per shell) | |
|---|---|
| 1 | certified Morse index per shell = [1,2,2,1,2,1,2] — matches Lemma 2 signatures + Remark 2 edge index |
| 2 | index totals {1: 10, 2: 14, 0: 0, 3: 0} — paper Remark 2: m1 = 10, m2 = 14; no index-0/3 (V is harmonic: no interior extrema) |
A verifier that cannot reject a false claim is not evidence, it is decoration. So each one carries mutation controls: the claim is deliberately corrupted and the verifier must refuse it. All 3 were rejected in the run that built this page, and a control that stopped firing would refuse the page instead.
| the corruption, and how it was caught | |
|---|---|
| 1 | M1 (q_eps -> +eps^5 term) rejected: only 1/24 boxes certify at the claimed points |
| 2 | M2 (candidate #8 replaced by a copy of #6) rejected: 24/24 boxes certify but disjointness fails |
| 3 | M3 (q_eps -> (3/4)eps^3) rejected: origin box refuses (float Hessian singular — no approximate inverse); 0/24 certify |
The theorem’s asymptotic statement — "for all sufficiently small ε" — and every proof in the paper. One ε is certified, not a range.
This is the part worth reading twice. The verdict at the top of this page is true inside its scope line and false outside it, and the sentence above is where the scope line comes from. Across all six claims in this set the same split appears: the computational fragment certifies and the analytic core does not — the flagship page measures exactly that.
git clone https://github.com/carlostoledo1891/cert-machine cd cert-machine node legacy/research/challenges/lane/laneb-maxwell/verify.js
Plain Node, no packages. It prints the ledger above, runs its own falsifiers, and exits non-zero if anything fails. node instruments/laneaudit/audit.js runs all six.