Critical points and modes of the Kabata–Matsumoto–Okuno seven-mode Gaussian mixture

Abstract

Kabata, Matsumoto and Okuno (arXiv:2608.01776) constructed an equally weighted three-component bivariate Gaussian mixture p_(r,M) with at least seven modes for r ∈ [97/100,99/100] and M ≥ 10⁴, refuting the conjectured maximum C(d+k-1, d)=6 recorded by Améndola, Engström and Haase, and explicitly declined to say whether the seven modes are all of them. We settle two parts of that question. First, no mode lies on the three long ridges of the mixture: for r ∈ [97/100,99/100] and 1000 ≤ M ≤ 10⁶, no point x with |nⱼ · x-r| ≤ 1/2 and 16 ≤ tⱼ · x ≤ M-1 is a local maximum. Second, for r=97/100 and every M ≥ 10⁴, the mixture has exactly seven critical points in the rhombus Ω={|n₁ · x| ≤ 2, |n₂ · x| ≤ 2}, which contains the closed disc of radius 2 and hence the central triangle bounded by the three major axes; exactly four of them are modes (the origin and one near each pairwise intersection of the major axes), and the other three are saddle points. The count is uniform in M because a change of coordinates makes the gradient system rational in (n₁ · x, n₂ · x, 1/(sqrt3M)); the certificate is an exclusion covering by 1451 boxes under a preconditioned (Krawczyk) test, together with 7 × 4 Krawczyk existence-and-uniqueness certificates. Every statement, including the seven-mode existence theorem itself (re-proved by a derivative-free argument valid from M ≥ 1000), is verified in Lean 4 with Mathlib. The mode count on the whole plane, and the count at other values of r, remain open; a numerical Newton census, reported but not proved, finds thirteen critical points in the plane—seven maxima and six saddles—for every tested r ∈ [0.9,0.99].

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-09-07 03:53 UTC

   File fingerprint

   2f6d90971102e231654927ada06cd145b8b3702fe1dbd826115a6cf80cb6f65b