Modes of a mixture of three homoscedastic Gaussians: log-concavity at diameter two, the ghost-mode threshold, and a certified asymmetric four-mode example
Abstract
Let f(x)=Σⱼ₌₁³ wⱼexp(-norm(x-μⱼ)²/2) be a mixture of three Gaussian densities with a common covariance (normalised to the identity), nonnegative weights of positive sum, and centres μⱼ in a real inner product space. The largest possible number of modes (local maximisers) of f is known to lie between 4, attained by the equilateral construction of Carreira-Perpiñán and Williams, and 8, the unconditional bound of Okuno and Kabata. We prove that if the centres have pairwise distance at most 2 then f is log-concave, whatever the weights and the dimension, so every mode is a global maximiser and all modes take the same value; the constant 2 is exact, being the classical two-component threshold. For the equilateral equal-weight configuration of side L the second derivative of log f at the barycentre is exactly (L²/3)(L²/6-1), so the central "ghost" mode disappears at L²=6, the threshold of Carreira-Perpiñán–Williams and of Edelsbrunner–Fasy–Rote in side-length form. Finally we give two explicit rational configurations in the plane, each with four rational balls of radius 10⁻³ that provably contain exactly one critical point of f, a maximiser of f over the ball: Duistermaat's equilateral example, and a scalene one with three distinct weights and one pair of centres further apart than sqrt6, so the four-mode region is not confined to the equilateral window; every published four-mode example we know of is equilateral with equal weights. All statements are machine-checked in Lean 4, the configuration-specific rational checks of the two examples by compiled code. A search over 9,288,663 configurations of the exact five-parameter reduction, reported as a computation, found no configuration with five modes or with more than seven critical points.
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Claim ledger
Stated results
HM1candidate2026-09-03
Three homoscedastic Gaussian components (common covariance normalised to I, any nonnegative weights with positive sum, any real inner product space) whose centres have pairwise distance at most 2 have a log-concave density; the constant 2 is sharp
HM2candidate2026-09-03
Under the diameter-2 hypothesis of HM1 every mode is a global maximiser and any two modes take the same value, so the mixture has one modal value and a convex set of maximisers
HM3known2026-09-03
For an equilateral triangle of side L with barycentre at the origin and equal weights, the log-density has vanishing first derivative at the origin in the direction of a vertex and second derivative exactly (L²/3)(L²/6 - 1)
HM4known2026-09-03
If L² > 6 the barycentre of the equilateral equal-weight homoscedastic three-component mixture is not a mode, in every ambient dimension; explicit witness (2,-1,-1), (-1,2,-1), (-1,-1,2) in EuclideanSpace R (Fin 3), an equilateral triangle of side sqrt 18
HM5routine2026-09-03
Too-large control: for a centred equilateral triangle with L² > 6 the log-density is not concave (explicit R³ witness of side sqrt 18), so the constant 2 of HM1 cannot be replaced by anything exceeding sqrt 6
HM6routine2026-09-03
Non-vacuity and too-small controls: three coincident centres satisfy the diameter-2 hypothesis and the density does have a mode; and 0, e, 2e with |e| = 1 meets the hypothesis with equality, so it does not secretly force the centres together
HM7measurement2026-09-03
Search: over 9,288,663 configurations of the exact five-parameter reduction (gammaᵤ, gammaᵥ, psi, kappaᵤ, kappaᵥ), no homoscedastic three-component mixture with 5 or more modes, and none with more than 7 critical points against the proved bound of 15; the four-mode region is confined to within about 6 percent of equilateral with weights within about 5 percent of equal; a Morse-parity audit shows the enumeration is complete on 97.3 percent of the samples
This ledger entry is reported in prose and is not bound to a Lean theorem.HM8known2026-09-03
The constant 2 of HM1 is exactly optimal: at the midpoint of two centres carrying positive weight, the second derivative of the log-density along the line joining them is d²(d²/4 - 1) with d their distance, so the log-density fails to be concave as soon as d > 2; explicit witness, two unit-weight components at distance 3 on the real line
HM9measurement2026-09-03
The four-mode region, measured: with exactly equal weights, all 44 four-mode configurations found in a 400,000-sample isoceles scan have both sides in [2.327, 2.393], apex angle in [59.32, 60.57] degrees and diameter in [2.353, 2.394]; and at Newton grid 60 the Morse-parity audit reports 0 of 400,000 detection failures on that arm and 1 of 400,000 on the near-equilateral arm, against 18.3 percent in the extreme-scale, extreme-weight regime
This ledger entry is reported in prose and is not bound to a Lean theorem.HM10known data2026-09-03
Certified four-mode witness at Duistermaat's configuration: an exact rational three-component homoscedastic mixture in R² (one side of squared length exactly 5.76, equal weights) for which four explicit pairwise-disjoint rational balls of radius 1/1000 each provably contain EXACTLY ONE critical point of the density, and that critical point maximises the density over the whole ball – so the mixture has at least four modes, kernel-checked
HM11candidate2026-09-03
A certified four-mode homoscedastic three-component mixture whose centre diameter EXCEEDS the equilateral ghost-mode threshold: a scalene triangle with squared side lengths 5.491666, 5.785263, 6.158248 (pairwise different, and the largest above 6, the exact equilateral threshold of HM3/HM4) and three distinct weights 1: 1.051434: 1.107307 has at least four modes, each localised to an explicit rational ball of radius 1/1000 containing exactly one critical point of the density; so the four-mode region of the five-parameter problem is not contained in diameter² <= 6
HM12routine2026-09-03
Negative controls for the mode certificate: with the preconditioner +Id instead of -Id the contraction constant at a genuine mode is 1.051973 > 1; at radius 1/50 instead of 1/1000 it is 1.066766 > 1 (and 1.731133 at radius 1/10); and at a genuine SADDLE of the same density, same radius and same preconditioner, it is 1.046502 > 1, so the test discriminates modes from other critical points
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A Gaussian mixture is homoscedastic when all components share one covariance matrix Σ. The affine change of variables x ↦ Σ^(-1/2) x is a bijection on modes, so we always normalise Σ = I and study, in a real inner product space E,
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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