Non-Gaussianity of the Stagnation Law in Particle Swarm Optimization
TL;DR
It is proved that throughout the open mean-square stability region, no invariant position marginal, and hence no limiting position marginal, can be Gaussian.
Abstract
We study one-dimensional particle swarm optimization during stagnation, with two fixed distinct attractors and equal independent uniform acceleration ranges. The position then satisfies a second-order random affine recurrence. For inertia $w$ and acceleration range $c$, we prove that throughout the open mean-square stability region \[ -10,\qquad 12(1-w^2)-c(7-5w)>0, \] no invariant position marginal, and hence no limiting position marginal, can be Gaussian. This solves the open Problem 18 in \cite{ParticleSwarmProblems}. The proof compares the stationary moment equations with the Gaussian moment identities through order eight. A Hermite-polynomial formulation gives explicit fourth- and sixth-order compatibility conditions whose common solutions lie on a degree-$107$ polynomial branch. Exact eighth-order equations exclude every point on that branch. The final certificate is verified using arithmetic modulo $23$ and independently modulo $1{,}000{,}003$. A separate raw-moment implementation produces exact polynomials $Q_4,Q_6,Q_8$ in $(w,c)$ and verifies the same obstruction over the rational numbers. The fourth- and sixth-order curves have a genuine admissible intersection, but the eighth-order condition removes it, showing why low-order Gaussian diagnostics are insufficient. All code, exact polynomials, logs, and plot-validation data are supplied as online resources.