Almost-Sure Lyapunov Stability of PSO-Based Social Consensus on the Neutrosophic Simplex via Doob Supermartingales

Abstract

Almost-sure Lyapunov stability is established for Particle Swarm Optimization (PSO) acting as a distributed social consensus protocol on the neutrosophic 2-simplex Δ². The system is modelled as a stochastic hybrid system H = (C, f, D, g) with a PSO flow (Mode 1) and anti-stagnation escape resets (Mode 2). With the Lyapunov candidate V(t) = 1 − (1/N)ΣᵢΨᵢ(t), Ψᵢ = (1−Fᵢ)(1−Iᵢ) being the epistemic reliability of agent i, three results are proved. Lemma 1 gives an explicit reliability-separation bound εgap = LΨ · Dnet,min > 0. Theorem 2 establishes E[V(t⁺)|ℱt⁻] ≤ V(t⁻) at every escape step size, once the escape is clipped at the chord joining the stagnating agent to the reliability-weighted centroid. Theorem 3 proves V(t,j) → 0 almost surely over the hybrid time domain via Doob's non-negative supermartingale convergence theorem, the first almost-sure result for a hybrid social consensus protocol on the neutrosophic simplex, as far as the literature indicates. Proposition 3 derives tₛ ∝ 1/(1−Ī)², linking settling time to corpus mean indeterminacy Ī. Validation on six corpora (n = 17,662; health, politics, science denial, social movements) over 180 Monte Carlo runs yields V(T) = 0 in every run, tₛ = 5.4 ± 0.3, and one deviation of magnitude 3×10⁻⁴ across 25,000 hybrid transitions. Hegselmann–Krause, DeGroot and Friedkin–Johnsen all plateau above V∞≥ 0.584 (paired t-test p < 10⁻¹⁰, Cohen's d > 14). The framework suggests a reusable proof template for stochastic hybrid consensus protocols.