A Structural Law of Swarm Intelligence: Path-Theoretic Unification of Navigability, Motion, and Direction

Abstract

Swarm intelligence (SI) algorithms—ant colony optimisation, particle swarm optimisation, bacterial foraging optimisation, and their descendants—share a striking structural regularity: agents navigate heterogeneous cost landscapes by locally coupling differentiation capacity to resistance gradients. Yet the field lacks a single, falsifiable, domain-spanning law that explains why these algorithms work and when they must fail. This paper proposes such a law. We introduce a path-theoretic framework built on three primitives—Perception (P), the capacity to differentiate among continuation paths; Distortion (D), the patterned resistance that makes some paths costlier than others; and Freedom (F), the resulting structural navigability—and derive F = P/D. We prove this is the unique ratio form satisfying scale-covariance, monotonicity, and dimensional separability. The law is developed through three formally distinct movements: (i) Navigability, establishing that usable continuation requires differentiation capacity divided by resistance; (ii) Motion, deriving the gradient law dx/dt = −P(x)·∇D(x) that governs directional displacement across all SI algorithms; and (iii) Direction, showing how discrete path selection and path filtration emerge from the scalar law. We demonstrate structural identity of the gradient law across ACO This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6304936 Preprint not peer reviewed A Structural Law of Swarm Intelligence Page 2 pheromone dynamics, PSO velocity updates, BFO chemotaxis, and backpropagation. Stigmergy is reinterpreted as Distortion-to-Perception conversion, with a formal dominance condition specifying when it outperforms structural modification. Testable computational predictions are derived for smart building scheduling optimisation: the framework predicts that sensor-augmented scheduling (Pintervention) achieves 3–7× greater Freedom improvement per unit cost than physical restructuring (Dintervention), and that convergence rate in scheduling algorithms scales as O(P/D̄). Seven falsification criteria specify conditions under which the framework must be abandoned. We argue that F = P/D offers the swarm intelligence community a structural language for algorithm analysis, design, and crossdomain transfer that complements the No Free Lunch theorems. Author ORCID: 0009-0008-6255-7724. License: Creative Commons Attribution 4.0 International.