From Empirical Description to Analytical Truth: Rectifying Systemic Errors in Mathematical Modeling of Swarm Morphogenesis

This paper presents a rigorous scientific and engineering protocol for the deconstruction and subsequent analytical solution of the swarm morphogenesis problem, using termite arch construction as a model. The study systematically uncovers and eliminates critical methodological errors inherent in classical mathematical frameworks, such as the reliance on stochastic Langevin white noise, dynamic adaptive perception thresholds, and the neglect of three-dimensional spatial physics. Based on a strict deterministic behavioral protocol of an isolated agent, we derive a continuous integro-differential convergence equation for structural consoles. This macro-model is intrinsically closed via the non-linear coupling of local environmental humidity and pheromone diffusion rates, allowing the analytical calculation of invariant phase transition bifurcation constants. The framework is validated through a comprehensive coordinate grid search and a Monte-Carlo stochastic verification protocol (N=5000 independent environment states).