S-AI-CITY: MATHEMATICAL FOUNDATIONS OF HORMONAL URBAN ORCHESTRATION

Abstract

S-AI-CITY Article I introduced a first-generation Sparse Artificial Intelligence architecture for urban orchestration based on six artificial hormones, a Hormonal Engine, parsimonious activation of 24 specialized agents, a Zone → District → City hierarchy, and symbolic Urban Engram memory. This Article II develops its formal mathematical specification. The city is modeled as a dynamic multi-layer graph on which six Gland Agents convert normalized observations into bounded self-limiting emissions that drive a stochastic graph reaction-diffusion system with leakage, cross-regulation, layer-dependent diffusion, external regulation, delay, and noise. We derive sufficient conditions for forward invariance of the normalized hormonal domain, global exponential stability of the frozen-input equilibrium, an explicit practical-stability bound for persistent inputs, and a second-moment bound for additive noise. A weighted loop-gain certificate shows that stability is governed by feedback loops of cross-regulation rather than by coupling magnitudes, yields explicit calibration guidelines, and guarantees incremental stability under time-varying inputs and undirected topologies. The second-moment bound is extended to the implemented projected Euler-Maruyama chain and to its reflected continuous limit. Agents are specified through formal contracts and a baseline 24 × 6 sensitivity matrix, and parsimonious orchestration is formulated as a constrained binary optimization problem whose extended operator provides a well-posed, authorizationpreserving fallback under infeasibility. A worked three-zone example illustrates the complete cycle.