Volume III extends the mathematical framework of The Architecture of Necessary Existence into the domain of cognition. Building on the proportional-delay field (Volume I) and the contained-time field (Volume II), this volume examines how recursive systems within cognitive processes can be modelled using bounded delay, feedback structure, and stability dynamics. The analysis develops the idea of cognitive containment: the way recursive patterns stabilise when proportional delay (δᴰ) regulates feedback and reduces collapse pressure in awareness-based systems. Using Lyapunov reasoning, contraction principles, and feedback-law structure, the volume outlines a general mathematical approach for understanding stability in recursive cognitive dynamics. The work is fully theoretical and presents no numerical simulations or empirical data. Its aim is to provide a systems-theoretic bridge between recursion, delay, and stability in cognition, forming the conceptual and mathematical foundation for the subsequent volumes on intelligent containment and universal containment.