DDSE-IV: Mathematical Foundations of Developmental Computation

This work presents the fourth stage of the Developmental Dynamics and Structural Evolution (DDSE) research programme. It develops the mathematical foundations of Developmental Computation as an autonomous, implementation‑independent theory. Beginning from a small set of axioms, the monograph systematically constructs the principal mathematical structures: Developmental Space, admissible subspaces, Primitive Developmental Transformations, a directed quasi‑metric geometry, and the central notion of Developmental Capacity. Capacity gives rise to a full dynamical theory—Momentum, Curvature, Genesis and Decay operators—and a classification of trajectories into expanding, saturated, and contracting regimes. The theory then lifts to the algebraic and categorical levels, yielding a category of developmental systems, a monoid of trajectories, and a Graph‑Rewriting Realization Theorem that faithfully embeds every DDSE‑system into a graph‑rewriting system while proving that DDSE is strictly richer than graph rewriting alone. Open‑ended development is characterised in terms of the asymptotic density of Genesis steps, and Developmental Generators are identified as sufficient algebraic conditions for open‑endedness. The monograph establishes a rigorous mathematical foundation—encompassing geometry, dynamics, algebra, and category theory—upon which future theoretical investigations, computational implementations, and empirical studies of Developmental Computation may be systematically developed. A structured list of open mathematical questions is provided to guide subsequent stages of the DDSE research programme. Related works DDSE‑I: Developmental Dynamics and Structural Evolution: A Formal Theory of Recursive Autonomous Developmenthttps://zenodo.org/records/21176088 DDSE‑II: Computational Realization of DDSE: From Formal Theory to Autonomous Developmental Systemshttps://zenodo.org/records/21205087 DDSE‑III: Experimental Evaluation of Developmental Computation — A Minimal Executable Reference Implementation of the Developmental Enginehttps://zenodo.org/records/21241321 AcknowledgementsThe author gratefully acknowledges the assistance of DeepSeek and ChatGPT (OpenAI) during the preparation, editing, formatting, and technical refinement of this manuscript. All scientific concepts, theoretical claims, mathematical definitions, and final editorial decisions remain the sole responsibility of the author.