Foundational Architectures for Self-Generative Mathematical Systems: Recursive Axiomatization and Meta-Logical Completeness

This research delineates a formal framework for Self-Generative Mathematics (SGM), char-acterized by the autonomous derivation of axiomatic sets and procedural theorem discoverywithin recursive meta-logical structures. By leveraging higher-order logic and algorithmic in-formation theory, the study examines the conditions under which a mathematical system candynamically expand its own ontological boundaries without compromising internal consis-tency. We investigate the convergence of recursive model-theoretic expansion and automatedheuristic synthesis, proposing a taxonomy for self-evolving formalisms. The paper furtheraddresses the computational complexity of verifying emergent properties within these non-stationary systems and evaluates the implications for autonomous proof-search architecturesand the limits of automated formal reasoning. We present the Axiomatic Saturation Engine(ASE) v11.0 as a prototype for this architecture.