The Function Model Architecture: Mathematical Foundations with an LLM-Specific Appendix

This work extends and formalizes the author’s earlier foundational release (DOI: 10.5281/zenodo.17786610) by providing a proof-complete mathematical treatment of the Function Model Architecture (FMA) and an explicit appendix addressing applicability to large language model (LLM) systems. In addition to full convergence and validity proofs, this release clarifies finite-difference bias bounds, Polyak–Łojasiewicz–based convergence rates, rollback semantics, and topological stability guarantees. The LLM-specific appendix formalizes interface conditions, determinism, probabilistic semantics, canonical learning records, and equivalence classes, under which LLM-based systems can participate in function-level learning without violating the core invariants established in the main text. This document is intended as the authoritative, proof-complete reference for the Function Model Architecture and supersedes the earlier release for purposes of formal citation, while preserving the original DOI as a foundational disclosure.