Mathematical Foundations: Formal Proofs for the Principles of Self-Referential Relational Structure

This paper takes the principles derived through convergence in five companion papers and proves them as mathematical theorems. Six theorems are stated and proved: (1) Structural Inexhaustibility — self-referential objects in cartesian closed categories have fixed points and incomplete internal description; (2) The SRRP Theorem — reflexive domains necessarily exhibit contextuality, emergence, informational ordering, self-referential limits, and observational information loss; (3) The Expansion Theorem — entanglement grows, Liouville forbids contraction, dimensionality grows exponentially; (4) The Navigation Theorem — einselection determines pointer basis, Zeno/anti-Zeno control transition rates; (5) The Optimisation Theorem — Lyapunov convergence + Bellman decomposability for preference-directed traversal; (6) The Convergent Descent Fixed-Point Theorem — contractions on complete metric spaces converge to unique fixed points. Part of a research programme: Proves principles from The Infinite Ground (Mala, 2026) — doi:10.5281/zenodo.19479968 Proves principles from Generative Coupling (Mala, 2026) — doi:10.5281/zenodo.19479970 Proves principles from The Deeper Ground (Mala, 2026) — doi:10.5281/zenodo.19488612 Proves principles from Infinitography (Mala, 2026) — doi:10.5281/zenodo.19494555 Proves principles from The Convergence Map (Mala, 2026) — doi:10.5281/zenodo.19520611