Mathematical Foundations of Reflexive Reality: A Unified Formalism of Self-Defining Systems, Transputation, and the Geometry of Coherence

We present the Mathematical Foundations of Reflexive Reality (MFRR), a unified framework demonstrating that a self-consistent, computable universe must be reflexive: its laws, description, and execution are coextensive. Through a suite of foundational closure theorems and the Two-Layer PSC Theorem, we prove that Perfect Self-Containment (PSC) necessitates a lawful, non-computable mechanism for resolving indeterminacy—Transputation (PT) —which functions not as an algorithm, but as a physical process of thermodynamic relaxation governed by the Reflexive Landauer Bound (EPT kB T n + _ _). The forcing of transputation as the unique internal adjudicator under closed-choice conditions is machine-proved in companion paper (closed\choice\forces\ₜransputation; zero sorry; Strong Transputational Universality, STU). At the constructive level, the TE₂. U experiment demonstrates a Strong Transputational Universality advantage: a reflexive DSAC architecture achieves (10⁴) speedup over classical solvers across diverse task families (Theorem, empirically justified). This framework unifies logic, energy, and geometry into a single causal structure. Key foundational discoveries include: The Quantum-Geometric Equivalence Theorem: We prove that quantum superposition is formally equivalent to a system dwelling on an "Adjudicative Manifold" of sustained, unresolved degeneracy. This identifies the "collapse" as a lawful optimization of the dissonance functional. The Information Profit Principle: We derive a universal self-organization threshold of Generation/Drain > 1. 13, analytically determined by the fundamental constant = / (2). This principle unifies quantum decoherence (re-framed as profit accounting corruption, providing the computational proof of the No-Go Theorem for Stochastic Resolution), biological metabolism, and economic viability as manifestations of a single profit-accounting requirement. . . .