Mathematical Foundations of Reflexive Reality: A Unified Information-Geometric Framework for Physics

The Mathematical Foundations of Reflexive Reality (MFRR) establishes a unified framework in which logic, computation, and physics are aspects of one self-defining process. A universe can exist consistently only if its laws are internal, reflexively executed, and energetically self-accounting. The central construct—Transputation (\( \))—realizes lawful internal adjudication of computational degeneracies, producing Perfect Self-Containment (\( \)). The forcing of transputation as the unique internal adjudicator under closed-choice conditions is machine-proved in a companion Lean development (zero sorry; Strong Transputational Universality, STU). Seventeen closure theorems connect logic, energy, geometry, thermodynamics, and theory space. The pinnacle structural result is the Two-Layer PSC Theorem: Layer~I (hard PSC axioms) uniquely forces gauge group SU(3)× SU(2)× U(1) with N_gen 3; Layer~II (Presentation Invariance/MDL) selects N_gen=3 as the unique PSC-optimal solution. The framework further resolves the black-hole information paradox via reversible transputation ( ^-1 =I), derives ensemble quantum decoherence as the origin of GKSL dynamics without external environment, and establishes three universal information-energetic laws: (i)~superlinear energy amplification (α: 1.01→ 1.83) quantifying the quantum-to-classical transition; (ii)~the Information Profit Principle (Gen/Drain >1.13=1+Λ/2) governing all self-organization; (iii)~decoherence as profit-accounting corruption by additive external noise. Part~V realizes MFRR constructively via steel-man experiments (TE_2.1: 11 experiments, 710 runs) deriving gravity, quantization, entanglement, and wormhole formation from information-theoretic constraints alone, and via five advanced theorems (TE_2.2–TE_2.6) proving PSC-optimality of SM, nuclear physics, unitarity, ΛCDM, and Δ-machine necessity. The framework is verified by the ΛΩ-RCP, TE_1, and TE_2 validation programs: 297 first-party Python modules, 57,337+ experimental runs, 15,894+ result files, confirming core predictions to sub-percent precision.