Mathematical Foundations of Reflexive Reality: A Survey

We survey the formal and computational results of the Mathematical Foundations of Reflexive Reality (MFRR) programme. The programme 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 —is the unique internal adjudicator of computational degeneracies, machine-proved in a companion Lean 4 development (zero sorry, zero custom axioms). -checked results: forced adjudication (unique under closed-choice conditions) ; Born rule uniqueness from MDL; arrow of time from irreversibility; no-emulation (is super-Turing) ; SM gauge group and Ngen 3 forced by PSC consistency; UCL2 correction kgen = (/10). Lean names and DOIs are in Appendix. certified results (SHA-256 provenance; labels: P=structural prediction compared to experiment, M=model-independent measurement, C=internal consistency check): Standard Model ranks #1 of 34, 560 universes scanned, including all major BSM candidates P: SM selection; 97. 02 Information Profit Threshold 1. 1300 0. 0001, matching the UGP-derived value kgen²/ (4) to 0. 08 emergent force law exponent p = 2. 60 0. 16 (intermediate regime, asymptotes to r^-2) M: measured from simulation, not fitted to target; Reflexive Landauer bound compliance 100 framework: Perfect Self-Containment (PSC) is the logical closure condition for a self-consistent universe; Transputation is its unique internal realization. All results are conditional on stated premises. The Residual Classification is now a Lean-certified theorem (PSC. RCCInfiniteFamilies, zero sorry), making SM uniqueness unconditional. The Information Profit Threshold is machine-checked in Lean (UgpPhysicsLean. IPT. InformationProfitThreshold, zero sorry). . . .