Mathematical Foundations of Reflexive Reality: A Survey

We survey the formal and computational results of the Mathematical Foundations of Reflexive Reality (MFRR) programme, which establishes a unified framework in which logic, computation, and physics are aspects of one self-defining process. The central construct, Transputation ( ), is the unique internal adjudicator of computational degeneracies (machine-proved in Lean~4, zero sorry, zero custom axioms). Machine-certified 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 N_ gen ≥ 3 forced by PSC consistency; UCL2 correction k_ gen = φ (π/10). Computationally certified results: SM ranks \#1 of 34,560 universes scanned (P: SM selection); 97.02\% SRRG attraction rate to the SM fixed point (C: flow-basin size); Information Profit Threshold 1.1300 ± 0.0001, matching the UGP-derived value to 0.08\% (P); emergent force law exponent p = 2.60 ± 0.16, asymptoting to r^-2 (M); Reflexive Landauer bound compliance 100\% (C). The Residual Classification is Lean-certified ( .RCCInfiniteFamilies, zero sorry), making SM uniqueness unconditional. The Information Profit Threshold is machine-checked ( .IPT.InformationProfitThreshold, zero sorry). Primary open fronts: gravity/gauge unification in the two-layer theorem and T8 holographic closure numerical certification (circularity noted; ).