Reflexive Reality: A Survey for Formal Theory Specialists New Impossibility Results, New Proof Methods on Classical Objects, and a Positive Science of What Happens After Impossibility

This paper is Paper C2 of the Reflexive Reality capstone series. It surveys the program for specialists in logic, computability theory, proof theory, type theory, and mechanized mathematics. The program's contributions to this audience are of four kinds: (1) new impossibility results—three independent engines for proving impossibility of complete internal self-capture, each closing a distinct class of escape routes, combined into a complete classification; (2) new machine-checked theorems on classical mathematical objects—splitting criteria for group extensions, the first machine-checked Quillen Theorem A for Galois connections, and a 12-tranche external validation of a positive-closure proof architecture; (3) a positive dynamical classification theorem—the Reflexive Development Law, characterizing the lawful structure of what reflexive systems must do after impossibility; and (4) a methodology for typed formal synthesis at program scale. Every load-bearing claim marked with a bracketed citation corresponds to a machine-checked theorem proved with zero sorry in Lean 4. To verify: lake update interpretive sentences are not read as further theorem statements unless so labeled.