The General Science of Reflexive Systems: Nonexhaustibility, Completion Failure, and Residual Structure

This paper develops a general science of reflexive systems: systems that, in a mathematically nontrivial sense, represent, evaluate, certify, locate, or attempt to complete themselves from within. The same structural tensions recur wherever serious self-representation, internal generation, and self-certification meet. Its central claim is that strong anchored internal completion is systematically obstructed: anchored completion limits, barrier families, residual aftermath, and the Reflexive Development Law are all proved as machine-checked Lean 4 theorems. The paper provides the first unified mathematical treatment of why reflexive systems cannot exhaust themselves, what structure remains after obstruction, and how this residual structure organizes the space of all possible completions.