Representational Incompleteness: No Self-Modeling System Can Capture Its Own Diagonal — A Machine-Checked Barrier with Six Closed Escape Routes

Any system that models itself — any s : A -> A -> B whatsoever — is necessarily incomplete with respect to its own diagonal behavior: the function a |-> f(s a a) is never equal to any row s a0 of the self-model, for every fixed-point-free f : B -> B and every a0 in A. No surjective self-model of this kind can exist. This paper proves six "no-escape" routes are formally closed as machine-checked Lean 4 theorems: no surjection, no injective tower, no idempotent loop, no parametric evaluator, no diagonal row, and no representational bypass. The topological complement proves the Mobius strip is not homeomorphic to the closed cylinder, the orientability boundary of the representational incompleteness thesis.