The Double Jump: Why Truth Must Transcend Itself Twice

We establish that the minimal depth of the reflection hierarchy required for sound, self-certifying arithmetic is exactly two. No single recursively axiomatizable theory extending Peano arithmetic can simultaneously (i) define a total truth predicate for a base theory T₀ via Tarski biconditionals, and (ii) prove the reflective soundness of that truth predicate. The proof combines Tarski's undefinability theorem, Gödel's second incompleteness theorem, and Löb's theorem to show that any attempt at complete formal self-verification necessarily stratifies into at least two distinct levels. This "double jump" is unavoidable: truth must transcend itself twice to achieve certification. The result holds even for local reflection principles, establishing a fundamental topological constraint on the structure of metamathematical self-reference.