Why Gödel's §4 Sentence and Turing's Diagonal Machine Are Ill‑Typed Before Any Arithmetization Under Pre‑Semantic Type Validation

Every classical presentation of Gödel’s incompleteness theorems and Turing’s halting problem begins with a single meta‑linguistic claim: that one can construct a sentence (or machine) that refers to its own provability or halting behavior. This initial natural‑language claim—the Step Zero assumption—is already a direct instance of the Liar paradox. Pre‑Semantic Type Validation (PTV), a stratified formation regime that prohibits implicit cross‑level coercion, rejects this sentence at the moment of its formation. Since the object intended for arithmetization never exists as a well‑typed entity, the subsequent Gödel numbering or diagonalization cannot proceed. As a consequence, no Gödel sentence, no undecidable halting instance, and no diagonal function can be formed in a PTV‑stratified foundation. Hilbert’s program is thereby restored in a strictly syntactic manner.