PAPER 3 in The UAP Gödel Obstruction Series This paper establishes the diagonal engine of the series, transitioning from arithmetized validity predicates to self-referential unprovability. Given a theory code e, we define positive and negative fixed-point families, Γ⁺ₑ and Γ⁻ₑ, such that these sentences equate their truth to the Paired Consistency (Pair) of their own attachment. Key Technical Formalisms: Diagonal Engine: Using the Parameterized Diagonal Lemma, we construct primitive recursive functions d_+, d_- that generate sentences satisfying the biconditionals: Q ⊢ Γ⁺ₑ ⟺ Pair (ē, ⌈Γ⁺ₑ⌉) Q ⊢ Γ⁻ₑ ⟺ ¬Pair (ē, ⌈Γ⁻ₑ⌉) One-Sided Incompleteness: We prove that for any theory code e of a consistent, recursively axiomatized theory, the fixed-point Γ⁺ₑ is unprovable (Tₑ ⊬ Γ⁺ₑ), and the negative counterpart is essentially "un-refutable" (Tₑ ⊬ ¬Γ⁻ₑ). Obstruction Transfer: We demonstrate that this behavior is not unique to paired consistency. Any obstruction predicate Obs (u, x) satisfying the implication Obs → Pair inherits the same non-provability results via the abstract incompleteness schema.
David Betzer (Sat,) studied this question.