Theory-Attached Disputed-Sentence Regimes: Arithmetized Validity and the Π₁ Threshold

PAPER 2 in The UAP Gödel Obstruction Series This paper develops an arithmetized construction for disputed-sentence regimes as a solution to the canonical extraction barrier. Given a theory code e (representing Tₑ) and a sentence code g (representing G), we define a primitive recursive constructor Reg(e, g) that outputs a 3-chart finite regime. Key Technical Formalisms: The Nerve: The overlap structure is isomorphic to the simplicial circle S¹, with charts Φ₀, Φ₁, Φ₂. Validity Predicate: We show that Valid(e, g) ⟺ Con(Tₑ + G) ∧ Con(Tₑ + ¬G). This places the validity of the bridge regime exactly at the Π₁ level of the arithmetic hierarchy. Cohomological Obstruction: The construction yields a ℤ/2-valued transition cocycle (0, 1, 0). We prove this represents the unique non-trivial class in H¹(S¹, ℤ/2). Invariance & Minimality: The paper proves that the regime is invariant under torsor relabeling and control-tautology replacement. Furthermore, we prove that 3 charts are the minimal requirement to carry a non-trivial degree-one binary obstruction in a connected regime. This layer provides the arithmetic machinery for the subsequent Fixed-Point and Realization papers in the series, bridging the gap between raw provability and topological obstruction theory.