PAPER 5 in The UAP Gödel Obstruction Series This paper constructs finite, bounded arithmetic bridge regimes to provide an explicit realization of the first obstruction mechanism within the Apophatic-Paraconsistent Multiverse Framework. While previous papers in the UAP series deal with theory-attached regimes, this work focuses on a setting where local data, transition cocycles, and gauge classes are all finitely accessible and effectively computable. Key technical formalisms: Bounded Provability: The paper uses bounded proof predicates to ensure that the validity and obstruction data of the regime are effectively recoverable from finite data. Binary Three-Chart Case: For the free binary case, the obstruction is completely classified by parity and reduced to a normal form. Finite Presented Setting: The paper demonstrates that in the general finite setting, the cocycle and its gauge class (the cohomological signature) remain effectively computable. This work exhibits a concrete lane of the UAP framework, showing how obstruction-theoretic structures are visible and computable at the bounded arithmetic level. This serves as a finite complement to the theory-attached and fixed-point results of the broader series.