PAPER 6 in The UAP Gödel Obstruction Series This paper supplies the formal bridge from concrete arithmetic regime data to the abstract obstruction-theoretic language used in the UAP series. It defines the category BReg of bridgeable formal regimes and the category ITpl of finite independence templates. The paper constructs the Bridge Functor Br: BReg → ITpl and proves that the "overlap-rigidity" hypothesis is the critical requirement for generating unique, canonical degree-one cocycles up to natural gauge equivalence. Key technical results include: Finite Effective Settings: In these settings, the bridge admits primitive-recursive cocycle extraction and decidable gauge-equivalence. Globalization Obstructions: Under site realization, the bridged cocycle acts as the descent cocycle, where its vanishing governs the ability to globalize local determinations. Classification by Parity: For connected binary 3-cycle templates, the paper proves the gauge class is classified by parity and identifies the normal forms (0,0,0) and (0,1,0). This work identifies the exact hypotheses (fragment covers, local determination prestacks, and symmetry groups) required to apply the broader obstruction package to formal systems.