Prime-Base Roof Coordinates in the 6-Wheel Exact Rank Translation, Defect Level Sets, and Sparse Twin-Cofactor Fans

This note proves that admissible prime-base roofs are translated local presentations of one global twin-cofactor coordinate. Apparent diagonal equalities become exact defect level sets, while the general chamber correspondence and synchronized carrier fan describe how the same cofactor edge is represented across prime bases. A classical sieve bound yields unconditional density-zero sparsity of active indices and fan cells; Hardy–Littlewood, when explicitly assumed, supplies sharp asymptotics and eventual negative defect drift. The paper does not establish twin-prime infinitude or the asymptotic coverage of the broader overlapping chambers.