Goldbach Intersection Geometry and Prime-Pair Correlations: Exact Identities, Crossed Bridges, Modular Selection, Singular Series, and Computational Evidence

For each even integer $2n$, let $S(n)=\{p\in\PP:2n-p\in\PP\}$ be the set of prime summands of its Goldbach representations, and let $I_k(n)=S(n)\cap S(n-k)$. These intersections encode prime pairs of every even separation. We develop exact correspondences, a zero-or-many theorem for nonempty intersection rows, consequences of finite and infinite fixed-gap prime-pair classes, and counterexamples showing why gap recurrence and neighboring-row overlaps alone cannot prove Goldbach. We then study crossed intersections $B_h(n)$ and prove that they are precisely fixed-gap autocorrelations of $S(n)$. On the residue class $n\equiv1\pmod3$ where the summand 3 is unavailable, modulo-3 selection restricts nonexceptional crossed bridges to gaps divisible by six. For gaps $6,12,18$ we derive their four-prime singular series and local modulation modulo 5 and 7, prove exact overlap exclusions and a pair-count conservation law, and provide reproducible computations through $n\le10^6$. There are 258,814 tested extinction indices and no simultaneous zero of the three crossed-bridge counts. The numerical and heuristic observations are explicitly separated from theorems: universal short-gap rescue, the twin-prime conjecture, and the binary Goldbach conjecture remain unproved.