Bridge Algebra: A Formal Framework for Cross-Domain Knowledge Transfer

We develop a formal algebraic framework for mathematical bridges — theorems that transfer results from one domain to another. A bridge \(B_{A \to B}\) is a theorem asserting that any object satisfying predicate \(P_A\) in domain \(A\) implies the existence of an object satisfying predicate \(Q_B\) in domain \(B\). We prove that bridges compose transitively, admit identity elements, and satisfy left and right identity laws. We further establish quadratic growth laws showing that for \(n \geq 3\) domains, the number of bridge pairs \(n(n-1)/2\) exceeds the number of within-domain results \(n\). All results are formally verified in Lean 4.