Fusion is the closure operation that enforces coherence between syntax and semantics in the semiotic universe.
It works in two directions. First, it identifies syntactic operators when their semantic interpretations agree on all fragments: if two operators f and g satisfy ⟦f⟧ ≡_F ⟦g⟧ for every fragment F, then fusion collapses them into a single equivalence class. This is quotienting — removing syntactic distinctions that make no semantic difference.
Second, fusion names existing semantic behaviors: if an operation on H is already definable from the existing operators and Heyting structure, fusion adds a syntactic representative for it. This is completion — ensuring that everything the semantics can do has a syntactic handle.
The two directions together form the fusion closure operator S_fus on the lattice of partial semiotic structures. Given a partial structure X = (H_X, Op_X), fusion produces a new structure where Op_X is quotiented by fragmentwise semantic equality and new operators are added for semantic behaviors that lack syntactic names.
Fusion is one of three closure operators whose composite S = S_fus ∘ S_syn ∘ S_sem defines the semiotic universe as its least fixed point. The semantic closure S_sem closes the semantic domain under algebraic operations. The syntactic closure S_syn closes the operator algebra under lambda-definability and composition. Fusion closes the gap between the two. At the fixed point, syntax and semantics are fully coherent: every semantic distinction has a syntactic name, and every syntactic distinction makes a semantic difference.