Structural approaches to myth and narrative are compelling in close reading but hard to compare across traditions, media, and scale. We propose a formal framework that renders Lévi-Straussian transformation as mathematics while remaining readable as narrative analysis. Variants, superhero continuities, and franchise arcs are modeled as typed rewrite programs on a coupled two-register state $(X,Y)$, abstracting an everyday/social channel and a symbolic/legitimation channel. The canonical formula becomes coherence data: a natural transformation $η:U\Rightarrow V$ between update endofunctors, where $U$ updates each register in place and $V$ performs a swap+inversion. Context is internalized by operator choice, turning naturality into a corpus-facing type check: failures diagnose mis-specified oppositions or illegal transport; successes witness coherent structural models. Order effects are summarized by a five-value invariant (Key). We apply the method to 80 narratives (20 folktales, 20 religious myths, 20 superheroes, 20 franchises), each encoded as $(a,b,x,y)$ with a Key. 59/80 (74\%) explicitly name a normative constraint in $y$ (law, taboo, contract, prophecy), supporting the two-register abstraction. The result is a testable bridge between structural anthropology and cultural analytics: stories remain interpretable yet become transportable objects for computation, comparison, and falsifiable constraints on transformation.