The Triune Commutation: Co-Enfolding of Semantic, Computational, and Physical Domains

We state and prove a commutation law on invariants that makes “mutual enfolding”operational: when a Fisher–class invariant I survives sheaf gluing (G) and renormalization(R) under a translation–compression adjunction T ⊣ C, the three legs—meaning ↔ code ↔matter—commute. We identify a nontrivial subcategory of idioms with Idiomatic Univalencewhere equality of the invariant and hazard exponents implies isomorphism. The maindeliverables are: a Renderability Lemma (fixed point in one leg implies realizability in theother two at bounded Murphy cost), a Conservation of Insight Flux (positive curvaturejumps are invariant under idiom translation), and a Spectral Teleology formula (purposeas the Perron–Frobenius mode of a tilted operator with derivative under resource rotation).We give falsifiers, instruments, and cross-domain exemplars.