Equilibrium-join coherence C1

Adopt C1 (equilibrium-join coherence): whenever $x{\sim }_{T}y$, the joined state $J_{ij}(x,y)$ is thermally equivalent to each constituent, $J_{ij}(x,y){\sim }_{T}x$ and $J_{ij}(x,y){\sim }_{T}y$, using the model’s declared nested-join identifications.

C1 is an additional postulate of this completed contact branch. The source’s Theorem 4.8 is stated with T1–T4; the local conditional calibrator route uses the constituent-to-join bridge in addition to the stronger T5 range assumption. Neither fact licenses deriving C1 from $u=u_i+u_j$, from T1–T4, or from transitivity alone. The displayed source supports the contact context, not the claim that its bare printed axioms already contain C1.