Operational content of the extra axiom

structure and translations. Hypothesis $\mathsf{Aff}$ is a declared representation hypothesis, not a consequence of equal local arity. An operational derivation would require coherent displacement composition and a compatible real affine completion. At the completed level, translations form a simply transitive action of the additive group $V$. If that action is to describe operations rather than coordinates, it must be lifted to a family of record-preserving experiments. An abstract abelian group alone does not supply real scalar multiplication or a topology. By Proposition prop:pastfinite, the full continuous action cannot be identified with a transitive action on the past-finite event set used in the countermodels.

. Interior homogeneity means that any two interior displacements are related by an invertible linear map preserving the completed cone. Its stronger operational reading is a corresponding symmetry of experiments, including channel interfaces, admissible interventions, and records. This lift is additional data. Positive diagonal maps make the real orthant homogeneous, but generally do not act bijectively on its integer event sample. Cone homogeneity, graph homogeneity, and uniform reduction shapes are three different conditions.

-duality. The pairing in self-duality is an explicitly assumed positive-definite inner product. A proposed derivation from record correlations or protocol costs would have to prove bilinearity, positive definiteness, and the stated primal–dual equality; none is proved here. The pairing is Euclidean auxiliary data, not the Lorentzian metric eventually obtained. The weaker published selection theorem in Theorem thm:weaker avoids assuming a pairing, but still requires homogeneous and strictly convex completed influence geometry.

and absence of faces. Irreducibility excludes a decomposition of the completed order into independent product orders. The orthant is the canonical counterexample. Strict convexity is stronger: any two nonproportional feasible displacements have an interior sum. Operationally, when displacement concatenation and robust endpoint control are available, two different limiting propagation directions compose to a robustly feasible displacement. This interpretation requires that completion and control structure; it is not an assertion about two arbitrary finite protocols. Neither exclusion follows from commuting reductions.