Abstract

Can consistency of local reductions force a Lorentzian causal geometry? We separate event order, operational influence, cone representation, and conformal reconstruction. Acyclic dependency generates a partial order, and compositional commuting diamonds give schedule-independent records. Neither conclusion selects a geometric cone.

We construct bounded-degree record networks with finite support, finitely many reduction shapes, terminating finite experiments, exact operational influence, and schedule-independent records. Their standard affine realizations have either Euclidean-ball one-step lattice sections or polyhedral reachability cones. We also exhibit a distinguishing chronology on a smooth three-dimensional manifold whose chronological automorphisms need not be differentiable.

A matching positive result identifies sufficient additional hypotheses. Translation and positive-dilation invariance produce convex cone orders. An irreducible, homogeneous, self-dual, strictly convex proper cone is linearly isomorphic to a future Lorentz cone; the proof applies the classical classification of symmetric cones. A stronger published characterization allows self-duality and irreducibility to be omitted. We state the operational interpretation and the unproved representation content of these assumptions separately. The theorem excludes the specified polyhedral affine realizations, not the underlying rewrite systems under arbitrary re-embeddings.

The cone determines a Lorentzian quadratic form up to positive scale. A causal-origin refinement gives an exact typing criterion, while general conformal reconstruction still requires a distinguishing Lorentzian realization. Scheduler consistency alone supplies neither the affine structure nor the cone symmetries used in the positive theorem.