Proposition: A non-Lorentzian scheduler countermodel

Acyclic dependency, finite local updates, and deterministic schedule-independent records do not imply a Lorentzian generated cone.

Proof. Take events indexed by ${\mathbb{N}}^4$, with directed edges $n\to n+e_i$, $i=1,\dots ,4$. Use a deterministic bit relay on any finite predecessor-closed subnetwork: the recorded value is the Boolean OR of incoming values and a local input bit. Every update occurs after its predecessors. The graph is acyclic, has bounded degree, and every finite experiment terminates. Associativity and determinism make records independent of the legal interleaving. A source bit can be detected at every reachable descendant by setting other inputs to zero.

The closed cone generated by edge displacements is

It is not linearly isomorphic to the four-dimensional Lorentz cone. Indeed, $K$ has four extreme rays, whereas the Lorentz cone has infinitely many. An invertible linear map preserves extreme rays.