In Haag–Kastler AQFT, locality asserts that algebras assigned to spacelike sepa-rated regions commute. The converse direction (commutation ⇒ spacelike separation)is not automatic, especially in background-free operational settings where commu-tation can arise from non-geometric tensor factorizations or sector restrictions. Wegive a sharp algebraic upgrade. Starting from a modularly closed admissible family(F, Ω), we define an intrinsic causal complement operation by taking the maximalcommuting completion inside F. We introduce a quantitative commutant-completiondefect measuring failure of this complement to exist (or be stable) at finite resolu-tion. New discovery: under a minimal maximal-commuting closure axiom plus asplit-independence condition, the commutation graph on F canonically induces anorthocomplemented causal poset on the causally-complete subfamily, and the “iff”locality statement becomes a theorem internal to the reconstructed causal structure.This turns “locality from commutants” into a checkable diagnostic for emergent causalorder.
SIKX HILTON (Tue,) studied this question.