We ask how Lorentzian causal structure can emerge from a pregeometric substrate. For a rigorously defined class of finite–range, ferromagnetically coupled “chronon” models with quartic norm pinning, we prove the existence, with strictly positive Gibbs probability, of a macroscopic percolating domain \ (D M\) on which the coarse–grained field \ (^\) is smooth, future–directed, unit–norm timelike (\ (^_=-1\), \ (⁰0\) ) and twist–free. We work on a smooth differentiable manifold but do not assume Lorentzian signature or a global time field a priori; these arise on \ (D\) from the dynamics. Under four operational axioms—well-posed local dynamics, finite-speed signalling, acyclic causal order, and stable memory/records—we further prove that no alternative (Euclidean or ultrahyperbolic) signature, nor a Lorentzian background lacking a globally unit–norm time field, can sustain such behavior; the Lorentzian, unit–norm phase is therefore exclusive. Finally, we show that “measurement” acts as a boundary-induced selector of this phase: an interface coupling to an aligned apparatus field \ (A\) admits a unique minimizer, pins the norm and alignment, suppresses twist, and drives any initial state to the aligned phase with exponential convergence; large-deviation bounds quantify high-fidelity selection. While our theorems hold for general \ ( (1, d) \) signatures with \ (d 1\), heuristic coarse-graining and stability considerations suggest \ (d=3\) as the most probable large-scale outcome. Together, these results provide a mathematically controlled foundation for the emergence and exclusivity of Lorentzian causal structure and for boundary-driven selection (measurement) in pregeometric ensembles.
Bin Li (2025) studied this question.