Emergence and Exclusivity of Lorentzian Signature and Unit-Norm Time from Random Chronon Dynamics

In this paper, 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 [Formula: see text] on which the coarse-grained field [Formula: see text] is smooth, future-directed, unit-norm timelike ([Formula: see text], [Formula: see text]) 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 signaling, 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 [Formula: see text] 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 [Formula: see text] signatures with [Formula: see text], heuristic coarse-graining and stability considerations suggest [Formula: see text] 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.