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

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 \(D\subset M\) on which the coarse–grained field \(\Phi^\mu\) is smooth, future–directed, unit–norm timelike (\(\Phi^\mu\Phi_\mu=-1\), \(\Phi^0>0\)) 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 \(\Phi_A\) 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 \(d\ge 1\), 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.