Deriving the Born Rule from Boundary–Induced Alignment in Chronon Field Theory

We derive the Born rule in Chronon Field Theory (ChFT) from first principles under realistic measurement assumptions. A recent work in ChFT shows that Lorentzian signature and causal structure emerge dynamically from a unit–norm constraint on the chronon field, providing the geometric foundation for the time–asymmetric alignment dynamics studied here. First, we obtain a diffusion limit for alignment order parameters on the outcome simplex and prove that the overlaps with apparatus eigen–domains form martingales up to the absorption time; optional stopping then yields single–shot Born probabilities. Second, we derive the stochastic limit from noisy chronon gradient flow with boundary coupling by a hydrodynamic limit (tightness, identification of the generator, and boundary layer analysis). Third, we establish a large–deviation principle for empirical frequencies in repeated measurements via Sanov’s theorem, with rate function minimized at the Born vector. We quantify robustness to finite temperature, imperfect interfaces, and basis degeneracies, and outline falsifiable predictions for alignment timescales and drift bounds. Conceptually, the analysis shows that measurement is not a collapse of the system into one of its own eigenstates, but stochastic absorption of the system field into pre–stabilized apparatus eigen–domains. This does not directly contradict the standard textbook view—since apparatus domains are engineered to correspond to system eigenbases—but offers a deeper dynamical interpretation of why outcomes are definite and Born–weighted.