Euclidean-Lorentzian Dichotomy and Algebraic Causality in Finite Ring Continuum

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) programme, showing that symmetry-complete prime shells \ (F ₏\) with \ (p = 4t + 1\) exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realised within a single space-like prime shell \ (F ₏\), since to split time from space one requires a time coefficient \ (c²\) in the nonsquare class of \ (F ₏^\), but then \ (c F ₏\). An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group \ (O (Q_, F₏ℂ) \) of split type (Witt index 1). These results demonstrate that the essential algebraic features of special relativity—the invariant interval and Lorentz symmetry—emerge naturally within finite-field arithmetic, which further provides a comprehensive justification for the ``relativistic algebra'' terminology in FRC. Finally, this dichotomy implies the algebraic origin of causality: Euclidean invariants reside within a space-like shell \ (F ₏\), while Lorentzian structure and causal separation arise in its quadratic (space-time) extension \ (F₏ℂ\).