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 \(\mathbb{F}_{\mathsf p}\) with \(\mathsf{p} = 4\mathsf{t} + 1\) exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realised within a single space-like prime shell \(\mathbb{F}_{\mathsf p}\), since to split time from space one requires a time coefficient \(c^2\) in the nonsquare class of \(\mathbb{F}_{\mathsf p}^{\times}\), but then \(c \notin \mathbb{F}_{\mathsf p}\). An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group \(O(Q_\nu,\mathbb{F}_{p^2})\) 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 \(\mathbb{F}_{\mathsf p}\), while Lorentzian structure and causal separation arise in its quadratic (space-time) extension \(\mathbb{F}_{\mathsf{p}^2}\).