Euclidean-Lorentzian Dichotomy and Algebraic Causality in Finite Ring Continuum

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells Fp with p=4t+1 exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realized within a single space-like prime shell Fp, since to split time from space one requires a time coefficient c2 in the nonsquare class of Fp×, but then c∉Fp. An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group O(Qν,Fp2) 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, thereby establishing an intrinsic relativistic algebra within FRC. Finally, this dichotomy implies the algebraic origin of causality: Euclidean invariants reside within a space-like shell Fp, while Lorentzian structure and causal separation arise in its quadratic spacetime extension Fp2.