Information-Complete and Paradox-Free Finite Ring Calculus

Finite Ring Calculus (FRC) is a first-order formal system whose sole intended model is a fixed finite ring~\(\mathbb Z_q\). By eliminating unbounded induction and restricting all syntactic numerals to residues modulo~\(q\), FRC blocks standard diagonal constructions, thereby evading the classical Gödel-Turing-Russell paradox pattern. The resulting theory is shown to be recursive, consistent, complete and decidable, hence \emph{information-complete} in the precise sense that every well-formed sentence admits an effective truth-value within~\(\mathbb Z_q\). We sketch how this finite arithmetic can act as a foundational layer for physics and computation without re-introducing classical paradoxes. We furthermore stress how these properties dovetail with the smooth-geometry and harmonic-analysis layers already constructed in the broader FRC framework.