Algebraic Learning in Finite Ring Continuum

The Finite Ring Continuum (FRC) models physical structure as emerging from a sequence of finite arithmetic shells of order \(q = 4t+1\). While Euclidean shells \(\mathbb{F}_{p}\) support reversible Schr\"odinger dynamics, causal structure arises only in the quadratic extension \(\mathbb{F}_{p^2}\), where the finite-field Dirac equation is defined. This paper resolves the conceptual tension between the quadratic expansion \(\mathbb{F}_{p} \to \mathbb{F}_{p^2}\) and the linear progression of symmetry shells by introducing an algebraic innovation-consolidation cycle. Innovation corresponds to the temporary access to Lorentzian structure in the quadratic extension; consolidation extracts a finite invariant family and encodes it into the arithmetic of the next shell via a uniform G\"odel recoding procedure. We prove that any finite invariant set admits such a recoding, and we demonstrate the full mechanism through an explicit worked example for \(p = 13\). The results provide a coherent algebraic explanation for how finite representational systems---biological, computational, and physical---can acquire, assimilate and preserve structure.