A Finite-Pattern Engine over Finite Chain Rings: Deterministic Evaluation, Valuation Stability, and Finite-Geometry Fingerprints

Submitted preprint of a manuscript submitted to Finite Fields and Their Applications (Elsevier, 2026). We develop a finite-pattern engine over finite chain rings R = Z/p^k Z: a deterministic framework in which a finite formal expression induces a well-defined map by retaining exactly the monomials whose p-adic valuation keeps them visible at the working ideal depth. We prove deterministic well-definedness and a bounded-valuation-loss law; a valuation-stability theorem for finite linear recurrences; a divided-power / denominator-visibility criterion including the von Staudt-Clausen dependency; a Smith-Normal-Form classifier deciding threat-image depth; and a finite-geometry fingerprint from exact conic point counts (including the Odd-Prime Circle Fingerprint theorem). The framework unifies these as instances of a single valuation gate and supports bit-exact, hardware-independent replay. This is the submitted version (preprint); the version of record, if accepted, will appear in Finite Fields and Their Applications.