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

We develop a finite-pattern engine over finite chain rings R = Z/pkZ: 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 image-depth; and a finite-geometry fingerprint from exact conic point counts. The framework unifies these as instances of a single valuation gate and supports bit-exact, hardware-independent replay.