Soluble State Tracking in Materialized Fixed-Point GDN+

Every finite soluble monoid admits a fixed-width, fixed-depth GDN⁺ network that computes its nonempty prefix products in the materialized fixed-point model defined here, with constant gates α = β = 1, zero initial memory, and either of two specified centered normalizations. StateBench establishes aperiodic and rounded-parity lower bounds; our construction supplies all cyclic factors and their full-network soluble composition in this explicit operational model. Individually rounded matrix products implement bounded modular counters. Cyclically shifted copies expose constant-norm codes that survive normalization and residual addition, and bias-free ReLU selectors return reusable finite labels. A first-position marker replaces a beginning-of-sequence token, while a parity-driven buffer supplies the exclusive prefix required by wreath composition. Exact certificates exhaust complete modulo-three and ℤ₃ ≀ U₂ networks under both normalizations; the universal claim follows from an algebraic induction.