Modular-addition MLPs factorize into a semi-ring algebraic structure

Tian trains 2-layer quadratic-activation MLPs (widths 20/512/1024, 5 seeds) on modular addition over Z_p (p in {23,71,127}) and factorizes the trained weights against a closed-form semi-ring / ring-homomorphism construction [tian-2024-composing-global-solutions-algebraic-objects] About 95% of gradient-descent solutions are factorable with small factorization error (~0.04 relative to solution norm) [tian-2024-composing-global-solutions-algebraic-objects] Each Fourier frequency admits both order-6 (6 hidden nodes) and order-4 (4 hidden nodes) solution components, not a fixed count, with the mix depending on the modulus [tian-2024-composing-global-solutions-algebraic-objects] For d=127 the overwhelming majority of frequencies take the order-4 form (only ~1.26 of 63 frequencies are pure order-6), so six-nodes-per-frequency is only the sufficient count for a pure order-6 construction [tian-2024-composing-global-solutions-algebraic-objects] The framing is purely algebraic (commutative semi-ring, ring homomorphisms); the paper makes no literal circular-geometry claim, treating the circle only as membership in the modular-addition cluster [tian-2024-composing-global-solutions-algebraic-objects] The measurement is observational weight structure with no activation-level causal intervention [tian-2024-composing-global-solutions-algebraic-objects]