The Mixed-Group Framework and Algebraic Residual Iteration (ARI): From Structural Unification to Efficient Learning

This work introduces the mixed group H_D(γ,p), which unifies the real space R^D and the finite field F_p^D at the group level via the unified operation ⊕_γ and the double-derivation operator D_γ. The exponential lift operator Eγ,g is further defined, establishing an exact algebraic correspondence between continuous exponential maps and the discrete logarithm problem over finite fields. We develop Algebraic Residual Iteration (ARI), a gradient-free universal approximation and solving paradigm. ARI uses closed-form least squares and algebraic residual iteration, achieving global optimality at each subproblem, avoiding local minima, reaching millisecond-level speed, and naturally supporting parallelization and meta-learning. Experiments verify the mixed-group homomorphism, ARI function approximation, discrete logarithm learning, XOR nonlinear classification robustness, and ARI's sensitivity to prime algebraic structure: on smooth-order groups (Fermat prime p=257), bitwise prediction reaches 100% accuracy; on safe primes, only the least significant bit is predictable, while the second bit is indistinguishable from random, consistent with the Blum-Micali hard-core predicate theorem. This work provides new theoretical tools and experimental paradigms for the intersection of algebraic machine learning and cryptography. All code and raw data are included for full reproducibility.