Dual-Derivative Power Theorem: Group Homomorphic Structure on Hybrid Characteristic Abelian Groups

Abstract

We define the hybrid characteristic Abelian group H_D(γ,p) = R^D × F_p^D with a unified nonlinear operator: a ⊕_γ b = a + b + γ·(a⊙b) acting simultaneously on real (characteristic 0) and finite field (characteristic p) components. We introduce the dual-derivative power operator D(x) = 1/(1+γx) and prove the core theorem: D(A ⊕_γ B) = D(A) ⊙_γ D(B) i.e., the dual-derivative power operator provides an exact homomorphism from the additive structure to the multiplicative structure of the hybrid group. This theorem circumvents the limitations of classical field homomorphism theorems and provides new tools for cross-characteristic algebraic structures. Experimental verification is performed at p=17 and the safe prime p=1019 (where 1019 = 2×509 + 1 and 509 is prime). For both primes, real component errors are below 10^-12 and all finite field components pass verification. The framework preserves all prior properties: exact linear term preservation, rank collapse elimination, and the unified algebraic structure over real and finite fields. This work is accompanied by complete Python code for reproducibility. Keywords: hybrid characteristic Abelian groups, dual-derivative power operator, group homomorphism, cross-characteristic algebra, discrete logarithm continualization