Hybrid-Characteristic Abelian Groups: A Unified Algebraic Framework over Real and Finite Fields

We construct a hybrid-characteristic Abelian group H_D(γ,p) = R^D × F_p^D equipped with a unified nonlinear operator: a ⊕_γ b = a + b + γ·(a·b) acting simultaneously on real and finite-field components. This structure preserves linear terms exactly (a+b) while introducing a deterministic nonlinear residual γ·(a·b) that prevents rank collapse. Three numerical experiments are provided: 1. Mixed linear system solving: X ⊕ Y = Z in R^D × F_p^D 2. Discrete logarithm in the hybrid group: solving k·G = P with O(1) complexity on the real component and DLP consistency on the finite-field component 3. Stability of continuous indistinguishability with discrete separation: states that are identical in the real component (error < 1e-10) but completely different in the finite-field component (distance > 10) remain stable under the unified operator. All numerical results are independently verified. The internal neuro-symbolic reasoning system (NLM-AGI) used in this work was designed and implemented by the author as a computational tool; all conclusions and interpretations are the author's sole responsibility. Keywords: algebraic neural networks, Abelian groups, hybrid characteristic, finite fields, real numbers, rank collapse, discrete logarithms, unified algebra