A Non-Trivial Abelian Group Operator: Breaking Rank Collapse and Preserving Linear Terms

We define a binary operator on a subset of R^D: a ⊕γ b = a + b + γ(a ⊙ b), where ⊙ denotes element-wise multiplication and γ is a real parameter. This operator is isomorphic to ordinary vector addition via the nonlinear bijection φγ(x) = (1/γ) log(1 + γx), and hence forms a genuine Abelian group on its domain. The operator has two notable properties: 1. The linear term a + b is exactly preserved (to floating-point precision). 2. The multiplicative term γ(a ⊙ b) introduces a nonlinear residual that can prevent rank collapse. We provide experimental validation: - 10,000 random tests in dimension D=256 confirm associativity, commutativity, identity, and inverse (all pass rate 100%, max error ≤ 1e-15). - A constructed example where standard addition gives rank 1, while the proposed operator yields full rank 4. - The linear term is recovered with reconstruction error < 1e-12, confirming zero-error preservation. This work presents a principled algebraic construction that combines exact linear information retention with controlled nonlinearity. The preprint includes all derivations, experimental code, and reproducibility materials.