The Operation: A Non-Associative Commutative Structure for Quadratic Interactions

This work introduces the binary operation , defined as , and explores its mathematical properties. The operation is commutative but non-associative and lacks an identity element, making it a novel structure beyond traditional algebraic groups. We analyze its applications in nonlinear differential equations, where it naturally arises in modeling quadratic interactions. A matrix representation of is provided, along with its potential implications for number theory and alternative algebraic systems. The operation offers new perspectives in mathematical modeling, particularly in contexts where independent squared contributions must be preserved.