The Addponent Operator (\underline{\times}): A Non-Commutative Magma for Generative Accretion Models

Standard multiplication provides a highly accurate model for inert conservation and proportional scaling. However, modeling generative, compounding physical processes where interactions are strictly additive often requires recursive workarounds. This paper formalizes the "Addponent" operator, defined algebraically as x \underline{\times} y = x(y + 1). Algebraically, this operator defines a non-associative, non-commutative magma on \mathbb{R}. By encoding a clear history-dependent arrow of action and preserving the base state, the Addponent provides a streamlined mathematical shorthand for discrete accretion. Key Contributions: Algebraic Classification: Formalizes the operator as a non-commutative, non-associative magma. The Addponent Associator: Proves a unique outer-dependent associator factorization (x, y, z) = xz, which strictly isolates the interaction between the base origin and the final chronological actor, exhibiting symmetric outer-dependence. Machine Learning Application: Demonstrates how the operator provides a purely multiplicative syntax for additive residual connections in deep neural networks (\mathbf{x} + \mathcal{F}(\mathbf{x})) via Hadamard products. Physical Modeling: Extends the scalar operator to tensor generalizations for continuum mechanics and mechanical strain. This paper establishes the algebraic foundations of the operator and demonstrates its utility across discrete growth models, neural network architectures, and geometric physics.