Unified Field Theory via Monstrous Moonshine and A₂ⁿ -Norm Geometry

This paper develops a unified field theory based on the Cayley–Dickson construction of hypercomplex algebras. Each algebra of dimension 2ⁿ is assigned a fundamental physical role: the reals (R) correspond to gravity, the complex numbers (C) to time, quaternions (H) to spacetime, octonions (O) to magnetism, sedenions (16D) to electricity, and higher algebras to the weak and strong nuclear forces, culminating in a 1024-dimensional spacetime-light continuum. Algebraic norms give rise to generalized metrics, while nonassociativity and zero divisors are reinterpreted as physical couplings and confinement mechanisms. We review the mathematical properties of these algebras, explore their representations, and propose explicit Lagrangians. Connections to exceptional groups, string theory, and monstrous moonshine are discussed. The formalism suggests testable predictions such as nonlinear electrodynamics and new particles at high energies. This work aims to provide a coherent algebraic foundation for all fundamental interactions, extending beyond the Standard Model while maintaining mathematical consistency through careful handling of nonassociative structures and zero divisors.