Division as a radial vector relationship – Alpha group

In this paper we present the formulation of a hypercomplex algebra called the Alpha number algebra, defined by four fundamental elements: 1, i, μ, and i.μ, with the following operational properties: The imaginary number μ is idempotent and non-commutative with the other imaginary number i. This non-commutative and partially idempotent structure provides an algebraic basis for the matrix M(θ), which parameterizes a class of operators that exhibit internal oscillatory symmetries with explicit dependence on the angle θ. The spectral analysis of M(θ) reveals the emergence of discrete natural frequencies at θ = π/2 rad and its multiples, pointing to a spontaneous geometry. The eigenvalues and associated eigenvectors characterize internal modes of vector rotation and propagation, allowing us to describe an internal vector dynamic geometry. This geometry suggests an alternative model of the internal structure of topological space and geometry, in which interactions and vector oscillations emerge directly from the algebra of the Alpha group. This formulation provides support for a new mathematical language applicable to the study of fundamental fields.