Spectral Decomposition and Analytic Continuation of the Geometric Product for Non-Integer Exponents

The geometric product x⃗⃗ in Clifford algebra plays a central role in extending fractal dynamics to higher dimensions. For non-integer exponents p, however, the power (x⃗⃗)ᵖ is not defined by repeated multiplication. This paper provides the algebraic foundation for such an extension. We prove that x⃗⃗ decomposes into a commuting pair $(a, B)$, where a = x⃗⃗ is a scalar and B = x⃗⃗ is a bivector satisfying B² = -\|x⃗_⊥\|². This pair generates a subalgebra isomorphic to the complex numbers C via the mapping φ(u+iv) = u + v B/\|x⃗_⊥\|. Under this isomorphism, x⃗⃗ corresponds to the complex number a + i\|x⃗_⊥\| = reiθ, where r = \|x⃗\| and θ = atan2(\|x⃗_⊥\|, a). This polar representation permits the analytic continuation\[(x⃗⃗)^p = r^p eipθ, p > 1 real,\]which is continuous in x⃗ and reduces to the usual geometric product power when p is an integer. The decomposition provides the algebraic machinery needed for the which will be developed in forthcoming works on non-integer Clifford dynamics and phase transitions.