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

The geometric product $\vec{x}\diamond\vec{n}$ in Clifford algebra plays a central role in extending fractal dynamics to higher dimensions. For non-integer exponents $p$, however, the power $(\vec{x}\diamond\vec{n})^p$ is not defined by repeated multiplication. This paper provides the algebraic foundation for such an extension. We prove that $\vec{x}\diamond\vec{n}$ decomposes into a commuting pair $(a, B)$, where $a = \vec{x}\cdot\vec{n}$ is a scalar and $B = \vec{x}\wedge\vec{n}$ is a bivector satisfying $B^2 = -\|\vec{x}_\perp\|^2$. This pair generates a subalgebra isomorphic to the complex numbers $\mathbb{C}$ via the mapping $\phi(u+iv) = u + v B/\|\vec{x}_\perp\|$. Under this isomorphism, $\vec{x}\diamond\vec{n}$ corresponds to the complex number $a + i\|\vec{x}_\perp\| = re^{i\theta}$, where $r = \|\vec{x}\|$ and $\theta = \operatorname{atan2}(\|\vec{x}_\perp\|, a)$. This polar representation permits the analytic continuation\[(\vec{x}\diamond\vec{n})^p = r^p e^{ip\theta}, \qquad p > 1 \text{ real},\]which is continuous in $\vec{x}$ 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.