The Non-Integer Clifford-Mandelbrot Set Vector Invariance and Fundamental Properties

What happens to the Mandelbrot set when the exponent $p$ varies continuously, rather than being restricted to integers? This paper answers that question within the Clifford algebra framework, where the classical quadratic map $z\mapsto z^2+c$ becomes $F_p(\vec{x}) = (\vec{x}\diamond\vec{n})^p\diamond\vec{n}+\vec{c}$. The key technical obstacle is defining $(\vec{x}\diamond\vec{n})^p$ for non-integer $p$. We overcome it by revealing that $\vec{x}\diamond\vec{n}$ splits into a commuting scalar–bivector pair $(a,B)$ with $B^2 = -\|\vec{x}_\perp\|^2$, generating a subalgebra isomorphic to $\mathbb{C}$. This spectral decomposition yields the polar representation $(\vec{x}\diamond\vec{n})^p = r^p e^{ip\theta}$, where $r=\|\vec{x}\|$ and $\theta = \operatorname{atan2}(\|\vec{x}_\perp\|,\vec{x}\cdot\vec{n})$. Our main result proves that for every real $p>1$, $F_p(\vec{x})$ remains a vector — despite intermediate multivector terms — extending the vector invariance theorem beyond the integer case. With this, we define the non-integer Clifford–Mandelbrot set $\mathcal{M}_{p,\vec{n}}$ and establish its fundamental properties: rotational symmetry about $\vec{n}$, an explicit escape radius, boundedness, and compactness. When $p=2$ and $V=\mathbb{R}^2$, we recover the classical Mandelbrot set. This paper provides the analytical foundation for a companion study of phase transitions, where we show that $\mathcal{M}_{p,\vec{n}}$ changes topology at $p=2,3,4,\ldots$ and at rational resonances.