Analytic Properties of the Multivector Zeta Function: Natural Boundaries and the Euler Gap

We introduce and analyze the analytic properties of the Multivector Zeta Function, $\zetaPCA(s)$, defined over the Prime Clifford Algebra. Unlike the classical Riemann zeta function, which encodes integers as scalars, this function encodes integers as multivector states within a non-commutative algebra. We prove that the abscissa of absolute convergence is shifted to $\sigma_0 = 1/\ln 2 \approx 1.44$, a constant we identify as the ``Entropy Threshold'' governing the information density of the prime basis. We explicitly characterize the singular structure of $\zetaPCA(s)$ as a ``Spectral Staircase'' of meromorphic barriers. Specifically, we show that the function exhibits an infinite sequence of vertical singular lines $L_p$ at $\mathrm{Re}(s) = 1/\ln p$ for each prime $p$, containing a discrete lattice of simple poles. While the component subseries admit analytic continuation into the strips between these lines, the sequence accumulates at the imaginary axis ($\mathrm{Re}(s) = 0$), forming a dense natural boundary that prevents continuation into the left half-plane. Furthermore, we rigorously quantify the ``Euler Gap,'' a measure of the deviation from the classical Euler product formula. We demonstrate that the non-commutative geometric product of the algebra introduces spinor interaction terms that act as ``quantum interference,'' breaking the multiplicative symmetry required for an Euler product. Finally, we prove that this structural obstruction, combined with the natural boundary, precludes the existence of a functional equation of the form $\zetaPCA(s) \leftrightarrow \zetaPCA(1-s)$. These results suggest that $\zetaPCA(s)$ behaves analytically less like a modular L-function and more like a lacunary series with a spectral barrier.