Nilpotent Quantum Cosmology: Resolving the Initial Cosmic Singularity via j-Algebraic Mini-Superspace Metrics

This manuscript provides a dedicated regularization of mini-superspace quantum cosmology by mapping the Wheeler-DeWitt (WdW) equation over a fifth-order nilpotent quotient ring extension, $\mathcal{R} = \mathbb{R}[j]/\langle j^5 \rangle$. By replacing the standard discontinuous classical reciprocal scale factor with a smooth algebraic interpolation operator, the catastrophic initial Big Bang singularity ($a \to 0$) is completely resolved. The explosive $1/a^2$ classical potential barrier is replaced by a bounded, non-singular algebraic potential plateau at the origin ($V_0 = 2mkj^2$). The paper explicitly proves the strict conservation of the total probability current density ($\frac{\partial J}{\partial a} = 0$) across the core boundary, demonstrating a smooth, deterministic "Big Bounce" turnaround that preserves quantum information without requiring manual loop quantum holonomy corrections or exotic matter fields.