Clifford Tabular Algebra: A Multivector Data Model for Relational Tables, and the Measured Limits of Geometric Reformulation (A Rust Prototype, Part II) | [wAI ~ wErrors]

Abstract / Description for Zenodo (v0.30.3/4) This note presents a multivector ontology for relational tables—mapping columns to basis generators ($e_1, \dots, e_d$) and rows to grade-1 multivectors in $\mathcal{C}\ell(d, 0)$—alongside empirical benchmarks on numerical tabular regression. Key Findings & Measured Results Scalar vs. Bivector Geometry: The scalar part of the geometric product of rows equals the standard linear kernel ($x_i \cdot x_j$). The bivector component ($x_i \wedge x_j$) captures strictly antisymmetric information, vanishing if and only if rows/columns are collinear. Closed-Form Bivector Energy: The pairwise bivector energy $\frac{1}{2}\left[(\text{tr}\,G)^2 - \Vert{}G\Vert{}_F^2\right]$ (with $G = X X^T$) computes in $\mathcal{O}(n d^2)$ rather than $\mathcal{O}(n^2 d^2)$, functioning as an effective-rank statistic of the covariance spectrum. Rectangular SVD vs. Gram Squaring: Avoiding the squared condition number ($\kappa^2$) via direct rectangular QR reduction (solve_rectangular) achieves exact machine precision ($< 10^{-10}$). On ill-conditioned features, direct factorization outperforms the Gram-based method ($3.3 \times 10^{-9}$ vs. $1.1 \times 10^{-6}$ residual). Bivector-Aware Anisotropic Regularization: Using inverse pairwise wedge stress ($\Lambda_{jj} \propto 1 / \text{stress}_j$) penalizes collinear directions while preserving orthogonal ones, consistently outperforming isotropic Ridge regression on held-out test splits. Architecture A two-layer model: discrete semantics (keys, schemas) remain in a functorial/relational layer, while multivector Clifford geometry operates directly on the numerical feature substrate. NOTE: v0.30.4 WIP