Phase Algebra II: Clifford Tabular Algebra - A Multivector Data Model for Relational Tables, Subspace-Coupled Joint Diagonalization across Heterogeneous Dimensions, and Measured Limits (A Rust Prototype, Part II) | [wAI ~ wErrors]

Abstract

Description / Abstract Part I of this work read a matrix as a network of Clifford multivectors and built phase-rotor solvers on that reading. This note asks whether the same mental model extends to the foundational object of modern data systems: the relational table. We establish the geometric ontology: Columns are basis generators ($e_1, \dots, e_d$). Rows are grade-1 multivectors in a Euclidean $\mathcal{C}\ell(d, 0)$ Clifford algebra. Schemas are blades. Rather than asserting theoretical equivalence, we implement and empirically measure the consequences of this representation. Key Findings & Measured Results Row vs. Column Geometry (Dual vs. Primal): Which generator family is wedged decides the contraction index. The geometric product of two rows yields the classical linear kernel ($x_i \cdot x_j$), providing sample–sample structure ($n \times n$). Conversely, the product of two columns provides the feature–feature correlations ($d \times d$) required by the normal equations. Closed-Form Bivector Energy: The total pairwise bivector energy computed in $\mathcal{O}(n^2 d^2)$ has an exact closed form: $$\mathcal{E}_{\wedge} = \frac{1}{2} \left[ (\text{tr}\,G)^2 - \Vert{}G\Vert{}_F^2 \right]$$ where $G = X X^T$ (or $X^T X$). This equals the second elementary symmetric polynomial of the covariance spectrum, reducing computation to $\mathcal{O}(n d^2)$. It is invariant to whether rows or columns are wedged, and the normalized column-wedge ratio is shown to equal $\text{PR} - 1$ (where $\text{PR}$ is the participation ratio). Rectangular SVD vs. Gram Squaring: The natural "avoid squaring the condition number" route initially showed a $50\text{--}96\%$ reconstruction error. Diagnosing this revealed a missing rectangular exact solver (rather than a flaw in the principle). Introducing direct rectangular QR reduction (solve_rectangular) resolved this, achieving machine precision ($< 10^{-10}$) and outperforming the Gram route by three orders of magnitude on ill-conditioned features ($3.3 \times 10^{-9}$ vs. $1.1 \times 10^{-6}$ residual). Subspace-Coupled Joint Diagonalization (Heterogeneous JADE): When multiple tables share only a subset of columns, they form a family of matrices sharing only a subset of generator axes. We formulate Subspace-Coupled JADE: a shared axis pair receives a single rotation angle computed and applied only to matrices observing both axes at their local indices. We demonstrate that the alternative—padding into a single ambient space—artificially fabricates energy on unmeasured axes. Architecture & Tabular Geometry We argue for a two-layer architecture: Discrete Layer: Relational semantics, key lookups, and schemas are rigorously maintained within a functorial data model. Geometric Layer: Clifford multivector geometry operates directly on the numeric feature substrate. Practical capabilities enabled by this geometry include: Dual/Primal Dispatch: Automatic routing based on table aspect ratio ($n \gg d$ vs. $d > n$). Missingness by Subspace Projectors: Robust pairwise deletion preserving multivector norms. Label-Free Domain Transfer: Orthogonal Procrustes rotor alignment across shared subspaces. Directed-Flow Diagnostics: Temporal circulation statistics utilizing bivector orientation rather than magnitude. Code: https://github.com/skovnats/lie_cliffalg_analog_svd