Machine Learning the Phase Structure and Algebraic Invariants of Clifford- and Grassmann-Valued Cellular Automata

Technical Summary: Machine Learning the Phase Structure and Algebraic Invariants of Clifford- and Grassmann-Valued Cellular Automata

Problem Statement
While machine learning has increasingly been applied to discover mathematical structures in fields ranging from string theory to topological invariants, Clifford and Grassmann algebras have largely been utilized within learning architectures (e.g., as inductive biases) rather than being the object of study themselves. This paper addresses the inverse problem: Can machine learning recover the underlying algebraic structure (specifically the generating algebra and its metric-signature class) solely from the dynamical data generated by a system driven by that algebra? The authors investigate this by treating multivector-valued cellular automata (CA) as generators of algebra-labeled dynamical data, asking whether the "fingerprint" of the algebra is encoded in the system's phase transitions and statistical observables.

Methodology
The study constructs a family of normalized multivector-valued cellular automata on a one-dimensional periodic lattice ($N=96$).

• Dynamical System: The state at each site $i$ is a multivector $X_i(t)$ belonging to one of five algebras: $Cl(2,0)$, $Cl(3,0)$, $Cl(1,2)$, $Cl(1,3)$, and the Grassmann algebra $\Lambda(\mathbb{R}^3)$. The local update rule is a normalized map involving memory, neighbor diffusion, geometric product nonlinearity, grade involution, and pseudoscalar chirality terms, controlled by five parameters $(\alpha, \beta, \gamma, \delta, \eta)$.
• Observables: To resolve the dynamics beyond scalar magnitudes, the authors introduce grade-resolved observables. These include grade energy density, grade entropy ($\bar{H}_{grade}$), a grade-mixing index ($C_{mix}$), stabilized Clifford energy, and perturbation-growth descriptors (effective Lyapunov exponent $\lambda_{eff}$). Additional metrics include spectral entropy, recurrence quantification analysis (RQA) descriptors, and correlation lengths.
• Phase Definition: A rule-based classification scheme assigns trajectories to five phases: Ordered, Periodic, Constrained, Intermittent, and Chaotic, based on thresholds of the observables.
• Learning Protocol:
  • Unsupervised Learning: PCA, UMAP, and Gaussian Mixture Models are used to recover phase organization without labels.
  • Supervised Learning: Random Forests, Gradient Boosted Trees, SVMs, and MLPs are trained to classify phases and, crucially, to identify the generating algebra and its signature class (definite, indefinite, degenerate).
  • Blind Controls: To ensure the learning is not trivially exploiting algebra dimension or metric definitions, the authors employ "genuinely blind" protocols. These use descriptor sets ($D_{Euc}$) computed entirely in Euclidean coefficient space, excluding any descriptors that explicitly contain the algebra dimension ($n$) or the Clifford metric. Cross-validation is performed with folds grouped by parameter coordinates to prevent data leakage.
  • Conjecture Formulation: Symbolic regression and decision trees are used to derive compact mathematical relations and interpretable surrogates for the phase rules.

Key Contributions and Results

1. Phase Structure and Order Parameters:

   • The study maps two-parameter phase diagrams for five algebras, revealing distinct regimes of ordered, periodic, chaotic, constrained, and intermittent behavior.
   • Grade entropy and the effective Lyapunov exponent are identified as coincident order parameters; their transition contours align across all algebras, suggesting that phase transitions in these systems are fundamentally reorganizations of grade content rather than scalar amplitude changes.
   • Cross-Algebra Comparison: Phase diversity is not monotonic with algebra dimension. While definite and degenerate algebras show rich phase structures, indefinite algebras ($Cl(1,2)$ and $Cl(1,3)$) appear almost entirely chaotic under the standard metric-dependent normalization.

2. The Role of Normalization:

   • A critical finding is that the collapse of indefinite algebras into chaos is driven by the interaction between the indefinite bilinear form and the normalization step, not the signature of the product itself.
   • Under a common Euclidean coefficient normalization, the indefinite algebras recover phase diversity comparable to definite ones. This indicates that the "chaotic" behavior of indefinite algebras in the primary analysis is an artifact of the stabilization near the null cone, where the pseudo-norm vanishes.

3. Recovery of Algebraic Invariants (Experimental Mathematics Core):

   • Supervised Recovery: Classifiers can recover the generating algebra and its metric-signature class from dynamical descriptors with high accuracy.
     • Using a grade-component-excluded set ($D_{14}$), accuracy reaches 95.8% for algebra identification and 97.3% for signature classification.
     • Under a genuinely blind protocol (using only 8 descriptors in Euclidean space, excluding dimension and metric information), the signature class is recovered with 87.2% accuracy (chance 33%), and the algebra with 72.2% (chance 20%).
   • Transferability: A classifier trained on one pair of algebras (e.g., $Cl(2,0)$ vs. $Cl(1,2)$) transfers to a held-out pair ($Cl(3,0)$ vs. $Cl(1,3)$) with 81–94% balanced accuracy, demonstrating that the dynamics carry a learnable, transferable algebraic fingerprint.
   • Same-Dimension Control: Even when restricted to three algebras of identical dimension (8), the signature classes remain separable with 96.3% accuracy, proving the signal is not merely a function of dimension.

4. Automated Conjecture Formulation:

   • Symbolic Regression: Proposes a candidate scaling relation for the onset of chaos: $\lambda_{eff} \approx 0.41 H_{spec}^{3.33}$, linking the finite-time perturbation growth rate to spectral entropy. The authors present this as a testable conjecture rather than a proven law, noting the exponent varies by algebra.
   • Interpretable Surrogates: A depth-three decision tree reproduces the rule-based phase labels with 98.3% fidelity, providing a human-readable logic for the phase boundaries.

Significance and Claims
The paper positions itself within the program of experimental mathematics, where machine learning serves as a tool for discovering mathematical structure and formulating conjectures.

• Inverse Problem: The primary claim is that one can "hear the signature of a Clifford algebra from the drum of its automaton." The dynamics generated by the algebra contain sufficient information to reconstruct the algebra's identity and its metric signature, even when the descriptors are stripped of explicit algebraic definitions.
• Methodological Insight: The work demonstrates that for multivector-valued systems, grade-resolved statistics are the necessary order parameters, as scalar observables fail to resolve the phase structure.
• Caveats and Modesty: The authors are explicit about limitations. The phase labels are rule-based conventions, not absolute physical truths. The symbolic relation is an empirical regularity with modest predictive power ($R^2 \approx 0.66$) and algebra-dependent exponents. The study is limited to one-dimensional lattices and finite time horizons, precluding claims about critical exponents or thermodynamic limits. The "null-cone mechanism" for the instability of indefinite algebras is offered as a hypothesis consistent with controls, not a proven cause.

In summary, the paper establishes multivector-valued cellular automata as a tractable sandbox where machine learning can successfully invert dynamical data to recover algebraic invariants, provided that the analysis respects the graded structure of the underlying algebra and controls for normalization artifacts.