Abstract
<title>Abstract</title> <p>The application of data-science methods to the discovery of mathematical structure, from the string landscape to topological invariants, representation theory and automated conjecture generation, has grown into a distinct branch of experimental mathematics. In the spirit of that programme, and of the interests of this collection, we ask what machine learning can reveal about Clifford and Grassmann algebras when those algebras are used to drive a discrete dynamical system. We introduce a family of normalized multivector-valued cellular automata whose sites carry states in the Clifford algebra Cl(p, q) or in the exterior (Grassmann) algebra Λ(V), and treat the automaton as a generator of algebra-labeled dynamical data. Using grade-resolved algebraic statistics, grade energy and entropy, a grade-mixing index, a stabilized Clifford energy and perturbation-growth descriptors, we (i) compute two-parameter phase diagrams for Cl(2,0), Cl(3,0), Cl(1,2), Cl(1,3), and the nilpotent Grassmann algebra Λ(V), exposing ordered, periodic, chaotic grade-mixing, algebraically constrained, and intermittent regimes; (ii) show that grade entropy and an effective Lyapunov exponent act as coincident order parameters; and (iii) demonstrate that unsupervised learning recovers the phase organization directly from trajectory descriptors. Crucially for the experimental-mathematics setting, supervised learning recovers the generating algebra and its metric-signature class from dynamical observables alone. Under a genuinely blind protocol, using eight descriptors computed in Euclidean coefficient space that contain neither the algebra dimension nor the Clifford metric, and folds grouped so that every algebra at a parameter coordinate shares a fold, the signature class is recovered with 87% accuracy (chance level 33%), and a definite-versus-indefinite classifier trained on one algebra pair transfers to the other at 81–94% balanced accuracy (chance level 50%); the dynamics carry a learnable, transferable algebraic fingerprint, although a normalization control shows that the metric normalization contributes part of it. Finally, symbolic regression proposes a compact candidate scaling relation for the onset of chaos, λ_eff ≈ 0.41 · H_spec^3.33, and a depth-three decision tree reproduces all five phase labels with 98.3% fidelity as an interpretable surrogate. A matched-protocol comparison across scalar, complex, quaternionic, Clifford, and Grassmann automata shows that phase diversity is not monotone in algebra dimension. A normalization control, however, shows that the collapse of the indefinite algebras is produced by the interaction of the indefinite bilinear form with the normalization step rather than by the signature of the product itself: under a common Euclidean coefficient normalization, the indefinite algebras recover a phase diversity comparable to the definite ones.</p>