From Symmetry to Invariance: Learning Across Basis Representations in Finite Fields

Neural models can learn algebraic operations from finite examples, but it remains unclear whether such generalization transfers across mathematically equivalent representations of the same operation. We study this question through multiplication in F₈ under changes of basis, where Galois symmetry induces a natural equivalence structure among representations. This setting allows us to construct controlled train--test splits that separate learning the underlying algebraic operation from generalizing across basis representations. We compare several ways of exposing or recovering this structure, including explicit invariant labels, raw basis information, direct recognition objectives, algebraically decomposed computation, and symmetry-based canonicalization. Across these settings, we observe a consistent gap between learning with explicit invariant information and recovering that information from representation-dependent inputs. We further investigate whether the Frobenius action can serve as a mechanism for canonicalization and evaluate this approach under strict held-out-input protocols. More broadly, our framework provides a controlled setting for studying representation-level generalization in finite algebraic systems.