Associativity as Occam's Razor: Differentiable Measure of Algebraic Complexity

Discovering discrete algebraic structures from data is a fundamental challenge in machine learning. However, standard deep learning models often struggle to learn generalizable algebraic rules, suggesting that their intrinsic inductive biases may be ill-suited for such tasks. Formalizing this intuition reveals a deeper theoretical issue: the literature currently lacks a precise, quantitative definition of algebraic simplicity. In this work, we resolve this gap by analyzing the global objective landscape of a canonical model for recovering discrete operation tables. We establish a lower bound for this objective and prove that attaining this bound entails a geometric alignment (collinearity) between the model's latent matrix representations. Furthermore, this lower bound implements an inverse ${\ell }_2$ penalty, which induces a full-rank unitarity bias—fundamentally opposing the standard low-rank bias in deep learning models. We establish that the objective over the feasible set is bounded below by an absolute floor determined solely by the target size. Attaining this absolute floor is structurally rigid, requiring the latent factorization to be perfectly unitary and collinear, which imposes a rigid algebraic homomorphism. Crucially, this homomorphism requirement transfers the associativity of matrix multiplication directly to the target operation. The global optimum ${\mathcal{H}}_{\inf }(\delta )$ exhibits a strict associativity gap, attaining the absolute minimum floor if and only if the target is a group isotope. Every global minimizer uniquely encodes the underlying group's left-regular representation (up to isotopy and unitary gauge). We empirically confirm that the minimum objective monotonically increases with the target's latent associativity violation ratio. These results formalize a novel differentiable inductive bias for deep learning—structural simplicity via latent associativity up to isotopy—offering a provably distinct alternative to low-dimensional representation. All foundational theorems are formalized in Lean 4.