Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations

This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $Γ$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show t...

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