WHEN ATTENTION FAILS ON FINITE FIELDS: GALOIS ALGEBRAIC REASONING NETWORKS WITH HARD COSET EXECUTION

Exact algebraic relations need not align with similarity in a learned embedding space. We study this mismatch through a controlled coset-classification task over AES $GF(2^8)$: labels depend on multiplicative ratios, while both classes have identical single-byte marginals. We introduce GARN, a Galois Algebraic Reasoning Network that separates learned interaction weights from exact field execution. Attention proposes interactions, fixed inverse and multiplication tables compute ratios, and learnable membership masks provide evidence for prediction and state updates. Across five seeds, GARN reaches 99.5% test accuracy with 256 labeled examples, whereas the tested Transformers, MLPs, and discrete relaxations remain near chance through 1,024 examples. With shuffled partners, GARN reaches 79.9%, compared with 67.0% for a fixed-adjacency probe. Template-free masks, alternative byte encodings, and corrupted-operator controls isolate the role of exact execution. These results identify operator access as a decisive source of sample efficiency and establish a practical interface between continuous interaction learning and discrete finite-field computation.