# Discrete Logarithm Problem

A two-layer, 427k-parameter transformer shows grokking on the finite-field discrete logarithm map: after a long memorization phase it reaches 99.8% held-out accuracy at p = 113. A 12-model capacity sweep from 90k to 14.4M parameters finds grokking recurring at multiple scales, with a non-monotonic region whose outcome changes when training data increase; hidden states grow aligned with multiplicative-character structure, and prime-order and disjoint-generator controls also generalize.

A two-layer, 427k-parameter transformer shows grokking on the finite-field discrete logarithm map: after a long memorization phase it reaches 99.8% held-out accuracy at p = 113. A 12-model capacity sweep from 90k to 14.4M parameters finds grokking recurring at multiple scales, with a non-monotonic region whose outcome changes when training data increase; hidden states grow aligned with multiplicative-character structure, and prime-order and disjoint-generator controls also generalize.