Fourier-circuit synchronization completes before grokking, causally advanceable

Sivasankar introduces Frequency Synchronization Degree (FSD), a permutation-tested metric of Fourier-circuit synchronization across a modular-addition transformer's embedding, MLP and unembedding components [sivasankar-2026-circuit-synchronization-precedes-generalization] The model is a 2-layer transformer (d_model=128, 4 heads, d_mlp=512) trained across five primes p in {53,71,97,113,131} over 11 configurations and 3 seeds [sivasankar-2026-circuit-synchronization-precedes-generalization] FSD reaches its ceiling 500-3,000 training steps before grokking (mean lead +1,722 steps, sign-test p~0.004), so the Fourier circuit is computation-complete well before generalization [sivasankar-2026-circuit-synchronization-precedes-generalization] Forking training at the FSD ceiling with higher weight decay (lambda in {1,2,3,4,5,10}) produces monotonically earlier grokking (delta-t proportional to 1/lambda), causally tying synchronization to accelerated generalization [sivasankar-2026-circuit-synchronization-precedes-generalization]