Persistent-homology H1 cycle strength rises sharply at grokking

Tang et al. apply persistent homology (Vietoris-Rips, degrees 0-1) to token-embedding and hidden-state point clouds from modular-addition models mod {113,149,197}, using a 2-layer transformer and a 3-hidden-layer MLP [tang-etal-2026-topological-signatures-of-grokking] H1 (loop/cycle) persistence rises sharply and reproducibly at the grokking transition (e.g. transformer p=197 max persistence ~0.075 to 0.20-0.25) [tang-etal-2026-topological-signatures-of-grokking] This is an independent topological confirmation of the circular structure Nanda et al. (2023) found via Fourier analysis, now also demonstrated in an MLP architecture [tang-etal-2026-topological-signatures-of-grokking] Local intrinsic dimension (TwoNN) collapses from ~20-25 to ~5 at the transition, alongside the topological cycle strengthening [tang-etal-2026-topological-signatures-of-grokking] Persistence statistics correlate with test accuracy (max H1 rho up to 0.81; total H0 rho down to -0.91), a correlational not causal linkage [tang-etal-2026-topological-signatures-of-grokking] A label-permutation ablation breaks the topology-generalization link above ~10-20% corruption, and an MNIST control shows no sharp topological transition [tang-etal-2026-topological-signatures-of-grokking]