037 Trained to grokking on multiplication mod p, which characters does a small network learn — the additive characters of Z/p, or the multiplicative (Dirichlet) characters, i.e. Fourier modes in discrete-log coordinates on (Z/p)^x?CONFIRMED 2026-09-28 by the repo owner, scored on the prediction's intent.
P-019 Does every computable magnitude come through a convex dual?
P-027 Can the staircase of a deep linear network's loss be read backwards, as a measurement of a chosen spectrum?
P-028 Is the period-doubling past the edge of stability Feigenbaum's cascade, and in what dimension does that stop holding?
P-029 Is the double-descent spike at p = n quantitatively the Marchenko–Pastur edge?
N-031 MPS and CPU training diverge for the same seed in R26_training_dynamics — so a retrain meant to reproduce 037's W_E must run on the same device, and be checked
P-031 Grokking multiplication on the units of Z/561 — which characters appear when the group is not cyclic and no discrete log exists?
P-034 Eta's coefficients indexed by n versus by m — is one coordinate trivial and the other near-impossible?
P-035 Are learning-curve shapes exactly two kinds — spectral readouts and competitions between coexisting solutions?
N-039 Loop-in from the grokking chat, 2026-10-03 — "define the geometry by what the shapes optimise" is a duality bridge run backwards plus the rigidity question, not a new term; candidate §67 entries; two candidate probes on merging grokked seeds; a speculative physics chain with every attribution unverified