Small point/question, Quintin -- when you say that you "can fully avoid grokking on modular arithmetic", in the colab notebook you linked to in that paragraph it looks like you just trained for 3e4 steps. Without explicit regularization, I wouldn't have expected your network to generalize in that time (it might take 1e6 or 1e7 steps for networks to fully generalize). What point were you trying to make there? By "avoid grokking", do you mean (1) avoid generalization or (2) eliminate the time delay between memorization and generalization. I'd be pretty interested if you achieved (2) while not using explicit regularization.
I checked whether this token character length direction is important to the "newline prediction to maintain text width in line-limited text" behavior of pythia-70m. To review, one of the things that pythia-70m seems to be able to do is to predict newlines in places where a newline correctly breaks the text so that the line length remains approximately constant. Here's an example of some text which I've manually broken periodically so that the lines have roughly the same width. The color of the token corresponds to the probability pythia-70m gave to predicting a newline as that token. Darker blue corresponds to a higher probability. I used CircuitsVis for this:
We can see that at the last couple tokens in most lines, the model starts placing nontrivial probability of a newline occurring there.
I thought that this "number of characters per token" direction would be part of whatever circuit implements this behavior. However, ablating that direction in embedding space seems to have little to no effect on the behavior. Going the other direction, manually adding this direction to the embeddings seems to not significantly effect the behavior either!
Maybe there are multiple directions representing the length of a token? Here's the colab to reproduce: https://colab.research.google.com/drive/1HNB3NHO7FAPp8sHewnum5HM-aHKfGTP2?usp=sharing