A small neural network trained on modular addition memorized answers early, then—after many more training steps—abruptly acquired a general, human-like method. That late emergence, called grokking, exposes how training curves can hide deep internal change and raises practical questions for model development.
In the modular-addition experiment, the network discovered a circular number representation and solved addition via rotation—effectively rediscovering trigonometric structure without being told.
Standard training monitoring and early stopping can cut off models before grokking appears, so flat external metrics do not guarantee stasis inside the model.
They trained a tiny neural network, far smaller than anything that gets called "AI" today, on one of the simplest tasks imaginable: modular addition, basically clock math. Within a thousand training steps it was answering every practice question correctly. It's a familiar failure in machine learning: the model had memorized the answer key instead of learning the rule behind it.
Give them a new question that tests the same idea in a slightly different form, and they freeze, because they memorized answers rather than the concept underneath them. They stop recalling flashcards and start actually understanding the topic well enough to solve problems they've never seen. That raises a question worth taking seriously well beyond toy math problems.
What is 8 plus 7 if the clock only goes up to 12? They can answer yesterday's practice questions perfectly. The click happens long after the student appears finished.
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