Why it is a question
THEORY-181 now states two things. The first is a theorem: in a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set whose same-answer equations have nullity 2 fixes the embeddings to E_k = a + kb. The second is one measurement: for non-modular addition with p = 10 and one-dimensional embeddings, Liu et al. (LIT-858, NOTE-662) find that the probability of nullity 2 jumps near a training fraction of 0.4, and that trained networks take diverging time to form the structure at the same fraction. That is one size and one embedding dimension. Whether the thresholds stay together as p grows is what would make the counting argument an account of the critical training fraction, and not a coincidence at one size.
What would answer it
Run the same toy, two learned embeddings summed and decoded with addition hard-coded, at several group sizes p. For each training set, report the nullity of the linear system its same-answer pairs impose, and whether the trained network generalises. It is answered yes if, at every p, full generalisation occurs with high probability exactly on training sets of nullity 2, and the empirical critical fraction follows the fraction at which nullity 2 becomes likely. It is answered no by networks in that setting that generalise fully on training sets of nullity above 2, or by a critical fraction that departs from the nullity threshold as p grows. More single-size runs, or pictures of structured embeddings in larger models, are the wrong kind of evidence: they show structure, not that the training set’s equations are what fix it.