Why it is a question
THEORY-195, narrowed on 2026-10-19 under ADR-047 v3, holds what Cagnetta et al. (LIT-877) measured: deep CNNs matched to the tree learn the Random Hierarchy Model from about n_c m^L examples, at the same training-set size at which their hidden layers become invariant to synonym swaps, while shallow and fixed-kernel learners need a number proportional to the whole input space. The paper’s explanation is a large-v scaling estimate: the class statistics of input patches rise above sampling noise at the same n_c m^L. Until that date THEORY-195’s title also asserted that explanation, and its promote_when asked for a way to test it. At the owner’s direction, from the promotion audit’s batch 027 (ADR-047), the THEORY was narrowed to the measured result, and the condition moved here unchanged.
It is open because, in the model as given, the threshold and the grammar’s size are the same number. Removing the correlations altogether (Fig. 9) shows they are necessary, not that their detectability fixes the count (NOTE-677).
What would answer it
The condition formerly on THEORY-195, unchanged: a result that separates the number of examples a deep network needs from the counting of the grammar, by moving the input–label correlations while n_c, m and L stay fixed. For instance, non-uniform production probabilities could make some tuples’ class statistics detectable earlier than others, with the training-set sizes at which each level becomes synonym-invariant, and at which test error falls, tracking the predicted detection thresholds rather than n_c m^L. Or a proof that gradient descent on a deep network learns the Random Hierarchy Model from a number of examples of order n_c m^L, polynomial in the input dimension. Further collapses of learning curves on n_c m^L at other parameter values cannot settle it, because the threshold and the grammar’s size are the same number there.