Why it is a question
THEORY-021, narrowed on 2026-10-19, holds that in the exact configurations the dimensionality-weighted projectors sum to the identity: Σᵢ Dᵢ ŵᵢŵᵢᵀ = I. The configurations are regular polygons, the tetrahedron, antipodal pairs, and spanning tegum products of these, with equal norms within each factor. So the features form a rank-one POVM compressed from the n-feature basis. Until that date, THEORY-021’s promote_when said “The algebra is settled for the exact configurations. What is open is whether trained models reach them”, and asked for the measurement. At the owner’s direction, from the promotion audit’s batch 010 (ADR-047), the THEORY was narrowed and promoted on the re-derived algebra, and the condition moved here unchanged.
It is open because the configurations come from the toy model’s best-of-many runs (LIT-323, read in NOTE-278), whose authors expect them to generalise least, and nothing the record holds measures the frame operator on trained weights. It refines QUESTION-058’s question of what a trained network’s representation carries.
What would answer it
The condition formerly on THEORY-021, unchanged: a measurement on trained weights, either the toy’s W or a sparse autoencoder’s decoder directions, that computes Σᵢ Dᵢ ŵᵢŵᵢᵀ over the represented features and finds it close to the identity on the span. Finding that the Dᵢ sum to m, or that the configuration looks like a polytope, does not settle it. Leverage scores always sum to the rank, and a polytope seen by eye is not a measured frame operator. Trained configurations whose dimensionalities are sticky at the polytope values while Σᵢ Dᵢ ŵᵢŵᵢᵀ stays far from the identity would refute the account for trained models.