Why it is a question
THEORY-233 holds the envelope CKA/√(r(K_X)r(K_Y)) ≤ NBS² ≤ min[r(K_X), r(K_Y)]·CKA (LIT-250, eq. 24). The rank factors leave room for the two measures to order pairs of representations differently. Until 2026-10-19 the THEORY’s promote_when also asked for “a measurement on trained networks of how far CKA and NBS disagree in practice”. At the owner’s direction, from the promotion audit’s batch 007 (the audit’s decision file for THEORY-233, ADR-047), the inequality was promoted and that clause moved here unchanged.
It is open because the paper shows the discrepancy only on random Wishart matrices, with factors of two to three (Fig. 2), and the record holds no comparison on trained networks.
What would answer it
A measurement of CKA and NBS on pairs of representations from trained networks, reporting how often and by how much the two order the pairs differently. Agreement on particular model pairs cannot settle it, because the envelope permits agreement. The answer needs the rate of disagreement across a stated family of pairs, at the ranks those representations have.