Why it is a question
THEORY-008 states Harvey, Lipshutz & Williams’s identities (LIT-257, Prop. 2 and Cor. 1) for M stimuli. The task-averaged readout agreement is Tr(K_X K_z K_Y). With K_z = I it gives linear CKA, GULP, mean squared CCA or ENSD. NOTE-834 read the proofs and re-derived them. Until 2026-10-19, THEORY-008’s promote_when also asked for the population version. At the owner’s direction, from the owner’s promotion audit of that date, the finite identities were promoted alone, and that clause moved here unchanged.
It is open because the source does not settle it. App. A.3 is an outline by its own account. It needs b > 0, so the CCA member is not covered, and it mis-states a constant (eq. 77). It also treats a trace-class task covariance, and at K_z = I, the white-noise limit, the quantity it tracks, (1/M³)Tr(K̂ I Ĥ) = (1/M)Tr(K_X K_Y), goes to 0. The non-degenerate form needs the normalization (1/M²)Tr(K̂Ĥ) → Tr[KH], which A.3 does not state.
What would answer it
A source that proves the identities at K_z = I in the population limit, such as GULP’s population analysis (Boix-Adsera et al., not held). Or a proof the record checks of the convergence of (1/M²)Tr(K̂Ĥ), with G estimated from the sample, for each member of the family, CCA included.