Data-free preconditioning builds the Kronecker factors of KFAC from synthetic probes instead of private data. DP-KFC validates a probe by matching the eigenvalue decay of the private factors; we turn this criterion into a random-matrix account of what a probe can reconstruct. A single dichotomy organizes it: in the factorization
A⊗G, the error-signal factor
G and the architecture-determined bulk of the activation factor
A are reconstructible without data, while only the data-induced spikes of
A are not. We prove the
G half is data-insensitive up to an explicit error-energy bound, with zero violations over a full training trajectory across four modalities and a natural-image benchmark. We locate the failure boundary at a Baik-Ben Arous-Péché threshold: a bulk-only probe suffers an exact damage
s2/(1−s2) from a data-induced spike, which becomes invisible once it exceeds
(d/n)1/4, confirmed on an injected-spike surrogate. Holding the probe spectrum fixed and varying only its eigenbasis, we show that spectral-shape matching is neither sufficient nor the operative object: a probe with a near-exact matched spectrum (log-ratio dispersion
0.02) still inflates the residual
2×-
90× when its eigenvectors are random. A closed-form whitened-volume law explains this and predicts the within-spectrum ordering of the critical volume, though not its cross-dataset magnitude. Downstream, per-layer tuning does not transfer to accuracy and the headroom is a small oracle-minus-white gap. Under a privacy mechanism the source ordering survives while learning does, then inverts at a noise cross-over where the flat probe overtakes the matched ones.