Is the double-descent spike at p = n quantitatively the Marchenko–Pastur edge?

Status: open · laid 2026-09-27 · type: lead

Statement. Fit n samples with p features by minimum-norm least squares — the pseudoinverse solution, no ridge, no early stopping. Sweep p through n. The test risk does not fall monotonically and it does not simply rise past the interpolation threshold: it spikes at p = n and then falls again as p grows beyond n (double descent). The mechanism usually named for the spike is that at p = n the n × p feature matrix X is square, its smallest singular value has nothing to keep it away from zero, and the pseudoinverse's variance term — which contains tr((XᵀX)⁻¹), i.e. a sum of 1/λ over the eigenvalues — is dominated by the smallest one.

The probe is whether that mechanism is quantitative rather than gestural. Marchenko–Pastur is exactly a statement about where the smallest eigenvalue goes: for X with iid entries and aspect ratio γ = p/n held fixed as n → ∞, the spectrum of XᵀX/n fills [(1 − √γ)², (1 + √γ)²], and the lower edge closes on zero precisely at γ = 1 — which is the location of the spike, arrived at from random matrix theory with no reference to learning at all. So the question:

1. Does the measured spike have the shape the MP edge forces, not just the position? The soft-edge density near the lower edge and the divergence rate of ∫dμ(λ)/λ as γ → 1 predict a definite curve for the risk on either side of the threshold, with the noise level as the only scale. Measure the risk curve and the empirical spectrum of the same feature matrices, and check the two against each other rather than the risk against a fitted shape.
2. Where does it stop matching, and why? Three candidates, each a different kind of departure: at finite n the peak cannot diverge, so its height is set by the fluctuation of λ_min (a Tracy–Widom-scale quantity, not an MP one) and the finite-n peak may not even sit exactly at p = n; with non-Gaussian or heavy-tailed features universality is expected to hold and is worth confirming rather than assuming; with correlated features the limit law is a free convolution of MP with the population spectrum, and the edge — hence the spike — moves with the population covariance. The third is the interesting one: it predicts a shifted spike, and a shifted spike measured where the covariance says it should be would settle the mechanism far more firmly than the γ = 1 coincidence does.

What is known, and therefore is not what this asks. That a closed form for the ridgeless risk exists in the proportional limit (p/n → γ with both large) is established in the literature, and the γ = 1 divergence is a consequence of it. This probe is not asking whether the asymptotic theory is right. It asks whether the spike as measured — at the finite sizes anyone actually computes at, with its height, width and location — is that object, and which of the three departures above is what a discrepancy would be.

Why a probe and not a case. The computation is trivial (a sweep of pseudoinverse fits and an eigenvalue histogram), but the branches are not yet written: "the spike is the MP edge", "the spike is a finite-n fluctuation object that only agrees with MP in the limit", and "the spike is set by the population covariance and MP is the isotropic special case" are three accounts and the discriminating measurement between them has to be chosen before it is a case.

No prediction and no falsifier are filed here — both are the human's (MANUAL.md, Decisions only you make).

Cross-refs. Siblings from the same session: P-027 (the deep-linear staircase as a spectral readout) and P-028 (the edge of stability and Feigenbaum's δ). The run that opened the track is lab/runs/037-grokking-multiplicative-characters.md. The common shape across the three: a training phenomenon whose explanation is already a spectral object — a singular value, a curvature eigenvalue, an edge of a spectral density — so each asks whether the phenomenon measures the spectrum or merely resembles it.