Non-Asymptotic Identification of the Edge of Stability by Southwell Extrapolation of the Step Variance

Preprint, not peer reviewed. Consolidated working draft of 8 November 2026 (revision 19). Hypotheses, limitations and exploratory results are stated explicitly in the text, and failed predictions are retained. Corrections will be released as new versions of this record. Structural engineers locate the buckling load of a column without loading it to failure: Southwell (1932) showed that sub-critical deflections, suitably plotted, fall on a line that gives the critical load. The same construction applies to gradient descent. For heavy-ball momentum β on a quadratic mode of curvature λ1 driven by additive noise, the ratio η²/D(η, β) of the squared learning rate to the stationary mean-square step is exactly linear in η and vanishes at ηcrit = 2(1+β)/λ1. Short bursts at a few learning rates no higher than 0.75 ηcrit therefore locate the stability threshold by linear extrapolation, without Hessian-vector products and without crossing it. The estimator is analysed as a statistical procedure: a non-asymptotic error bound; minimax lower bounds and a position–step estimator that attains them; minimax rates on non-quadratic losses; and burn-in, Adam and progressive-sharpening corrections. Further results include identification of the dominant Hessian eigenvector from first-order steps alone (by analogy with output-only modal analysis); the exact second-moment equation of heavy-ball SGD with curvature noise that need not commute with the Hessian; the exact mean-square ceiling for random-feature minibatches as the root of a rank-one secular equation; and an extension to features with heterogeneous fourth moments (Appendix C), applied to the least-squares vocabulary model of Everett and Paquette (2026). The estimators are evaluated on a synthetic AR(2) family, on two-layer tanh networks (P = 3 610 and 12 826) against analytical Hessian-vector products and destructive search, on a 33-parameter network with weight decay (a replication with predictions committed before the runs), and on a WebAssembly inference block. Sections 4.13–4.14 disclose exploratory development reports and subsequent corrections; their raw GPU artifacts are not included.