Directional Curvature from Armijo Backtracking: A Low-Cost Sharpness Probe and a Calibration-Free Learning-Rate Safeguard for Adam

The local sharpness of the loss, the top Hessian eigenvalue $λ_1$, determines the largest stable gradient step, but measuring it normally requires Lanczos or Hessian-vector products. A single Armijo backtracking line search already carries this information at the cost of a few forward passes: the accepted step $α$ brackets the directional curvature along the probed direction within the multiplicative band set by the backtracking factor: exactly the curvature averaged over the tested step, and empirically $q = g^\top H g/\|g\|^2$ to within that band. Across CIFAR-10, Fashion-MNIST and Imagenette, $\logα$ tracks $\logλ_1$ at Pearson $-0.91$ to $-0.95$, and the relation survives a per-run detrending check at $-0.60$ to $-0.70$, a low-cost online Edge-of-Stability reading of the slow sharpness component. Used as a safeguard rather than a faster optimiser, the reading caps a too-large initial learning rate. A single fixed protocol, probing along Adam's own update direction at initialisation and over the first fifty optimiser steps and capping the rate at twice the smallest reading, removes every divergence across learning-rate grids spanning $10^{-3}$ to $3.0$ and at GPT-2 pretraining scale, and all but one marginal case across the further architectures we test, at about $1\%$ overhead, and it leaves training bit-identical whenever the cap does not bind. No constant in the protocol is tuned per architecture; this is the sense in which the safeguard is calibration-free. The guarantee is divergence, not accuracy: where the productive range is narrow the capped run survives at strongly reduced accuracy (chance level on AG News at aggressive rates), and our measurements show why any cap frozen at initialisation must fail at pretraining scale: the loss surface sharpens within the first five optimiser steps, the gap warmup has always filled by convention.