Observation Does Not Imply Control: The Dynamics of Directional Curvature Interventions at the Edge of Stability

Curvature estimation has made sharp directions increasingly accessible in neural-network optimization, but a measured direction need not respond predictably to intervention. We study directional gradient edits at the Edge of Stability (EoS). In a full-batch MLP, fixed-divisor edits provide an interventional test of known preconditioned-stability algebra: conditional on trajectories that exhibit the corresponding sign-flip orbit, late raw sharpness tracks its predicted location $2d/\eta$, while orbit survival is separate and is not explained by the reduced model. In stochastic ResNet-18/CIFAR-100 training, an engaged rank-1 attenuation arm and wider raw-divisor top-eigenspace rescalings through $k=20$ show no statistically resolved reduction in the tested sharpness diagnostics at the canonical operating point; the wider rescaling rule is not unconditional pure damping because below-target directions receive bounded sign-reversed amplification. We use SAM as a sharpness-reduction control: the tested $\pm H\mathbf{g}$ first-order corrections do not reproduce its ResNet response, and the sampled linSAM-to-SAM interpolation associates lower sharpness with increasing weight on the finite re-query residual bundle. This pathwise decomposition does not isolate a unique nonlinear mechanism and does not transfer in the same form to the LayerNorm ViT control. These results distinguish observing curvature from predicting the response to a specific curvature intervention, within the tested systems, measurements, and intervention rules.