General-case energy evolution beyond the minimal example

Derive the slow evolution of the self-stabilization energy for the general codimension-one valley setting, extending the minimal-example calculation to multidimensional valley variables.

Background

The paper computes the slow energy variation only for the minimal example, where the valley has codimension one and the specific functions gg and hh permit explicit calculations.

The corresponding calculation for the more general codimension-one setting is explicitly left unresolved, despite being relevant for understanding the slow damping or growth of self-stabilization fluctuations when the valley itself has multiple dimensions.

References

We have done these derivations only in the minimal example of this section and leave the more general case of Sec.~\ref{sec:codim-one} for future work.

— The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow  (2609.01034 - Berthier, 1 Sep 2026) in Section 3.2, final paragraph of “Energy variations in self-stabilization”