Curvature Controlled Mixing on the Interpolation Manifold: Sharp Cutoffs in Late-Phase Training Dynamics

After a network interpolates its training data, weight decay drives a slow norm-minimising flow on the interpolation manifold. We study the stochastic version of this late phase and show that its approach to equilibrium is not gradual but abrupt. We analyse the effective diffusion on the interpolation manifold $M\subset {\mathbb{R}}^n$, whose invariant measure carries the Fixman $\frac{1}{2}\log \det$ correction produced by integrating out normal fluctuations. Our main result is a lower bound on the Bakry-Emery Ricci curvature of this measure, ${\mathrm{Ric}}_{\varphi }⪰\kappa g$ with $\kappa =\frac{2\alpha }{{\sigma }^2}(1-o(1))$ for weight decay $\alpha$ and noise scale $\sigma$, in which every correction — the intrinsic Ricci curvature via the Gauss equation, the extrinsic shape-operator coupling, and the Hessian of the $\log \det$ term — is controlled by quantities measurable on a trained network: the model Hessian norm, its third derivative, the tangential Hessian rank, and the spectrum of the empirical NTK Gram matrix. We also record what the hypotheses force: strong convexity of the log-density makes $M$ complete, noncompact and contractible, so the theorem describes mixing within a single well-conditioned basin rather than across the permutation-symmetric zero-loss set. Positive curvature has two consequences. Existing results on cutoff for non-negatively curved diffusions then give a total-variation cutoff, and we locate it: $t_c=\frac{1}{2\alpha }\log (d_0^2/{\xi }^2)$ with window $O(1/\alpha )$, where $d_0$ is the initial displacement and $\xi ={\kappa }^{-1/2}$ the equilibrium correlation length. The clock is set by weight decay alone, independent of noise temperature at leading order. We identify the equilibrium as the minimum-norm interpolator measure, concentrated to $O(\sigma /\sqrt{\alpha })$ in parameter norm, recovering deterministic norm-minimisation results as its zero-temperature mode. The resulting prediction — delay logarithmic in the ratio of initial displacement to equilibrium fluctuation scale — is distinct from existing norm-ratio delay laws. In simulations of constrained diffusions where every assumption holds by construction, the predicted location holds with regression slope $1.003$ ($R^2=0.99$) across independent variation of dimension, weight decay, temperature and initial displacement, and the total-variation profiles collapse under the predicted rescaling.