Gradient descent with momentum can be viewed as a particle moving with friction in the loss landscape. In deep learning the learning rate is usually large, so training is a discrete dynamical system: in many full-batch experiments the sharpness grows to the optimizer's stability limit, the edge of stability (EoS), and training continues there. Physics-inspired optimizers change this motion; Velocity-Regularized Adam (VRAdam), for example, shrinks the step at high speed. We study kinetic maps, in which the step is the gradient of a kinetic energy of the momentum, and ask what they do beyond the EoS. (i) In a reduced model, a kinetic map creates a symmetric period-two orbit beyond the edge, strictly linearly stable if and only if the map's elasticity (the log-log slope of step length versus momentum) lies between 0 and 1; on it, training sits at an effective edge of stability. VRAdam's damper fails above twice the critical learning rate, where its kinetic energy is not convex; the Legendre dual of VRAdam's own Lagrangian is convex and passes at every learning rate. (ii) Because the sharpness rises gradually, the loss of stability is delayed, as in Neishtadt's theory of slowly swept bifurcations. In a network with a controlled sweep, the dynamics match this theory without fitted parameters, including the memory of where the sweep started and its erasure by noise. (iii) With momentum, reaching the oscillation also requires elasticity at large momentum: maps that bound the step have competing period-four and chaotic attractors already at beta1 = 0.9, while the cubic-Legendre map reached the oscillation from every sampled start up to beta1 = 0.99. Experiments on an exact quadratic problem, a full-batch MLP and ResNet-32 support the theory; on ResNet-18, much of the apparent benefit of the maps is reproduced by matching the effective learning rate.
Stationary learning admits an ideal-gas description under restrictive noise assumptions. In neural networks whose gradient noise departs from these assumptions, however, an entropy-scaling relation resembling a Maxwell relation remains approximately valid. We ask which thermodynamic-looking relations follow from scale symmetry rather than equilibrium, and how departures from the ideal-gas description can be quantified, for discrete stochastic gradient dynamics with memory. An exact rescaling of trajectories explains the entropy relation without assuming thermal equilibrium. Within a local harmonic approximation, self-adjusting weight norms yield a spectral equation of state whose virial coefficients are set by moments of the curvature spectrum, and coupled parameter groups can sustain stationary probability currents. Our central result beyond this approximation is an exact curvature-transport identity. Together with a finite-step loss balance, it decomposes corrections to the harmonic fluctuation relation into measurable curvature changes, finite-step remainders, gradient-memory correlations and finite-window drift terms. A related displacement-weighted curvature change provides a diagnostic of the arrow of time; its mean and second moment give a lower bound on the relative entropy between forward and reversed state pairs. We also calculate the activation-boundary contribution to curvature for rectified teacher-student networks with Gaussian inputs. Tests in sphere models, teacher-student networks and convolutional networks on CIFAR-10 and CIFAR-100 support the transport description and identify limits of the harmonic and cubic-remainder approximations. These results separate symmetry-based relations from equilibrium assumptions and provide quantitative diagnostics of nonideal fluctuations in learning dynamics.