Dissipative Circulation Optimizer

1. Integration point: implement this as an optimizer wrapper around any PyTorch model. Partition the flattened parameter vector into blocks of size 2 or 4, and define a block-diagonal skew matrix \(J\); for every 2D block use \(J_b=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\). Do not materialize a full matrix: rotate each parameter-gradient pair in place. 2. Pseudocode: compute the minibatch gradient \(g\); for each pair \((g_i,g_{i+1})\), compute \(r_i=(-g_{i+1},g_i)\); update \(\theta\leftarrow\theta-\eta(g-\gamma\alpha r)\). Every \(K\) steps estimate the local Jacobian/Hessian action with two Hessian-vector products or finite differences, form a small block approximation to \(A=I-\eta(I-\gamma\alpha J)H\), and reduce \(\alpha\) or \(\eta\) if its estimated spectral radius exceeds 0.98. Use gradient clipping only as a fallback, not as the primary stability mechanism. 3. Computed from the paper: the force decomposition and the required nonzero circulation criterion. Estimated empirically: Hessian-vector products, the spectral radius, and loop work. Record checkpoints along a small rectangular loop in two selected parameter directions and numerically integrate \(F\cdot d\theta\); use the update direction as the force. 4. First cheap experiment: train a 2-layer MLP on MNIST or CIFAR-10 with SGD, Adam, and this optimizer at matched learning-rate budgets. Use a two-dimensional quadratic benchmark with known \(H=\operatorname{diag}(1,\kappa)\) first, sweeping \(\eta\) and \(\alpha\). The quantitative prediction is that divergence begins at the measured boundary \(\rho(I-\eta(I-\gamma\alpha J)H)=1\), within 20% of the observed boundary. On neural loss surfaces, nonzero loop work should scale approximately linearly with \(\alpha\) for small \(\alpha\), while setting \(\alpha=0\) must recover the baseline exactly. Compare time to reach a fixed loss and verify that any speedup disappears when loop work is statistically indistinguishable from zero.