Collective-Mode De-Gennes Optimizer

1. Integration point: implement this as a wrapper around AdamW or SGD for one or more selected layers. Construct a fixed orthonormal mode basis $U=[u_1,\ldots,u_m]$ for the parameter coordinates, using graph Laplacian eigenvectors for a neuron-connectivity graph, one-dimensional Fourier modes for sequence positions, or random orthogonal blocks for a cheap first test. At step $t$, project the raw update $g_t$ or actual parameter displacement $\Delta\theta_t$ into modes $z_{k,t}=u_k^T\Delta\theta_t$. 2. Pseudocode: maintain exponential averages $\bar z_{k,t}=\beta\bar z_{k,t-1}+(1-\beta)z_{k,t}$ and $\overline{|z|^2}_{k,t}=\beta\overline{|z|^2}_{k,t-1}+(1-\beta)|z_{k,t}|^2$. Estimate the static structure factor as $S_{k,t}=\overline{|z|^2}_{k,t}/(\overline{|z|}_{k,t}^2+\epsilon)$, or more robustly estimate it from the lag-zero covariance of a normalized mode signal $x_{k,t}$. Set $\eta_{k,t}=\eta_{\mathrm{base}}(S_{\mathrm{ref}}/(S_{k,t}+\epsilon))^\alpha$, clip it, apply the usual AdamW update in mode coordinates, and transform back with $U$. Use $\alpha=0$ as the baseline and $\alpha\in\{0.25,0.5,1\}$ for tests. 3. Computed versus estimated: the characteristic-function and structure-factor identities are inherited from the paper; $S_k$, $D_k$, and relaxation rates are empirical estimates. Estimate the neural ISF from $C_k(\tau)=\langle\exp(iq[\theta_{t+\tau,k}-\theta_{t,k}])\rangle_t$ for several small probe values $q$, then fit $C_k(\tau)/C_k(0)$ to an exponential and obtain $\widehat\Gamma_k$. 4. First cheap experiment: train a 2-layer MLP on MNIST or CIFAR-10 with SGD, AdamW, and this wrapper, using a 64- or 128-dimensional random orthogonal basis on the final layer. Log $S_k$, fitted $\widehat\Gamma_k$, loss, gradient norm, and the ratio $\widehat\Gamma_kS_k/|q_k|^2$. The quantitative prediction is a negative correlation between $\log\widehat\Gamma_k$ and $\log S_k$ with slope near $-1$ in the diffusive regime; across modes, $\widehat\Gamma_kS_k$ should be approximately proportional to the mode wavenumber squared or graph-Laplacian eigenvalue. Compare exponential-fit windows: systematic rate drift with window length predicts a non-negligible memory kernel, and the scheduler should then reduce momentum or increase $\beta$. A useful acceptance criterion is at least 0.7 correlation with the predicted inverse-$S_k$ law and a measurable reduction in the slowest-mode relaxation time without increasing final validation loss.