Odd-Drift, Symmetric-Noise Optimizer

Integrate this mechanism at the optimizer step for an MLP or small Transformer. Maintain a symmetric preconditioner \(S_t\), such as diagonal Adam or RMSProp scaling, and add a cheap antisymmetric block operator \(A_t\). Partition parameters into pairs \((i,j)\) and set \(A_{ij}=a\), \(A_{ji}=-a\) within each pair; use \(a=\alpha/(\sqrt{v_i}+\sqrt{v_j}+\varepsilon)\), where \(v_i\) are Adam second-moment estimates. For each minibatch, compute \(g=\nabla L\), update \(v_t=\beta v_{t-1}+(1-\beta)g^2\), form diagonal \(S_t=(\sqrt{v_t}+\varepsilon)^{-1}\), compute \(A_tg\) using two operations per parameter pair, and update \(\theta\leftarrow\theta-\eta(S_tg+A_tg)+\sqrt{2\eta T S_t}\,\xi\), with \(\xi\sim\mathcal N(0,I)\) and optional temperature \(T\). The paper supplies the exact cancellation of \(D_a\); the preconditioner, temperature, pairing, and \(\alpha\) are engineering choices. First test CIFAR-10 using a two-layer MLP, comparing SGD, Adam, and this optimizer at matched gradient evaluations and wall-clock cost. Record loss, validation accuracy, gradient norms, escape time from a quadratic basin, and lagged joint statistics of loss and gradient norm. The falsifiable prediction is that changing any formal antisymmetric diffusion term leaves scalar-observable statistics unchanged apart from discretization error, whereas changing \(\alpha\) changes relaxation. On a local quadratic \(L(\theta)=\frac12\theta^{\mathsf T}H\theta\), the deterministic update is \(\theta_{k+1}=[I-\eta(S+A)H]\theta_k\); divergence should begin when \(\rho(I-\eta(S+A)H)\ge1\), and the measured threshold should agree within 20 percent.