Polarized Generalized-Dual Curvature Regularization

Integrate this at the training-loss evaluation, not inside the ordinary forward activations initially. Flatten a sampled subset of model parameters into q, choose k=2 first, and draw two independent normalized directions v1 and v2 from the same distribution as the optimizer update or from Gaussian noise. Implement a generalized-dual scalar type containing coefficients [r0,r1,r2]; overload addition and multiplication using the paper's epsilon rule, and implement exp, log, tanh, and division through truncated Taylor coefficient propagation. Run the network and scalar loss with dual-valued parameters q+epsilon1*u for u equal to v1, v2, and v1+v2. Extract P2(u) as coefficient r2, compute the mixed curvature C12=(P2(v1+v2)-P2(v1)-P2(v2))/2, and add lambda*C12^2 or lambda*relu(abs(C12)-tau)^2 to the minibatch loss. The empty-set evaluation is zero for a second derivative and need not be run. The algebraic formulas and polarization coefficients are fixed by the paper; lambda, tau, direction sampling, and whether directions are full-parameter or layerwise are empirical. First test on a small ResNet-18 or 3-layer MLP on CIFAR-10 against baseline SGD or AdamW with identical minibatches and approximately matched forward FLOPs. Measure training-loss descent, validation accuracy, gradient norm spikes, and estimated Hessian-vector curvature. A useful success signal is fewer exploding-loss events or improved validation accuracy at equal final training loss; a failure signal is overhead exceeding the curvature benefit or no reduction in sharpness.