The CIFAR CNN — every parameter gradient node IS the loss's derivative, up to pool twins #

cifar_train_step_tied_certified ties each of the fourteen SGD updates to the certified per-layer Jacobian contracted with the cotangent the emitted chain threads to it. cifar_net_lossGrad states that the un-fused *Grad node of each layer, at the chain cotangent, is the gradient of the loss in that parameter, for any loss L of the logits with gradient g there; cifar_net_lossGrad_CE instantiates it at the softmax cross-entropy the render emits.

The two pools are handled as the MNIST CNN's one (CnnFold.cnn_net_lossGrad): each pool's clause allows ties between twins, cells equal at every weight upstream of that pool (CnnFold.CnnPoolTwin for the first, CifarPoolTwin2 for the second), and each pool's backward routes a window's cotangent to the one cell a selection names (CnnFold.cnnChainCotW2Sel, cifarChainCotW2Sel), as the rendered select_and_scatter does. At the first argmax of each window, the cell that op picks, the step tie's own chains are these (CnnFold.cnnChainCotW2_eq_sel, cifarChainCotW2_eq_sel). A parameter of the first stage sees both pools move; its germ rewrites the outer pool first, at the true pre-activation, then the inner one.

Hypotheses. Odd kernels, every ReLU off its kink, every pool window dead or tied only between twins, each selection naming a maximum of every window (CifarLossSmoothAt). Scope. One example (the emitted module batch-contracts; den is per-example).