Network parameters. Weights are stored as dense matrices so that Dense can be used for the linear maps; biases and activations remain as Array vectors.

Build NetParams from plain Array weights and biases.

Run the model with explicit parameters.

Mean-squared-error loss and its gradient w.r.t. the prediction.

L = (1/n) Σ (y - target)², dLdy = (2n)(y - target).

Training state: the parameter bundle. The optimiser is stateless plain SGD, represented as a Progress that is instantiated at step time with the current gradient and learning rate.

Initial training state.

One training step.

Returns the prediction and a function that consumes the learning-rate direction and produces the next state.

Polynomial coalgebra view of the network.

• State: NetState.
• Input position: (input, target); no input direction.
• Output position: prediction; output direction: learning rate.

Train the coalgebra for n steps on a fixed (x, target) pair.

Identity-mapping training oracle: train for n steps and return the loss before each update together with the parameters after each update.

Compute parameter gradients via the 'circuits-ad' DiffP model.

Hand-rolled reference gradients for the 2-layer network, kept to verify the 'circuits-ad' / DiffP gradients in the oracle.

This is the old skip-Para chain rule, written directly against the same Dense primitives used by the rest of the module.

A These-based boundary distinguishes inference-only, gradient-only, and combined inference-with-gradient traffic at a layer boundary.

• This prediction — inference only, no gradient.
• That gradient — training signal only, no prediction.
• These prediction gradient — training with prediction (common supervised case).

Batch scheduling stub. The intended implementation uses Tensor combinators (broadcast telecast indexWindows) to turn single-example layers into batch layers and fold gradients across the batch dimension.