River Learning: Forward-Only Path Carving Without Backpropagation

We present River Learning, a forward-only training algorithm grounded in a single principle: given a source (input) and a sink (output), flow finds the path. No gradient, no backward pass, and no global objective are required — only the knowledge of where information enters and where it must arrive. We introduce the concept of Natural Optimum: the optimal solution is not computed or designed — it is carved by flow alone. As a river does not calculate the shortest path to the sea but simply flows until it breaks through, a Natural Optimum network requires only a source, a sink, and the patience to let flow do its work. Critically, inference and learning occur within the same forward pass: there is no separate…