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 training phase, no stored computation graph, and no mode switch between learning and deploy ment. The network improves on every input it processes. We demonstrate that this procedure learns AND, OR, NOR, and XOR to 100% accuracy, and conjecture that any finite directed network problem with a defined source and sink is in principle solvable by this mechanism.