Riemannian Budget Manifold

Projection and vector transport are computationally cheap in the proposed manifold formulation. The formulation avoids heavier generic manifold operations by reducing retraction to scalar root-finding. Softmax logits induce a smooth expected-cost constraint surface when option costs are distinct. Retraction to the exact budget can be reduced to binary search because shifting logits along the cost vector changes expected cost monotonically. The monotonic retraction proof depends on cost being linear in assignment probabilities.