We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs onnvertices and branchwidth at mostk. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k⋅logk)⋅nsteps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition calledsurface cut decomposition, generalizing sphere cut decompositions of planar graphs, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of noncrossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidthk) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2O(k)⋅nsteps. That way, we considerably extend the class of problems that can be solved in running times with asingle-exponential dependenceon branchwidth and unify/improve most previous results in this direction.