This submission formalizes deterministic algorithms for the multicommodity flow–cut theorems of Leighton and Rao. The general statement allows rational product demands, zero-weight transit vertices, and directed or undirected capacities. One deterministic algorithm runs in time polynomial in the binary encoding size of the rational input and returns a positive-demand cut and an exact rational concurrent flow satisfying $\operatorname{ratio}(S)\le C\log_2(p)f$, where $p$ is the number of positive-weight vertices. Directed, undirected, and uniform variants are corollaries of this common guarantee. Arc-to-path conversion over real flows gives the classical bound $\operatorname{sparsestCut}\le C\log_2(p)\operatorname{maxFlow}$.

The submission also includes the deterministic uniform short-path flow construction of Theorem 18, with its simultaneous flow-value and path-length guarantees. Algorithms use the exact LP solver of lax-109476, the registered word RAM of lax-808846 and the polynomial-time predicate of lax-759944, measured in binary rational input size and requiring exact encoded outputs.