Network coding allows intermediate nodes to combine messages. Fractional routing instead splits each message among source-destination paths and forwards its parts without combining different messages. For undirected networks carrying independent messages between multiple source-destination pairs, the multiple-unicast conjecture asserts that the two methods achieve the same combinations of message rates. We give a finite counterexample and use its linear structure to accelerate external-memory integer sorting, addressing the question of whether integer operations can beat the comparison-sorting input/output (I/O) bound posed by Aggarwal and Vitter in 1988. We give a deterministic sorting algorithm: for $N$ integers of $Θ(\log N)$ bits, a memory holding $M$ integers, and block transfers of $B$ integers, where $2B\le M\le N$, the number of block transfers is $O\!\left(\frac{N}{B}\left[1+\frac{\log N}{(\log(M/B))^{1-\varepsilon}(\log M)^\varepsilon}\right]\right)$, where $\varepsilon=1.5\times10^{-5}$ is an absolute constant. In some parameter ranges, this bound is asymptotically smaller than the classical comparison-sorting bound. Under the external-memory model specified in this paper, we have formally verified the algorithm's correctness and I/O complexity bound in Lean.

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