A Planar Lower Bound 1.1397... > 9/8 for Cost-Preserving Single-Source Unsplittable Flows

For a single-source unsplittable flow, consider rounding a feasible fractional flow $\textbf{x}$ to one path per terminal while requiring both the cost constraint $\textbf{c}^\top\textbf{y}\le\textbf{c}^\top\textbf{x}$ and the overload constraint $y_a\le x_a+Cd_{\max}$ on every arc. Dinitz, Garg and Goemans proved that the congestion bound alone is achievable with $C=1$; Goemans conjectured that cost preservation comes for free, and this was disproved in July 2026 by a seven-vertex instance with critical constant 16/15.

We give a four-terminal planar acyclic family showing that every universal constant C in the overload constraint must satisfy
$$ C\ \ge\ C^\star:=\frac{299-41\sqrt{41}}{32}\ =\ 1.139747070789\ldots\ >\ \frac98 .$$

The construction is a directed spine with two private exits per terminal and exactly two simple source-terminal paths for each terminal. An integer one-parameter subfamily has exact critical constant $\frac{182359}{160000}-\frac 1n$, which is larger than $\frac98$ for $n\ge68$. We also prove that $9/8$ is the exact supremum of the three-terminal template underlying the known counterexample, so a different conflict system was needed; indeed, our new counterexample places four crossing intervals on a spine and drives the conflicts through the unavoidable baseline loads of the expensive routes. The graph is small and the results are easy to verify by hand, but we additionally certify all our claims in exact rational or symbolic arithmetic by programs that rediscover all simple source-terminal paths from the raw arc list; we make the certificate programs available at https://github.com/snikolenko/unsplittable-flows. We conjecture that the exact supremum of forced constants is $2$, and record some structural questions that stand between $C^\star$ and $2$.