Bond-dimension scaling of a local-refinement advantage over hyperoptimized tensor-network contraction on Sycamore like topologies

We identify a missing local-refinement stage in the cotengra tensor-network contraction pipeline and show that its impact grows monotonically with bond dimension on the \emph{connectivity graph} of Sycamore-like topologies. Appending a nearest-neighbor interchange (NNI) search to the \cotengra{} output at matched 8-s wallclock yields a median \emph{predicted} cost-model gap $\mathrm{\Delta }\text{fT}$ at $n=500$ that grows monotonically and approximately linearly in $\chi$, from $\sim \text{\hspace{0.17em}⁣}15$~bits at $\chi =2$ to $\sim \text{\hspace{0.17em}⁣}116$~bits at $\chi =16$ (Fig.~\ref{fig:chi_sweep}), with the refiner winning on $25\mathrm{/}25$ seeds at every tested $\chi$. Two control families – random $3$-regular and QAOA $p=2$ interaction graphs – show median $\mathrm{\mid }\mathrm{\Delta }\text{fT}\mathrm{\mid }\le 0.71$~bits across both controls at every $\chi$, with refiner win rate falling toward chance as $\chi$ grows; the signal is topology-specific, not a generic refinement-budget effect. An ablation establishes that refinement itself, not the four-axis Pareto acceptance rule, drives the gain ($\mathrm{\mid }\mathrm{\Delta }\text{fT}\mathrm{\mid }\lesssim 0.1$ bits between scalar and Pareto arms at $\chi =2$). The Sycamore-circuit envelope (App.~\ref{em:sec:results:syccirc}) reports the corresponding refinement on actual random circuits at depths $m\in \{4,6,8,10,12\}$, where the refiner wins on $5\mathrm{/}5$ instances at every depth. The advantage is therefore largest precisely in the bond-dimension regime relevant to physical contraction.