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 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 predicted cost-model gap ΔfT at n{=}500 that grows monotonically and approximately linearly in χ, from sim15 bits at χ{=}2 to sim116 bits at χ{=}16 Fig. ref{fig:chi_sweep}, with the refiner winning on 25/25 seeds at every tested χ. Two control families - random 3-regular and QAOA p{=}2 interaction graphs - show median |ΔfT| leq 0.71 bits across both controls at every χ, with refiner win rate falling toward chance as χ 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 $|ΔfT| lesssim 0.1$ bits between scalar and Pareto arms at $χ{=}2$. 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 5/5 instances at every depth. The advantage is therefore largest precisely in the bond-dimension regime relevant to physical contraction.