The Winning Edge: A universal Θ(log n) bound for Minimum-Congestion Cycle Bases

We establish tight $\Theta (\log n)$ bounds for minimum-congestion cycle bases, introduce a minimax characterization of fractional congestion and an exact generator-matrix reformulation, and study the resulting integrality gap. We also present Nibble-H, a practical constructor that attains the certified optimum on public quantum-code instances and improves maximum edge participation over the Wang–Irani Version-3 baseline on matched larger graphs.