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The basis number $\mathrm{bn}(G)$ of a graph $G$ is the minimum edge-congestion of a basis of its cycle space. We prove that every finite $n$-vertex multigraph satisfies $$\mathrm{bn}(G)=O(\log n),$$ resolving, for simple graphs, a question of Bazargani, Biedl, Bose, Maheshwari and Miraftab, subsequently stated as a conjecture by Miraftab, Morin and Yuditsky. The argument also yields the cycle-rank refinement $$\mathrm{bn}(G)=O(\log \beta(G)),$$ where $\beta(G)$ is the dimension of the cycle space, and a reduction of Lehner and Miraftab, based on a theorem of Richter and Shank, then gives $$\mathrm{bn}(G)=O(\log g)$$ for graphs of Euler genus $g$. These orders are best possible.
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The proof was found with the help of GPT-5.6 Sol. The model was also used to explore proof strategies, locate potentially relevant literature, and assist with drafting and revising the manuscript. Knauer independently checked the arguments and references and takes responsibility for the final result.
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Unreviewed. A nine-page preprint one day old at submission, with no peer review and no formal verification. The argument is conventional and self-contained - weighted cycle-basis bounds, minimax duality and dependent randomized rounding - so it is readable by any combinatorialist, and the author states he checked the arguments and references himself. Nobody independent has.
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Knauer proves that every finite $n$-vertex multigraph satisfies $\mathrm{bn}(G)=O(\log n)$, improving the previous general $O(\log^2 n)$ bound and matching the known $\Omega(\log n)$ order. He also proves the sharper cycle-rank bound $\mathrm{bn}(G)=O(\log\beta(G))$. Combined with a reduction of Lehner and Miraftab, this yields $\mathrm{bn}(G)=O(\log g)$ for graphs of Euler genus $g$, improving the previous $O(\log^2 g)$ bound to the optimal logarithmic order.
Source-reported tools: argument.
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.