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Precise statement
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison
$$\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).$$
The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then
$$\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).$$
The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT 5.6 Sol Ultra
The authors had already developed a strategy proving a weaker girth-dependent bound of the form $\exp(n/f(g))$, with $f(g)\to\infty$. GPT-5.6 Sol Pro developed this strategy into a proof with $f(g)=\Theta(g/\log g)$. Subsequent interactions with GPT-5.6 Sol Ultra, focused on understanding the source of the logarithmic loss and its relation to the Anari--Rezaei argument, led to the development of the final optimal $2^{2n/g}$ proof. Codex also assisted with manuscript preparation.
Unreviewed. A 21-page preprint two days old, no peer review, no formal verification. The optimality half is checkable by anyone: the paper exhibits the equality cases explicitly. The upper-bound argument is conventional and the authors take responsibility for it. Nobody independent has read it on the record.
For every nonnegative $n\times n$ matrix $A$ whose bipartite support graph has girth at least an even integer $g\ge4$, Dong and Jain prove the sharp inequality
$$\operatorname{Bethe}(A)\le\operatorname{per}(A)\le2^{2n/g}\operatorname{Bethe}(A).$$
The factor $2^{2n/g}$ is optimal whenever $g\mid2n$, attained by matrices whose support graphs are disjoint unions of $g$-cycles. Thus the result confirms Anari's conjecture, recovers the sharp $2^{n/2}$ universal Anari--Rezaei bound when $g=4$, and approaches exactness as the support-graph girth tends to infinity.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.