arXiv:2611.00129v1 Announce Type: new Abstract: For Hermitian matrices $\alpha,\beta$ let $\A_{n,m}(e^\alpha,e^\beta)$ be the average of the traces of all words with $n$ letters $e^\alpha$ and $m$ letters $e^\beta$. The refined Bessis--Moussa--Villani conjecture contains the lower bound $\A_{n,m}(e^\alpha,e^\beta)\ge\Tr e^{n\alpha+m\beta}$ (Tr denotes the trace), which for $n=m=1$ is the Golden--Thompson inequality. We prove this lower bound for all $n+m\le7$ when the matrices have size at most three, and for $\min(n,m)\le2$ in every dimension. In all these cases we prove the stronger inequality $\A_{n,m}(A,B)\ge\Tr(A^{n/m}B)^m$ for positive semidefinite $A,B$, or its mirror image with the roles of $A$ and $B$ exchanged; the lower bound of the refined conjecture follows from the Araki--Lieb--Thirring inequality. The cases with three letters $B$ in dimension three, $(n,m)=(3,3)$ and $(4,3)$, are reduced by an exact identity and the inequality of arithmetic and geometric means to a polynomial inequality in two variables, which is certified in exact rational arithmetic by Bernstein expansion on three boxes. The certificate is a short program in the ancillary files.
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