A commutator estimate for the Golden–Thompson gap in qubit chains

For a finite qubit chain with bounded nearest-neighbor interactions, split the interaction sum into its odd and even bond layers $A$ and $B$. We compare the logarithmic trace gap $F(\beta)=\log\{\tau(e^{\beta A}e^{\beta B})/\tau(e^{\beta(A+B)})\}$ with $\mathcal{E}=\tau([A,B]^*[A,B])$, where $\tau$ is the normalized trace. If every bond has operator norm at most $J$, then $\beta^4\mathcal{E}/48\le F(\beta)\le\beta^4\mathcal{E}/16$ for $0\le\beta J\le1/16000$. The constants and interval are independent of chain length and remain valid when local commutator contributions nearly cancel. The proof retains two occurrences of the full commutator in an ordered-cumulant expansion before applying local estimates. A spanning-tree bound and a matrix row-and-column estimate then control the full Taylor remainder relative to $\mathcal{E}$. The result gives a one-step partition-function certificate and a converse on the stated high-temperature interval. An exactly solvable four-site example illustrates the role of cancellation.