Structure before circuit tricks
Does the generator only preserve the target, or converge to it with a useful rate?
Connect invariance, reversibility/detailed balance, coercivity and error contraction only through explicit model-specific hypotheses.
This is an authored reusable derivation guide, not a claim that the full family has been source-assimilated.
No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.
This is a conceptual connection to Samplinglib, not a transfer of classical LSI results to arbitrary quantum generators.
A coercivity mechanism, not an equality of samplers — proposal
primary-text-checked Article abstract and Gibbs-sampler construction; Communications in Mathematical Physics 406, 67
Invariant Gibbs state and efficient mixing are different obligations. A low-temperature polynomial mixing claim needs additional model-specific evidence.
% Authored mechanism lesson; not a new theorem certificate.
\section*{Gibbs invariance versus mixing}
Does the generator only preserve the target, or converge to it with a useful rate?
\[
\mathcal L(\rho_\beta)=0\quad\not\Rightarrow\quad t_{\rm mix}=\operatorname{poly}(n,\beta)
\]
Connect invariance, reversibility/detailed balance, coercivity and error contraction only through explicit model-specific hypotheses.
\paragraph{Hypotheses and contracts.}
\begin{enumerate}
\item Specified Hamiltonian, temperature and target state
\item Primitive semigroup and quantitative convergence assumptions when used
\item Implementation and mixing error separated
\end{enumerate}
\paragraph{Mathematical proof mechanism.}
This is a reusable derivation guide; exact certified scope is given by the linked Lean signatures.
\begin{enumerate}
\item Prove the invariant-state equation.
\item Identify the quantitative coercivity/mixing certificate still missing.
\item Compose convergence with implemented-channel error.
\end{enumerate}
\paragraph{Boundary.} This is a conceptual connection to Samplinglib, not a transfer of classical LSI results to arbitrary quantum generators.