Read-back: ising_block_dynamics

Hypotheses. Let $V$ be a finite, nonempty type with decidable equality; $G$ a simple graph on $V$ with decidable adjacency; $\beta$ a real with $\beta > 0$; $b$ a natural number with $b > 0$; and $\mathcal{B} : \{0,\dots,b-1\} \to \mathrm{Finset}(V)$ a family of $b$ blocks of vertices such that

• (cover) every vertex $v \in V$ lies in at least one block $\mathcal{B}_i$;
• ($M$) $M \in \mathbb{N}$ bounds every block's size: $|\mathcal{B}_i| \le M$ for all $i$;
• ($M_s$) $M_s \in \mathbb{N}$ bounds the overlap multiplicity: each vertex $v$ belongs to at most $M_s$ of the blocks $\mathcal{B}_i$ (counting indices $i$, so a block listed twice counts twice).

Blocks may overlap and may be empty; $M$ and $M_s$ are arbitrary naturals subject only to these bounds (either could be $0$ only if all blocks are empty resp. no vertex is covered, which the cover hypothesis forbids for $M_s$).

Objects, unfolded. $\mu$ is the Ising measure on configurations $\sigma : V \to \{\pm1\}$, proportional to $\exp\big(\beta\sum_{\{v,w\}\in E(G)} s_\sigma(v)s_\sigma(w)\big)$ (the code's half-weighted ordered double sum). Two Markov-type matrices on configurations are compared:

• the block dynamics $Q = \mathrm{blockDynamics}(\mu, \mathcal{B})$, with $Q_{\sigma\tau} = \tfrac1b \sum_{i} \mathrm{blockUpdate}(\mu, \mathcal{B}_i)_{\sigma\tau}$, where $\mathrm{blockUpdate}(\mu, W)_{\sigma\tau}$ equals $\mu(\tau)\big/\sum_{\eta = \sigma \text{ off } W}\mu(\eta)$ when $\tau$ agrees with $\sigma$ outside $W$ and $0$ otherwise — "pick a uniform block, resample it from $\mu$ given the outside";
• the Glauber chain $P = \mathrm{glauber}(\mu)$, the single-site version: pick a uniform vertex, resample its spin from the conditional of $\mu$.

For any such matrix $R$, the spectral gap as defined here is $\gamma(R) = 1 - \sup\{\lambda \in \mathbb{R} : \exists f \ne 0,\ R f = \lambda f,\ \lambda \ne 1\}$, the real sSup over all real eigenvalues other than exactly $1$; if that set is empty or unbounded above the sSup is $0$ (junk), giving $\gamma = 1$.

The claim.

where $\Delta_G$ is the maximum degree of $G$ (a natural cast to $\mathbb{R}$), and $M, M_s$ are likewise cast to reals; the exponent $M+1$ is a natural-number power.

Direction and shape, exactly as written: the block-dynamics gap is bounded above by the constant $M^2 M_s (4e^{2\beta\Delta_G})^{M+1}$ times the single-site Glauber gap — not the reverse. The constant depends on the block structure only through the size bound $M$ and overlap bound $M_s$, and on the graph only through $\Delta_G$ and $\beta$. If $M = 0$ or $M_s = 0$ the right-hand side is $0$, so the claim then asserts $\gamma(Q) \le 0$. No reversibility, stochasticity, or positivity of either gap is assumed or asserted beyond the definitions above.