Diagonal quantum Ising interaction and classical conditional energies

Since $\mathrm{\Delta }$ is finite and every site couples only to sites within the fixed finite range $R$, the boundary $C_{\mathrm{\Delta }}$ and the set of pairs meeting $\mathrm{\Delta }$ are finite. Every $E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )$ is therefore real and finite. The configuration set $\{-1,+1{\}}^{\mathrm{\Delta }}$ is nonempty and finite, so $Z_{\mathrm{\Delta }}^{\eta }$ is a finite sum of strictly positive real numbers. When $\mathrm{\Delta }=\mathrm{\varnothing }$, that set has its single empty configuration, $E_{\mathrm{\Delta }}=0$, $Z_{\mathrm{\Delta }}^{\eta }=1$, and ${\gamma }_{\mathrm{\Delta }}$ is the identity kernel. In every case, ${\gamma }_{\mathrm{\Delta }}(\cdot \mid \eta )$ is a normalized probability distribution.

Write $X_{{\mathrm{\Delta }}^c}=\{-1,+1{\}}^{{\mathrm{\Delta }}^c}$ with its cylinder-generated Borel sigma-algebra, and let $r_{\mathrm{\Delta }}:X\to X_{{\mathrm{\Delta }}^c}$ be coordinate restriction. For a global input $x\in X$, the full-configuration kernel notation means ${\gamma }_{\mathrm{\Delta }}(f\mid x):={\gamma }_{\mathrm{\Delta }}(f\mid r_{\mathrm{\Delta }}x)$. For a bounded measurable $f$ on $X$, each replacement map $\eta \in X_{{\mathrm{\Delta }}^c}\mapsto {\sigma }_{\mathrm{\Delta }}\eta \in X$ is measurable and each weight in [F1] depends on only the finite boundary $C_{\mathrm{\Delta }}$. Thus the finite weighted sum defining ${\gamma }_{\mathrm{\Delta }}(f\mid \eta )$ is measurable. If $B\subseteq X$ is exterior-measurable, write $B=r_{\mathrm{\Delta }}^{-1}(D)$ for a measurable $D\subseteq X_{{\mathrm{\Delta }}^c}$. Since $r_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\eta )=\eta$, one has ${\mathbf{1}}_B({\sigma }_{\mathrm{\Delta }}\eta )={\mathbf{1}}_D(\eta )$ and therefore ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_B\mid \eta )={\mathbf{1}}_D(\eta )$ by normalization. Equivalently, for the full-configuration input $x$, ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_B\mid x)={\mathbf{1}}_D(r_{\mathrm{\Delta }}x)={\mathbf{1}}_B(x)$, proving properness. Its cylinder probabilities depend only on $C_{\mathrm{\Delta }}$ and the finitely many coordinates named by the cylinder, so it has finite range.

Let $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$ be finite. Split all interaction terms meeting $\mathrm{\Lambda }$ into terms meeting $\mathrm{\Delta }$ and terms disjoint from $\mathrm{\Delta }$. For fixed spins $\xi =s_{\mathrm{\Lambda }\setminus \mathrm{\Delta }}$ and exterior $\eta$, the latter terms contribute a finite $C(\xi ,\eta )$ independent of $s_{\mathrm{\Delta }}$, so $E_{\mathrm{\Lambda }}(\zeta \xi \mid \eta )=E_{\mathrm{\Delta }}(\zeta \mid \xi \eta )+C(\xi ,\eta )$ and $Z_{\mathrm{\Lambda }}^{\eta }={\sum }_{\xi \in \{-1,+1{\}}^{\mathrm{\Lambda }\setminus \mathrm{\Delta }}}{e}^{-\beta C(\xi ,\eta )}{Z}_{\mathrm{\Delta }}^{\xi \eta }$. Thus the ${\gamma }_{\mathrm{\Lambda }}$-marginal weight of each $\xi$ is $e^{-\beta C(\xi ,\eta )}{Z}_{\mathrm{\Delta }}^{\xi \eta }\mathrm{/}{Z}_{\mathrm{\Lambda }}^{\eta }$, and applying ${\gamma }_{\mathrm{\Delta }}$ gives $({\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}f)(\eta )=\frac{1}{Z_{\mathrm{\Lambda }}^{\eta }}{\sum }_{\xi }{e}^{-\beta C(\xi ,\eta )}{\sum }_{\zeta \in \{-1,+1{\}}^{\mathrm{\Delta }}}f(\zeta \xi \eta )e^{-\beta E_{\mathrm{\Delta }}(\zeta \mid \xi \eta )}=\frac{1}{Z_{\mathrm{\Lambda }}^{\eta }}{\sum }_{\xi ,\zeta }f(\zeta \xi \eta )e^{-\beta E_{\mathrm{\Lambda }}(\zeta \xi \mid \eta )}=({\gamma }_{\mathrm{\Lambda }}f)(\eta )\mathrm{.}$ The factor depending on $\xi$ remains in the outer sum, giving exactly the ${\gamma }_{\mathrm{\Lambda }}$ weights. Hence ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$.

Steps 1.1 and 2.2 prove that the general position-dependent kernels satisfy the specification axioms of [F2]. Now assume the translation-invariant subcase in the Statement. Choose a finite set $D$ containing exactly one of $v,-v$ for each nonzero displacement pair with $J(v)\mathrm{\ne }0$. Use the shape $\{0\}$ with potential $-h{s}_0$ and, for each $v\in D$, shape $\{0,v\}$ with potential $-J(v)s_0{s}_v$. Every field site is counted once, and every unordered interacting pair is counted once because $D$ contains one orientation per pair. The CSM finite-volume and conditional energies in [F3] are therefore exactly $H_{\mathrm{\Lambda }}$ and $E_{\mathrm{\Lambda }}$ in [F1], so the kernels coincide. Applying [F4] confirms that this same kernel is a valid CSM finite-range specification under the matching hypotheses.

Only finitely many singleton and pair supports meet the finite set $X$, because the interaction range is finite and each site has finitely many neighbors. Hence $K_X$ is a finite self-adjoint diagonal matrix. Choose any increasing finite exhaustion ${\mathrm{\Lambda }}_n$ that eventually contains every support $S$ meeting $X$. For all sufficiently large $n$, decompose $H_{{\mathrm{\Lambda }}_n}=K_X+L_n$, where $L_n$ is the sum of the remaining terms contained in ${\mathrm{\Lambda }}_n$. Every support in $L_n$ is disjoint from $X$, so $[L_n,A]=0$; every term is diagonal, so $[L_n,K_X]=0$. Therefore $e^{it{H}_{{\mathrm{\Lambda }}_n}\mathrm{/}\mathrm{\hslash }}A{e}^{-it{H}_{{\mathrm{\Lambda }}_n}\mathrm{/}\mathrm{\hslash }}=e^{it{K}_X\mathrm{/}\mathrm{\hslash }}A{e}^{-it{K}_X\mathrm{/}\mathrm{\hslash }}$ for all real $t$.

The right side of step 1.1 is independent of $n$. Passing to the norm limit supplied by [F2] gives the claimed exact formula for ${\alpha }_t(A)$. If $A$ is diagonal, it commutes with the diagonal matrix $K_X$, so the formula reduces to ${\alpha }_t(A)=A$.

Let $P$ be any local rank-one diagonal projection and $A$ any local observable. By [F1], ${\alpha }_t(P)=P$, so the lower boundary of the KMS function for the pair $(A,P)$ is the constant $\omega (AP)$. The constant function with that value is bounded, continuous, holomorphic, and has the same lower boundary. By [F3] it equals the KMS function. Evaluating the upper boundary in [F2] at $t=0$ gives $\omega (PA)=\omega (AP)$.

Fix a finite $\mathrm{\Lambda }$ and a product-basis matrix unit $E_{\sigma ,\tau }$ with $\sigma \mathrm{\ne }\tau$. Take $P=P_{\tau }$, the diagonal rank-one projection onto the basis vector $e_{\tau }$. Matrix-unit multiplication gives $E_{\sigma ,\tau }{P}_{\tau }=E_{\sigma ,\tau }$ and $P_{\tau }{E}_{\sigma ,\tau }=0$. Applying step 1.1 with $A=E_{\sigma ,\tau }$ yields $\omega (E_{\sigma ,\tau })=0$. By [A2], the restricted positive functional has a unique density matrix whose off-diagonal entries are precisely these matrix-unit expectations, so the finite local density matrix is diagonal.

For each finite cylinder assignment $\sigma$ on $\mathrm{\Lambda }$, put ${\mu }_{\omega }(\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }})=\omega (P_{\sigma })$, where $P_{\sigma }$ is the product-basis rank-one projection. Positivity of $\omega$ gives nonnegative weights, ${\sum }_{\sigma }{P}_{\sigma }=I_{\mathrm{\Lambda }}$ gives normalization, and for $\mathrm{\Lambda }\subset {\mathrm{\Lambda }}^{\mathrm{\prime }}$ the identity $P_{\sigma }={\sum }_{\tau {\mathrm{\mid }}_{\mathrm{\Lambda }}=\sigma }{P}_{\tau }$ gives consistency. By [A1] and [F9], these cylinder weights extend uniquely to a Borel probability measure ${\mu }_{\omega }$.

Fix a finite $\mathrm{\Delta }$, configurations $\sigma ,\tau$ on $\mathrm{\Delta }$, an assignment $\eta$ on $C_{\mathrm{\Delta }}$, and a cylinder projection $Q_{\zeta }$ on finitely many sites outside $\mathrm{\Delta }\cup C_{\mathrm{\Delta }}$. Let $Q_{\eta }$ be the product projection for $\eta$ and let $E_{\sigma ,\tau }$ be the local matrix unit sending $e_{\tau }$ to $e_{\sigma }$. Since every interaction term that meets $\mathrm{\Delta }$ has all its exterior sites in $C_{\mathrm{\Delta }}$, changing the $\mathrm{\Delta }$ configuration from $\tau$ to $\sigma$ while keeping $\eta$ fixed changes the finite diagonal Hamiltonian by exactly $\mathrm{\Delta }E:=E_{\mathrm{\Delta }}(\sigma \mid \eta )-E_{\mathrm{\Delta }}(\tau \mid \eta )$. Terms not meeting $\mathrm{\Delta }$ commute with $E_{\sigma ,\tau }{Q}_{\eta }$, so [F3] yields ${\alpha }_t(E_{\sigma ,\tau }{Q}_{\eta })=e^{it\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}{E}_{\sigma ,\tau }{Q}_{\eta }$.

For a complex-valued cylinder function, define its integral as the real integral of its real part plus $i$ times the real integral of its imaginary part. For a DLR measure $\mu$, define ${\omega }_{\mu }(A)=\int ⟨A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}},e_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}⟩d\mu (\sigma )$ for $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$, using the inner product linear in its first argument from [F13]. The value is independent of enlarging $\mathrm{\Lambda }$ because the added factors are identities. It is linear, equals $1$ on the identity, and ${\omega }_{\mu }(A^{\ast }A)=\int \mathrm{\parallel }A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}{\mathrm{\parallel }}^{2}d\mu (\sigma )\ge 0$; Cauchy–Schwarz gives $\mathrm{\mid }{\omega }_{\mu }(A)\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }$. Hence it defines a bounded positive normalized functional on ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. If local $A_n\to A$ in the completion, continuity of product and adjoint from [F12] gives $A_n^{\ast }{A}_n\to A^{\ast }A$, while each ${\omega }_{\mu }(A_n^{\ast }{A}_n)\ge 0$; the continuous extension is therefore positive and normalized and is a state.

Apply [F5] to $A=E_{\tau ,\sigma }{Q}_{\eta }{Q}_{\zeta }$ and $B=E_{\sigma ,\tau }{Q}_{\eta }$. The lower edge is $e^{it\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}\omega (P_{\tau }{Q}_{\eta }{Q}_{\zeta })$. The function $g(z)=e^{iz\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}\omega (P_{\tau }{Q}_{\eta }{Q}_{\zeta })$ is entire and bounded on the closed strip because $\mathrm{\Delta }E$ is real and $0\le \mathrm{Im}z\le \beta \mathrm{\hslash }$; it has the same lower edge. By [F6], $g$ is the KMS function, and its upper edge equals the reversed product boundary from [F5]. At $t=0$ this gives ${\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })=e^{-\beta \mathrm{\Delta }E}{\mu }_{\omega }(P_{\tau }{Q}_{\eta }{Q}_{\zeta })\mathrm{.}$

Let $A,B$ be local and choose finite $\mathrm{\Delta }$ containing their supports and every interaction support that meets the support of $B$. For each boundary assignment $\eta$ on $C_{\mathrm{\Delta }}$, let $H_{\mathrm{\Delta }}^{\eta }$ be the diagonal matrix with eigenvalues $E_{\mathrm{\Delta }}(\sigma \mid \eta )$ and let ${\rho }_{\mathrm{\Delta }}^{\eta }=e^{-\beta H_{\mathrm{\Delta }}^{\eta }}\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$. For bounded real measurable $f$, the DLR fixed-point equation [F14] gives $\mu (f)=\int {\gamma }_{\mathrm{\Delta }}(f\mid \xi )d\mu (\xi )$; extend this identity to complex $f$ by real and imaginary parts. If $D$ is exterior-measurable, properness gives ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_{D}f\mid \xi )={\mathbf{1}}_D(\xi ){\gamma }_{\mathrm{\Delta }}(f\mid \xi )$. For a bounded cylinder function $f$ depending only on spins in $\mathrm{\Delta }\cup C_{\mathrm{\Delta }}$, the kernel depends on the remaining exterior only through $C_{\mathrm{\Delta }}$, so for every finite-boundary assignment $\eta$, ${\int }_{Q_{\eta }}fd\mu =\mu (Q_{\eta }){\sum }_{{\sigma }_{\mathrm{\Delta }}}{\gamma }_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )f({\sigma }_{\mathrm{\Delta }}\eta )\mathrm{.}$ Apply this identity to the diagonal cylinder function $f=\mathrm{diag}(A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }})$. For each fixed $\eta$, this diagonal cylinder function depends only on the spins in $\mathrm{\Delta }$ and the specified boundary assignment on $C_{\mathrm{\Delta }}$: $A,B$ are supported in $\mathrm{\Delta }$, and $H_{\mathrm{\Delta }}^{\eta }$ is a matrix on that finite factor with coefficients determined by $\eta$. Thus it meets the support condition of the preceding finite-mixture identity. The finite sum of diagonal entries is the trace against ${\rho }_{\mathrm{\Delta }}^{\eta }$, so ${\int }_{Q_{\eta }}\mathrm{diag}\text{\hspace{0.17em}⁣}\left(A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)d\mu =\mu (Q_{\eta }){\mathrm{Tr}}_{\mathrm{\Delta }}\text{\hspace{0.17em}⁣}\left({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)\mathrm{.}$ Here the trace identity follows by summing ${\gamma }_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )=e^{-\beta E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )}\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$ against the diagonal matrix coefficients. By [F3], on $Q_{\eta }$ the global local dynamics of $B$ equals conjugation by $H_{\mathrm{\Delta }}^{\eta }$: terms in the global generator not meeting $\mathrm{\Delta }$ commute with $B{Q}_{\eta }$, and the terms in $H_{\mathrm{\Delta }}^{\eta }$ disjoint from the support of $B$ commute with $B$. Summing the displayed identities over the finitely many boundary assignments gives, for real $t$, ${\omega }_{\mu }(A{\alpha }_t(B))={\sum }_{\eta \in \{-1,+1{\}}^{C_{\mathrm{\Delta }}}}\mu (Q_{\eta }){\mathrm{Tr}}_{\mathrm{\Delta }}\text{\hspace{0.17em}⁣}\left({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)\mathrm{.}$

For fixed $\mathrm{\Delta },\eta ,\zeta$, step 2.1 implies $e^{\beta E_{\mathrm{\Delta }}(\sigma \mid \eta )}{\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })$ is constant as $\sigma$ varies. Summing over $\sigma$ and using ${\sum }_{\sigma }{P}_{\sigma }=I_{\mathrm{\Delta }}$ shows that the constant is ${\mu }_{\omega }(Q_{\eta }{Q}_{\zeta })\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$, including when this numerator is zero. Therefore ${\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })={\gamma }_{\mathrm{\Delta }}(\sigma \mid \eta ){\mu }_{\omega }(Q_{\eta }{Q}_{\zeta })\mathrm{.}$

For each $\eta$, define $F_{A,B}^{\eta }(z)={\mathrm{Tr}}_{\mathrm{\Delta }}({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{iz{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-iz{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }})$. By [F7] under [A2]–[A3], this is the finite Gibbs KMS strip function for $H_{\mathrm{\Delta }}^{\eta }$ and the pair $A,B$; it is bounded and continuous on the closed strip, holomorphic inside, and has lower and upper boundaries given by the finite Gibbs traces in their indicated orders. The finite sum ${\sum }_{\eta }\mu (Q_{\eta })F_{A,B}^{\eta }(z)$ has the same regularity and is bounded because the weights are nonnegative and sum to one. Step 2.2 identifies its boundaries with ${\omega }_{\mu }(A{\alpha }_t(B))$ and ${\omega }_{\mu }({\alpha }_t(B)A)$, proving the KMS condition for local pairs.

Every exterior cylinder event is a finite disjoint union of events $Q_{\eta }{Q}_{\zeta }$ after refining it to specify all spins in $C_{\mathrm{\Delta }}$. Summing step 3.1 gives the conditional identity for each interior assignment and every exterior cylinder. Every full cylinder is a finite disjoint union of events $P_{\sigma }{D}_{\sigma }$ with $D_{\sigma }$ an exterior cylinder; properness in [F2] then gives $({\mu }_{\omega }{\gamma }_{\mathrm{\Delta }})(P_{\sigma }{D}_{\sigma })={\mu }_{\omega }(P_{\sigma }{D}_{\sigma })$, so the two probability measures agree on all full cylinders. The class of Borel sets on which they agree contains the whole space, is closed under relative complements of nested members, and under disjoint countable unions, so it is a lambda-system. Full cylinders together with $\mathrm{\varnothing }$ and $X$ form a pi-system generating the Borel sigma-algebra. [F10] states that every lambda-system containing a pi-system contains its generated sigma-algebra, hence the two measures agree on that sigma-algebra. Thus ${\mu }_{\omega }{\gamma }_{\mathrm{\Delta }}={\mu }_{\omega }$ and ${\mu }_{\omega }\in {\mathcal{D}}_{\beta }$.

For arbitrary $A,B\in \mathcal{A}$, choose local $A_n\to A$ and $B_n\to B$ using [F12], and let $F_n$ be their local-pair KMS functions from step 3.2. On each strip boundary, the difference $F_n-F_m$ is bounded by the state norm and the isometric automorphism property [F15] by quantities tending to zero, for example on the lower edge by $\mathrm{\parallel }{A}_n-A_m\mathrm{\parallel }\mathrm{\parallel }{B}_n\mathrm{\parallel }+\mathrm{\parallel }{A}_m\mathrm{\parallel }\mathrm{\parallel }{B}_n-B_m\mathrm{\parallel }$; the analogous upper-edge bound has the factors reversed. The three-lines bound [F6] makes $F_n$ uniformly Cauchy on the entire closed strip. Its uniform limit is bounded and continuous there, has the required two boundary values by norm continuity, and is holomorphic inside by [F11]. Thus ${\omega }_{\mu }\in {\mathcal{K}}_{\beta }$.

For a KMS state $\omega$, [F4] and [A2] make every local density matrix diagonal, so for local $A$ its value is ${\sum }_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}\omega (P_{\sigma })⟨A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}},e_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}⟩={\omega }_{{\mu }_{\omega }}(A)$; density gives equality on $\mathcal{A}$. Conversely, ${\omega }_{\mu }(P_{\sigma })=\mu (P_{\sigma })$ for every cylinder, so uniqueness in [F9] makes $\mu \mapsto {\omega }_{\mu }$ injective. Steps 4.1 and 4.2 prove that the two maps take values in ${\mathcal{D}}_{\beta }$ and ${\mathcal{K}}_{\beta }$, and these identities show they are inverse affine maps.

Under these parameters, the conditional energy in Diagonal quantum Ising interaction and classical conditional energies is the one-dimensional nearest-neighbor Ising energy used by [F1]: each unordered edge is counted once and the same constant field term is included at every site. Thus the DLR sets in [F1] and [F2] are identical.

By [F1], that DLR set has exactly one measure. The bijection [F2] is both injective and surjective, so it sends this singleton to exactly one full-algebra $\beta$-KMS state. This is a model deduction conditional on [A1]–[A2], [F1], and [F2].

The energy formula in [F3] agrees term by term with the pair Hamiltonian in [F1], including the unordered-pair convention, finite range, translation symmetry, constant field, and the zero transverse component. Hence the DLR specifications addressed by [F1] and [F2] coincide.

By [F1], the common DLR set contains exactly one measure. The affine bijection [F2] transfers this singleton to exactly one full-algebra $\beta$-KMS state. In the homogeneous nearest-neighbor case there are $2d$ nonzero neighboring couplings at each site, so the displayed row sum is $2d\tanh (\beta \mathrm{\mid }J\mathrm{\mid })$; the strict bound remains only a sufficient criterion.

Let ${\mu }_{+},{\mu }_{-}$ be the translation-stationary DLR measures from [F1]. By the bijection [F2], each has a unique associated full-algebra $\beta$-KMS state ${\omega }_{{\mu }_{\pm }}$. Since $Z_i{e}_{\sigma }={\sigma }_i{e}_{\sigma }$, the diagonal lift gives ${\omega }_{{\mu }_{\pm }}(Z_i)=\int {\sigma }_{i}d{\mu }_{\pm }={\mu }_{\pm }({\sigma }_i)$, so the magnetization bounds hold and the states are distinct. For local $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$, the basis relabeling in [F8] gives $\mathrm{diag}({\tau }_k(A))(\sigma )=\mathrm{diag}(A)(T_k\sigma )$, where $(T_k\sigma {)}_i={\sigma }_{i+k}$. Stationarity means $(T_k{)}_{\ast }{\mu }_{\pm }={\mu }_{\pm }$, so the lift formula gives ${\omega }_{{\mu }_{\pm }}({\tau }_k(A))={\omega }_{{\mu }_{\pm }}(A)$. Density of local observables and continuity of both the state and ${\tau }_k$ extend the equality to all $A\in \mathcal{A}$.

For a uniform field parameter $h$, the finite matrix $H_{{\mathrm{\Lambda }}_n}(h)$ is diagonal in the joint $Z_i$ eigenbasis. Its eigenvalue on each spin configuration ${\sigma }_{{\mathrm{\Lambda }}_n}$ is exactly the CSM free-boundary energy $H_{{\mathrm{\Lambda }}_n}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}({\sigma }_{{\mathrm{\Lambda }}_n})$: [F9] identifies the translation-invariant pair energy with one CSM shape per unordered displacement pair, and [F3] counts each internal pair and site field once, matching [F5]. The ordinary trace is the sum over this product basis by [F4]. Therefore ${\mathrm{Tr}}_{{\mathrm{\Lambda }}_n}(e^{-\beta H_{{\mathrm{\Lambda }}_n}(h)})={\sum }_{{\sigma }_{{\mathrm{\Lambda }}_n}\in \{-1,+1{\}}^{{\mathrm{\Lambda }}_n}}{e}^{-\beta H_{{\mathrm{\Lambda }}_n}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}({\sigma }_{{\mathrm{\Lambda }}_n})}$ for every $n$ and real $h$. By [F6], the limit defining $p_Q(h)$ exists and equals the CSM pressure $p(h)$.

Applying [F7] to the pressure equality from step 1.2 gives the asserted secant-slope bounds and nondifferentiability at $h=0$. The coexistence threshold remains the sufficient $\beta J\ge 2$ from [F1]; no critical-point or extremality claim is used.