Finite Statistical Bridge

A finite statistical microstate model is a pair $(\mathrm{\Omega },ϵ)$ where $\mathrm{\Omega }$ is a finite nonempty set and $ϵ:\mathrm{\Omega }\to \mathbb{R}$ assigns a finite numerical energy to each microstate. The physical energy of $x\in \mathrm{\Omega }$ is $E(x)=ϵ(x)\mathrm{J}$, with its zero and any retained internal offsets fixed by the declared material model. For a physical internal-energy interpretation, the declared companion mechanical model supplies, for every admitted microstate preparation $x$, a positive total mass and a center-of-mass position map with a defined componentwise velocity at that preparation, obeying the Galilean transformation conditions of Galilean inertial charts and the center-of-mass rest frame. That velocity selects the statewise instantaneous rest chart in which $E(x)$ is recorded; a finite set and energy map alone do not construct these mechanical data or a common rest chart for a time interval. Bulk translational kinetic energy and separately tracked external-field energy are excluded from these internal-energy values. The energy zero and any internal offsets are fixed consistently across admitted states. If the statistical model also assigns a single equilibrium macrostate to a preparation, $U$ is its internal-energy coordinate of Equilibrium state domains; the pair $(\mathrm{\Omega },ϵ)$ alone does not assert that macroscopic identification. Equal energy values at distinct elements of $\mathrm{\Omega }$ count as distinct microstates.

When two such models are declared independent, their joint microstate set is the Cartesian product ${\mathrm{\Omega }}_A\times {\mathrm{\Omega }}_B$, and additivity of the microscopic energy is the explicit condition ${ϵ}_{AB}(x,y)={ϵ}_A(x)+{ϵ}_B(y)$. Independence and this energy equation are model data; they do not follow merely from the two systems being separated. The definition itself assigns no weights, temperature, probability law, or dynamics.

Assume the Axiom of Choice (The Axiom of Choice). Fix an integer $n\ge 1$ and an open phase-coordinate domain $O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\subseteq {\mathbb{R}}^{2n}$ with physical canonical coordinates $q_i$ in metres and $p_i$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$. The canonical symplectic form on this chart is ${\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\sum }_{i=1}^{n}d{q}_i\wedge d{p}_i$. Let ${\widehat{q}}_i=q_i\mathrm{/}(1\mathrm{m})$ and ${\widehat{p}}_i=p_i\mathrm{/}(1\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$, and let $O\subseteq {\mathbb{R}}^{2n}$ be the open image of the chart in these dimensionless coordinates. The corresponding dimensionless canonical form is $\widehat{\omega }={\sum }_{i=1}^{n}d{\widehat{q}}_i\wedge d{\widehat{p}}_i$, so ${\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(1\mathrm{J}\mathrm{s})\widehat{\omega }$.

Let $\widehat{H}:O\to \mathbb{R}$ be a numerical energy function with continuous partial derivatives through order two, and let $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to \mathbb{R}\mathrm{J}$ be $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(q,p)=(1\mathrm{J})\widehat{H}(\widehat{q},\widehat{p})$. Where defined, its physical Hamiltonian vector field $X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is determined by ${\iota }_{X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}}{\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=d{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. For each elapsed time $t$ in its domain, the physical flow map is a function ${\varphi }_t:D_t\to O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, where $D_t\subseteq O_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is the set of initial data whose trajectories exist through time $t$, and it satisfies $d{\varphi }_t\mathrm{/}dt=X_{H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\circ {\varphi }_t$. Equivalently, in the dimensionless chart, the flow map ${\widehat{\varphi }}_{\widehat{t}}:{\widehat{D}}_{\widehat{t}}\to O$ is generated by $X_{\widehat{H}}$ satisfying ${\iota }_{X_{\widehat{H}}}\widehat{\omega }=d\widehat{H}$, at dimensionless time $\widehat{t}=t\mathrm{/}(1\mathrm{s})$; here ${\widehat{D}}_{\widehat{t}}$ is the set of initial coordinates whose trajectories exist through $\widehat{t}$. These definitions specify the flow maps when they exist but impose no global-existence or completeness hypothesis on the shell preparation.

A specified open interval $J_E\subseteq \mathbb{R}$ is a regular compact-shell band when ${\widehat{H}}^{-1}(K)$ is compact in $O$ for every compact $K\subset J_E$ and $\mathrm{\nabla }\widehat{H}$ is nonzero at every point of ${\widehat{H}}^{-1}(J_E)$.

For $\widehat{E}\in J_E$, the energy shell is ${\mathrm{\Sigma }}_{\widehat{E}}:=\{x\in O:\widehat{H}(x)=\widehat{E}\}$. The graph-chart definition of the weighted shell integral is Regular energy shell and weighted area measure: in a local graph chart $x_i=h(z,\widehat{E})$, its coarea measure is $d{\mu }_{\widehat{E}}=d{A}_{\widehat{E}}\mathrm{/}\mathrm{\mid }\mathrm{\nabla }\widehat{H}\mathrm{\mid }$, with graph-area factor and gradient in the dimensionless Euclidean coordinates. Set $D(\widehat{E}):={\int }_{{\mathrm{\Sigma }}_{\widehat{E}}}d{\mu }_{\widehat{E}}$ and admit the preparation only when ${\mathrm{\Sigma }}_{\widehat{E}}\mathrm{\ne }\mathrm{\varnothing }$ and $0<D(\widehat{E})<\mathrm{\infty }$. Its normalized shell law is ${\nu }_{\widehat{E}}(A):=D(\widehat{E}{)}^{-1}{\mu }_{\widehat{E}}(A\cap {\mathrm{\Sigma }}_{\widehat{E}})$. The current draft theorem Coarea and invariant measure on compact regular energy shells proves the coarea identity, chart independence and positive finite shell mass under its stated $C^2$, compact-regular-band hypotheses; its statement and proof are present, while its acceptance remains unresolved. Multiplying the unnormalized shell measure by a fixed positive phase-volume normalization does not change ${\nu }_{\widehat{E}}$.

Assign a positive temperature $T$ in kelvins and the exact framework constant $k_{\mathrm{B}}$ in $\mathrm{J}{\mathrm{K}}^{-1}$ from SI quantities and unit conventions, and put $\beta =(k_{\mathrm{B}}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$. For a finite microstate model $(\mathrm{\Omega },ϵ)$, adopt the canonical preparation $P_{\beta }(x)=\frac{e^{-\beta E(x)}}{Z(\beta )},Z(\beta )={\sum }_{x\in \mathrm{\Omega }}{e}^{-\beta E(x)}\mathrm{.}$ Here $Z$ is dimensionless, positive, and finite because $\mathrm{\Omega }$ is finite and nonempty.

For a specified nonempty finite or countably infinite state list $I$ with a stated enumeration and finite physical energies $E_i$ in joules, the counted-once branch takes reference weights $g_i=1$. More generally, the model may explicitly assign fixed positive dimensionless reference weights $g_i>0$ and adopts $Z(\beta )={\sum }_{i\in I}{g}_i{e}^{-\beta E_i},P_{\beta }(i)=\frac{g_i{e}^{-\beta E_i}}{Z(\beta )},$ only at parameters with $0<Z(\beta )<\mathrm{\infty }$. Thus the counted-once law is exactly $P_{\beta }(i)=e^{-\beta E_i}\mathrm{/}Z$. To match Countable canonical model, set $E_{\ast }=1\mathrm{J}$, $e_i=E_i\mathrm{/}{E}_{\ast }\in \mathbb{R}$ and $b=E_{\ast }\beta$; its real-valued energy and parameter coordinates are $e_i,b$, and its admissible set is $J_Z$ for the declared parameter interval. Its nonnegative sums are suprema of finite subsums. The weights $g_i$ are declared model data, not inferred degeneracies or measurements.

For a measurable model $(X,\mathrm{\Sigma },\mu ,E)$ with a dimensionless reference measure $\mu$ and measurable finite physical energy $E:X\to \mathbb{R}\mathrm{J}$, adopt the probability density $p_{\beta }=e^{-\beta E}\mathrm{/}Z$ relative to $\mu$, with $Z(\beta )={\int }_X{e}^{-\beta E}d\mu$, only when $0<Z(\beta )<\mathrm{\infty }$. Measurability of $E$ means measurability of its numerical coordinate $E\mathrm{/}{E}_{\ast }$. The integral convention is Measure space and integral at its current draft definition.

This is an adopted statistical-mechanical preparation rule. It is not derived from the macroscopic thermodynamic laws, a finite bath, entropy maximization, or a time-evolution equation. It does not assert that a physical sample has this preparation; that identification is an additional model assumption to be tested against a specified preparation and apparatus.

For the regular-shell branch below, assume AC as required by Regular Hamiltonian shell preparation data and its coarea-measure supplier. No choice assumption is needed for the separate finite-window branch.

For a finite microstate model and a declared energy value $E_0$ in joules and resolution $\delta \ge 0$ in joules, define the admitted window ${\mathrm{\Omega }}_{E_0,\delta }:=\{x\in \mathrm{\Omega }:\mathrm{\mid }E(x)-E_0\mathrm{\mid }\le \delta \}\mathrm{.}$ When this set is nonempty, adopt the uniform preparation $P(x)=\mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }{\mathrm{\mid }}^{-1}$ on it and zero outside. Its dimensionless Boltzmann entropy is $\log \mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }\mathrm{\mid }$, and its physical entropy convention is $k_{\mathrm{B}}\log \mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }\mathrm{\mid }$ in $\mathrm{J}{\mathrm{K}}^{-1}$. A zero-width window is allowed only when it contains at least one microstate.

For a continuous classical model, when the regular-band, nonempty-shell, and normalization hypotheses in Regular Hamiltonian shell preparation data hold, the microcanonical preparation is the normalized shell measure ${\nu }_{\widehat{E}}$ defined there. The shell preparation does not require a globally defined flow. These finite-window and regular-shell rules are adopted model preparations. They do not assert Hamiltonian ergodicity, equality with long-time averages, canonical sampling, or equilibration.

Under the canonical sampling assumption Canonical sampling assumption for finite and normalized statistical models, let $I$ be a specified nonempty countable microstate list with finite energies $E_i$ in joules, counted once ($g_i=1$), and let $J_{\beta }\subset (0,\mathrm{\infty })$ be an open inverse-temperature interval in ${\mathrm{J}}^{-1}$. Set $E_{\ast }=1\mathrm{J}$, $e_i=E_i\mathrm{/}{E}_{\ast }\in \mathbb{R}$, $b=E_{\ast }\beta$, and $J_b=E_{\ast }{J}_{\beta }\subset (0,\mathrm{\infty })$; $e_i$ and $b$ are dimensionless numerical coordinates. For $\beta \in J_{\beta }$, define $Z(\beta )={\sum }_i{e}^{-\beta E_i}$ and, when $0<Z(\beta )<\mathrm{\infty }$, $p_i(\beta )=e^{-\beta E_i}\mathrm{/}Z(\beta )$. Suppose every compact interval $K\subset J_{\beta }$ admits a summable sequence $a_i\ge 0$ with $(1+\mathrm{\mid }{e}_i{\mathrm{\mid }}^2)e^{-\beta E_i}=(1+\mathrm{\mid }{e}_i{\mathrm{\mid }}^2)e^{-b{e}_i}\le a_i$ for all $i$ and $\beta \in K$. Define ${\bar{E}}_{\beta }={\sum }_i{p}_i{E}_i$ and ${\mathrm{Var}}_{\beta }(E)={\sum }_i{p}_i(E_i-{\bar{E}}_{\beta }{)}^2$. Then these moments are finite, $Z$ is $C^2$, $-{\mathrm{\partial }}_{\beta }\log Z={\bar{E}}_{\beta }$, ${\mathrm{\partial }}_{\beta }^2\log Z={\mathrm{Var}}_{\beta }(E)$, and the dimensionless Gibbs entropy $s_{\beta }:=-{\sum }_i{p}_i\log p_i=\log Z+\beta {\bar{E}}_{\beta }$ is finite. Define the physical Gibbs entropy by $S_{\beta }:=k_{\mathrm{B}}{s}_{\beta }$ in $\mathrm{J}{\mathrm{K}}^{-1}$. Then $F_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}:=-k_{\mathrm{B}}T\log Z={\bar{E}}_{\beta }-T{S}_{\beta }$ for $\beta =(k_{\mathrm{B}}T{)}^{-1}$. A lower bound on the energies alone does not supply the summable envelope or finite partition function.

For the lattice model and canonical law of Bulk log-partition pressure for a bounded finite-range lattice model, let $p_L(\gamma )=L^{-d}\log Z_L(\gamma )$ converge pointwise on an open interval $J_{\beta }\subset (0,\mathrm{\infty })$ to $p(\gamma )$. If $p$ is differentiable at $\beta \in J_{\beta }$, set $e=-p^{\mathrm{\prime }}(\beta )$ in joules per site. Then the canonical mean energy density satisfies ${\mathbb{E}}_{\beta ,L}[H_L]\mathrm{/}{L}^d\to e$, and for every $\epsilon >0$ there exist $c>0$ and $L_0$ such that ${\mathbb{P}}_{\beta ,L}\text{\hspace{0.17em}⁣}(\mid H_L\mathrm{/}{L}^d-e\mid \ge \epsilon )\le 2e^{-c{L}^d}(L\ge L_0)\mathrm{.}$ The probability is the model's finite-volume canonical probability, not an observed frequency. No conclusion is asserted when $p$ is nondifferentiable.

Assume AC for the regular-shell reading of Microcanonical sampling assumption. Let the subsystem have a finite nonempty state set $\mathrm{\Omega }$ and internal energies $E_x$ in joules with $\mathrm{\mid }{E}_x\mathrm{\mid }\le L$, where $L$ is in joules. The bath model supplies an open interval $U$ containing $[u_0-L,u_0+L]$, with bath energies $u,u_0\in U$ in joules, and a dimensionless shell-log-weight function $s_B:U\to \mathbb{R}$; define the positive shell weight by $W_B(u):=e^{s_B(u)}$. Assume $s_B\in C^2(U)$ and $\mathrm{\mid }{s}_B^{\mathrm{\prime }\mathrm{\prime }}(u)\mathrm{\mid }\le M$ on the displayed interval, where $M$ is in ${\mathrm{J}}^{-2}$. The model adopts $\beta =s_B^{\mathrm{\prime }}(u_0)>0$ in ${\mathrm{J}}^{-1}$, $k_{\mathrm{B}}$ in $\mathrm{J}{\mathrm{K}}^{-1}$, and $T=(k_{\mathrm{B}}\beta {)}^{-1}$ in kelvins. Assume weak additive coupling and separately stipulate that microcanonical shell sampling induces subsystem weights $P_B(x)\propto W_B(u_0-E_x)=e^{s_B(u_0-E_x)}$; this induced-marginal rule is an additional bath-model assumption, not a consequence of the microcanonical sampling postulate alone. Let $P_C(x)=e^{-\beta E_x}\mathrm{/}Z$ be the corresponding canonical law. With $\delta =M{L}^2\mathrm{/}2$ (dimensionless), the shell log-weight remainder obeys $\mathrm{\mid }{s}_B(u_0-E_x)-s_B(u_0)+\beta E_x\mathrm{\mid }\le \delta ,$ and the normalized laws satisfy ${\sum }_{x\in \mathrm{\Omega }}\mathrm{\mid }{P}_B(x)-P_C(x)\mathrm{\mid }\le e^{2\delta }-1.$ This is an ${\mathrm{\ell }}^1$ probability-vector bound; it does not assert a universal error or a thermodynamic limit.

Let $(\mathrm{\Omega },ϵ)$ be a finite microstate model with energies $E_x=ϵ(x)\mathrm{J}$ from Finite statistical microstate model. Let $T>0$, $\beta =(k_{\mathrm{B}}T{)}^{-1}$, and $P_{\beta }(x)=e^{-\beta E_x}\mathrm{/}Z(\beta )$ be the adopted preparation of Canonical sampling assumption for finite and normalized statistical models. With ${\bar{E}}_{\beta }={\sum }_x{P}_{\beta }(x)E_x$, ${\mathrm{Var}}_{\beta }(E)={\sum }_x{P}_{\beta }(x)(E_x-{\bar{E}}_{\beta }{)}^2$, and $S_{\beta }=-k_{\mathrm{B}}{\sum }_x{P}_{\beta }(x)\log P_{\beta }(x)$, the finite partition sum is positive and $-{\mathrm{\partial }}_{\beta }\log Z={\bar{E}}_{\beta },{\mathrm{\partial }}_{\beta }^2\log Z={\mathrm{Var}}_{\beta }(E),S_{\beta }=k_{\mathrm{B}}(\log Z+\beta {\bar{E}}_{\beta })\mathrm{.}$ Consequently the statistical free-energy value $F_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}:=-k_{\mathrm{B}}T\log Z$ satisfies $F_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}={\bar{E}}_{\beta }-T{S}_{\beta }$. The derivatives hold the finite state set and its energy list fixed. No equivalence with a macroscopic entropy or Helmholtz potential is asserted.

Let $(\mathrm{\Omega },ϵ)$ be a finite nonempty microstate model, let $T>0$, and let $q_x=e^{-\beta E_x}\mathrm{/}Z$ be its canonical probabilities at $\beta =(k_{\mathrm{B}}T{)}^{-1}$. Set $\bar{E}={\sum }_x{q}_x{E}_x$. Among all probability vectors $r=(r_x{)}_{x\in \mathrm{\Omega }}$ with ${\sum }_x{r}_x=1$ and ${\sum }_x{r}_x{E}_x=\bar{E}$, the dimensionless Shannon entropy $h(r):=-{\sum }_{x:r_x>0}{r}_x\log r_x$ is uniquely maximized by $r=q$. The corresponding physical Gibbs entropy is $S(r):=k_{\mathrm{B}}h(r)$ in $\mathrm{J}{\mathrm{K}}^{-1}$ and has the same unique maximizer. This is a finite-model variational theorem; it does not establish that a physical sample is prepared canonically.

Assume the Axiom of Choice and Countable Choice and the canonical Gibbs-preparation assumption The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature. Restrict to its $G=H$ branch: the declared energy operator $H$ is self-adjoint, bounded below, and measured in joules; $T>0$ is in kelvins, $\beta =(k_{\mathrm{B}}T{)}^{-1}$ is in ${\mathrm{J}}^{-1}$, and $K_{\beta }=e^{-\beta H}$ has $0<Z_{\beta }=\mathrm{Tr}{K}_{\beta }<\mathrm{\infty }$. Suppose also that on an open interval $J_{\beta }\subset (0,\mathrm{\infty })$ containing $\beta$, every compact $C\subset J_{\beta }$ admits a summable family $a_j\ge 0$ with $(1+\mathrm{\mid }{E}_j{\mathrm{\mid }}^2)e^{-\gamma E_j}\le a_j$ for all energy eigenvalues $E_j$ and $\gamma \in C$. Let $(u_j)$ be the finite or countable complete orthonormal eigenfamily supplied by Trace-class Gibbs normalization for a semibounded self-adjoint operator, repeated according to multiplicity, with $H{u}_j=E_j{u}_j$, and put ${\lambda }_j=e^{-\beta E_j}\mathrm{/}{Z}_{\beta }$. The supplied trace identity gives ${\sum }_j{\lambda }_j=1$, and these ${\lambda }_j$ are the eigenvalues of ${\rho }_{\beta }$. Define the spectral mean ${\bar{E}}_{\beta }:={\sum }_j{\lambda }_j{E}_j$, the variance ${\mathrm{Var}}_{\beta }(H):={\sum }_j{\lambda }_j(E_j-{\bar{E}}_{\beta }{)}^2$, the dimensionless entropy $s_{\beta }:=-{\sum }_j{\lambda }_j\log {\lambda }_j$, and the physical von Neumann entropy $S_{\mathrm{v}\mathrm{N}}({\rho }_{\beta }):=k_{\mathrm{B}}{s}_{\beta }$. These are spectral-sum definitions for this commuting Gibbs density; no operator trace of an unbounded product is used. The supplied moment envelope makes these sums finite. Then $-{\mathrm{\partial }}_{\beta }\log Z_{\beta }={\bar{E}}_{\beta },{\mathrm{\partial }}_{\beta }^2\log Z_{\beta }={\mathrm{Var}}_{\beta }(H),S_{\mathrm{v}\mathrm{N}}({\rho }_{\beta })=k_{\mathrm{B}}(\log Z_{\beta }+\beta {\bar{E}}_{\beta })\mathrm{.}$ Thus $F_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}=-k_{\mathrm{B}}T\log Z_{\beta }={\bar{E}}_{\beta }-T{S}_{\mathrm{v}\mathrm{N}}({\rho }_{\beta })$. This statement does not extend the Helmholtz interpretation to an effective generator $H-\mu \mathcal{N}$.

Let two finite microstate models from Finite statistical microstate model have nonempty state sets ${\mathrm{\Omega }}_A,{\mathrm{\Omega }}_B$, energies $E_A,E_B$ in joules, and the same $T>0$. Assume their declared composite model has state set ${\mathrm{\Omega }}_A\times {\mathrm{\Omega }}_B$ and energy $E_{AB}(x,y)=E_A(x)+E_B(y)$, with no interaction term. Under the canonical sampling assumption Canonical sampling assumption for finite and normalized statistical models, the partition function factorizes as $Z_{AB}=Z_A{Z}_B$, the joint canonical law is $P_{AB}=P_A{P}_B$, the Gibbs entropy satisfies $S_{AB}=S_A+S_B$, and the statistical free-energy value satisfies $F_{AB}=F_A+F_B$.

Let a physical Hamiltonian model satisfy Regular Hamiltonian shell preparation data on a nonempty compact regular energy band, and adopt its normalized shell law by Microcanonical sampling assumption. In addition to the definition's $C^2$ regularity, assume its numerical Hamiltonian $\widehat{H}$ is $C^{\mathrm{\infty }}$. Let ${\varphi }_t$ be the physical Hamiltonian flow defined there, with $t$ measured in seconds, and assume it is an invertible smooth map of the whole band for every real time. Then, conditional on the current batch-12 draft mathematical interfaces Hamiltonian volume, recurrence, and ergodic scope and Coarea and invariant measure on compact regular energy shells, every shell probability ${\nu }_{\widehat{E}}$ is invariant: ${\nu }_{\widehat{E}}({\varphi }_t^{-1}(A))={\nu }_{\widehat{E}}(A)$ for every Borel set $A\subseteq {\mathrm{\Sigma }}_{\widehat{E}}$ and every $t\in \mathbb{R}$. This result asserts no ergodicity, time-average equality, or thermalization.

Use the finite-alphabet, bounded, translation-invariant finite-range lattice model of Bounded finite-range lattice model, with physical interaction energies in joules, inverse temperature $\beta =(k_{\mathrm{B}}T{)}^{-1}>0$, and partition sum $Z_L(\beta )$. Suppose each site represents a fixed cell of volume $v_0>0$ in ${\mathrm{m}}^3$. Define the reduced log-partition density $p_L=L^{-d}\log Z_L$ and the lattice pressure convention $P_L=(k_{\mathrm{B}}T\mathrm{/}{v}_0)p_L$ in pascals. Then $p_L$ and $P_L$ have finite limits as $L\to \mathrm{\infty }$. If a second boundary-energy model on the same configuration sets differs pointwise by at most $C_b{L}^{d-1}$ joules for fixed $C_b$, it has the same limit.