Conditional entropy chain and disjoint-block bounds

Assume AC. Let $X_1,\dots ,X_m$ be standard-Borel spaces, let $Q_i$ be probability measures on $X_i$, and let $P$ be a probability on ${\prod }_{i=1}^m{X}_i$. Write $P_i$ for its $i$th marginal and use the directional extended relative entropy $D(\cdot \mathrm{\parallel }\cdot )$ of Relative Entropy of Probability Measures. Then $D(P\mathrm{\parallel }{Q}_1\otimes \cdots \otimes Q_m)\ge {\sum }_{i=1}^{m}D(P_i\mathrm{\parallel }{Q}_i)\mathrm{.}$ If the left side is finite, the conditional chain rule gives the stronger equality $D(P\mathrm{\parallel }{⨂}_i{Q}_i)=D(P_1\mathrm{\parallel }{Q}_1)+{\sum }_{i=2}^m{E}_{P}D(P_{X_i\mid X_1,\dots ,X_{i-1}}\mathrm{\parallel }{Q}_i),$ and each conditional increment is at least $D(P_i\mathrm{\parallel }{Q}_i)$. If the left side is infinite, the displayed inequality is understood in the extended order and is automatic. For an integer $d\ge 1$ and disjoint spatial blocks with fixed particle counts $n_i\in \{0,1,2,\dots \}$ and fixed positions, apply this result to their mark spaces $X_i=({\mathbb{R}}^d{)}^{n_i}$, with Borel sigma-algebras and probability mark references $Q_i$. The zero-fold product $X_i=({\mathbb{R}}^d{)}^0$ is the singleton consisting of the empty mark tuple. On that singleton both $P_i$ and $Q_i$ are its unique probability measure, so a zero-occupancy mark block contributes zero. No zero-divergence claim is made for an unconditioned occupancy law.

Let $d\ge 1$, mass $m>0$ in kilograms, reference momentum $p_{\ast }>0$ in kg m s${}^{-1}$, energy scale $E_0>0$ in joules, and dimensionless inverse temperature $b>0$. On ${\mathbb{R}}^d$ use $r(dp)=dp\mathrm{/}{p}_{\ast }^d$ and $K(p)=\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m{E}_0)$. Then ${\kappa }_b:=\int e^{-bK(p)}r(dp)={\left(\frac{2\pi m{E}_0}{b{p}_{\ast }^2}\right)}^{d\mathrm{/}2},{\nu }_b(dp)={\kappa }_b^{-1}{e}^{-bK(p)}r(dp)$ is a probability with $E_{{\nu }_b}{p}_i=0$, $E_{{\nu }_b}{p}_i{p}_j=(m{E}_0\mathrm{/}b){\delta }_{ij}$, and $E_{{\nu }_b}K=d\mathrm{/}(2b)$. For each $g$ among $K,p_i,p_i{p}_j$, and tilt parameter $t$ in reciprocal units to $g$, the centered logarithmic moment-generating function $c_g(t):=\log E_{{\nu }_b}\exp (t(g-E_{{\nu }_b}g))$ is finite and twice continuously differentiable on a neighborhood of $0$, with $c_g(0)=c_g^{\mathrm{\prime }}(0)=0$. In particular $c_g(t)=O(t^2)$ as $t\to 0$. Here $p_i$ has momentum units and $K$ is energy in units of $E_0$. For the optional physical-temperature parametrization, fix a positive scalar conversion parameter $k_B$ with units J K${}^{-1}$ and take $T>0$ in kelvin; then $b=E_0\mathrm{/}(k_{B}T)$.

Assume AC. Let $P,Q$ be probability measures on a measurable space and let $T$ be a measurable map to another measurable space. Then $D(T_{\ast }P\mathrm{\parallel }{T}_{\ast }Q)\le D(P\mathrm{\parallel }Q),\mathrm{\parallel }P-Q{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}:={\sup }_A\mathrm{\mid }P(A)-Q(A)\mathrm{\mid }\le \sqrt{D(P\mathrm{\parallel }Q)\mathrm{/}2}\mathrm{.}$ The entropy convention is the directional KL definition of Kullback–Leibler divergence, and the total-variation convention is the event-supremum convention of Total variation distance for probability laws. The inequality remains valid when $D=+\mathrm{\infty }$.

Assume AC. Fix $d\ge 1$ and let $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be the numerical even Borel pair potential. Assume the stability, cell-superstability, lower-regularity, and decreasing-integrable-tail conditions of Stability, superstability and regular tails, with its fixed cell side $\mathrm{\ell }$, and the superstable regular interaction, unit-coordinate, and periodic-image conventions of Superstable regular pair interactions and configurational ensemble notation. These are model hypotheses; the definitions do not prove that a particular material satisfies them. Use the finite-region phase-space and statistical-state conventions of Statistical states, densities, observables and expectations.

Let $E_0>0$ joules be the energy unit, ${\mathrm{\ell }}_0>0$ metres the length unit, $m>0$ kilograms the particle mass, and $p_{\ast }>0$ kg m s${}^{-1}$ the momentum reference. A physical position vector ${\mathbf{q}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\in {\mathbb{R}}^d$ in metres has dimensionless numerical coordinate $x={\mathbf{q}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\mathrm{\ell }}_0$. From here on, $q\subseteq {\mathrm{\Lambda }}_L=[0,L{)}^d$ denotes a finite configuration of these numerical coordinates, so the physical cube has side ${\mathrm{\ell }}_{0}L$ and its points are ${\mathrm{\ell }}_{0}x$. Each momentum mark $p_x\in {\mathbb{R}}^d$ is an SI momentum vector in kg m s${}^{-1}$. Write $V_L=L^d$ for numerical volume. Fix also a primitive positive scalar conversion parameter $k_B$ with units J K${}^{-1}$, called the Boltzmann constant: the thermal-energy scale assigned to temperature $T$ is $k_{B}T$ joules. Its value is fixed as part of the model data, not derived by this mathematical definition. If $T>0$ kelvin, put ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=1\mathrm{/}(k_{B}T)$ in J${}^{-1}$ and $b={\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{E}_0=E_0\mathrm{/}(k_{B}T)>0$.

The potential $\varphi$ in the two interaction suppliers is numerical in units of $E_0$ and takes dimensionless displacements as input. For a physical displacement $\mathbf{r}\in {\mathbb{R}}^d$ measured in metres, its pair-potential map is $V_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}(\mathbf{r}):=E_0\varphi (\mathbf{r}\mathrm{/}{\mathrm{\ell }}_0)$ joules. For a finite positional configuration $q\subseteq {\mathrm{\Lambda }}_L$, set $U_{L,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}(q):=H_{\varphi }(q)$ using Superstable regular pair interactions and configurational ensemble notation, and set $U_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(q):=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(q)$ using that item's periodic-image sum. The periodic definition is used only for periods for which that sum is defined; the large-$L$ results below use sufficiently large periods as supplied there. Thus $E_0{U}_{L,bc}$ is the dimensional pair energy in either boundary convention.

Let $Q_L$ be the unit-intensity Poisson probability on positional configurations in ${\mathrm{\Lambda }}_L=[0,L{)}^d$, defined by Poisson Configuration Reference. Its $N$-particle conditional law $Q_{L,N}$ is uniform on the unordered simple $N$-point configurations; in labelled coordinates it is $L^{-dN}d{x}_1\cdots d{x}_N$ on the no-collision set. A finite marked configuration in this sector is a simple set $\xi =\{(x,p_x):x\in q\}$ with $q\subseteq {\mathrm{\Lambda }}_L$, $N(q)=N$, and $p_x\in {\mathbb{R}}^d$. Equip the finite marked sector with the sigma-algebra generated by the counts $N_{A\times C}(\xi )=\mathrm{\#}\{(x,p_x)\in \xi :x\in A,p_x\in C\}$ for bounded Borel $A\subseteq {\mathbb{R}}^d$ and Borel $C\subseteq {\mathbb{R}}^d$. Let ${\lambda }_d$ be Lebesgue measure on the SI numerical momentum coordinates and set $r:=p_{\ast }^{-d}{\lambda }_d$. For each fixed $q=\{x_1,\dots ,x_N\}$, define $r^{\otimes q}$ as the image of $r^{\otimes N}$ under $(p_1,\dots ,p_N)\mapsto \{(x_j,p_j):1\le j\le N\}$; this is independent of the enumeration of $q$ because $r^{\otimes N}$ is permutation-invariant. The cell-code representation in Poisson Configuration Reference supplies a measurable enumeration on each finite sector, so $q\mapsto r^{\otimes q}$ is a measurable kernel for the marked-count sigma-algebra. The finite phase reference is $R_{L,N}(dq,dp):=Q_L(dq)r^{\otimes q}(dp)1_{\{N(q)=N\}}\mathrm{.}$ It is sigma-finite, meaning the marked sector is covered by countably many measurable sets of finite reference measure: for each integer $k\ge 1$, restrict to marks satisfying $\mathrm{\mid }{p}_x\mathrm{\mid }\le k{p}_{\ast }$ for every one of the $N$ points. These sets exhaust the sector, and their reference measures are at most $Q_L(N(q)=N)r(B_k{)}^N<\mathrm{\infty }$, where $B_k=\{p:\mathrm{\mid }p\mathrm{\mid }\le k{p}_{\ast }\}$ satisfies $r(B_k)\le (2k{)}^d$ because it lies in the box $[-k{p}_{\ast },k{p}_{\ast }{]}^d$ of Lebesgue volume $(2k{p}_{\ast }{)}^d$ by the box-volume supplier in the dependency closure of Poisson Configuration Reference. The factor $e^{-V_L}\mathrm{/}N!$ from the Poisson sector is retained. In physical coordinates the corresponding one-particle product scale is $h_{\mathrm{r}\mathrm{e}\mathrm{f}}^d={\mathrm{\ell }}_0^d{p}_{\ast }^d$, an action to the power $d$; it is a reference convention, not a derived quantum constant.

For a marked configuration define the total dimensionless energy ${\mathcal{H}}_{L,bc}(q,p)=U_{L,bc}(q)+{\sum }_{x\in q}K(p_x),K(p)=\frac{\mathrm{\mid }p{\mathrm{\mid }}^2}{2m{E}_0}\mathrm{.}$ The corresponding physical Hamiltonian is $H_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=E_0{\mathcal{H}}_{L,bc}$ joules, so ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=b{\mathcal{H}}_{L,bc}$. For $b>0$, Gaussian mark normalization, moments, and exponential controls gives ${\kappa }_b=\int e^{-bK}r(dp)=(2\pi m{E}_0\mathrm{/}(b{p}_{\ast }^2){)}^{d\mathrm{/}2}$ and the probability kernel ${\nu }_b(dp)={\kappa }_b^{-1}{e}^{-bK(p)}r(dp)$. Whenever the positional canonical partition is finite and positive, nonnegative product integration Tonelli's theorem for nonnegative measurable functions on a sigma-finite product gives exact identities $Z_{Q,L,N,bc}^{\mathrm{p}\mathrm{h}}(b)=\int e^{-b{\mathcal{H}}_{L,bc}}d{R}_{L,N}={\kappa }_b^N{Z}_{Q,L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}(b),C_{L,N,bc}^{\mathrm{p}\mathrm{h}}=C_{L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}{\nu }_b^{\otimes q}\mathrm{.}$ Here $Z_{Q,L,N,bc}^{\mathrm{p}\mathrm{o}\mathrm{s}}=E_{Q_L}[1_{\{N(q)=N\}}{e}^{-b{U}_{L,bc}}]$ and the canonical probabilities are the normalized respective integrands. Under a full grand preparation of activity $z_{\mathrm{p}\mathrm{h}}$, integrating each momentum changes the positional activity to $z=z_{\mathrm{p}\mathrm{h}}{\kappa }_b$; thus a positional activity $z=e^{\alpha }$ corresponds to $z_{\mathrm{p}\mathrm{h}}=e^{\alpha }\mathrm{/}{\kappa }_b$. These are exact definitions and normalizations; they do not assert dynamical equilibration or finite-volume grand convergence. Write ${\mathbb{T}}_L^d:={\mathbb{R}}^d\mathrm{/}L{\mathbb{Z}}^d$ for the periodic position space, represented by ${\mathrm{\Lambda }}_L$; translations act modulo $L$, and the translation-average is Lebesgue measure on ${\mathrm{\Lambda }}_L$ divided by $V_L$. For periodic boundary conditions, simultaneous translation of all positions modulo the period preserves $R_{L,N}$ and ${\mathcal{H}}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$: the conditional position law is uniform on ${\mathbb{T}}_L^d$, the mark reference is invariant under relabelling points, and the image energy depends only on pair displacements. Hence normalized canonical and finite band restrictions are invariant under every such translation.

Let $\mathrm{\Omega }$ be the positional configuration space of Poisson Configuration Reference. The marked configuration space ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ consists of marked point sets $\xi \subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ with spatial projection $q=\{x\in {\mathbb{R}}^d:\mathrm{\exists }p\in {\mathbb{R}}^d,(x,p)\in \xi \}$ that is simple and locally finite, and with exactly one mark $p_x$ paired to each $x\in q$. Its sigma-algebra is generated by the marked counts $N_{A\times C}(\xi )=\mathrm{\#}\{(x,p)\in \xi :x\in A,p\in C\}$ for bounded Borel $A\subseteq {\mathbb{R}}^d$ and Borel $C\subseteq {\mathbb{R}}^d$. The positional projection is $\pi (\xi )=q\in \mathrm{\Omega }$. For a bounded Borel set $\mathrm{\Delta }$, the marked local restriction is ${\xi }_{\mathrm{\Delta }}=\{(x,p)\in \xi :x\in \mathrm{\Delta }\}$ with the induced count sigma-algebra; write $N_{\mathrm{\Delta }}(\xi )=\mathrm{\#}{\xi }_{\mathrm{\Delta }}$. A marked-local observable on $\mathrm{\Delta }$ is a real measurable function of ${\xi }_{\mathrm{\Delta }}$; it is bounded when its supremum norm is finite.

Let $d\ge 1$, $L>0$, and ${\mathrm{\Lambda }}_L=[0,L{)}^d$. A finite marked configuration consists of a finite simple set of positions $q\subseteq {\mathrm{\Lambda }}_L$ and a mark map $p:q\to {\mathbb{R}}^d$, as in Interacting marked phase-space and reference measures. Its marked periodic continuation is ${\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}:=\{(x+Lk,p_x):x\in q,k\in {\mathbb{Z}}^d\}\mathrm{.}$ The spatial positions in this set are distinct: if $x+Lk=y+Lj$ for $x,y\in [0,L{)}^d$, then coordinatewise $x-y=L(j-k)$ forces $j=k$ and $x=y$. Every bounded spatial region meets finitely many lattice translates of the finite set $q$, so the continuation has a locally finite spatial projection.

For a bounded Borel set $\mathrm{\Delta }\subseteq {\mathbb{R}}^d$, define its marked local restriction by ${\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}{\mathrm{\mid }}_{\mathrm{\Delta }}:=\{(y,p_x)\in {\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}:y\in \mathrm{\Delta }\}\mathrm{.}$ This is an element of ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$, with the induced sigma-algebra of marked counts above. If $\mathrm{\Delta }\subseteq {\mathrm{\Lambda }}_L$, the restriction is exactly the original marked points with positions in $\mathrm{\Delta }$.

Assume AC. Let $P$ be any probability law on the simple locally finite positional configuration space $\mathrm{\Omega }$ of Poisson Configuration Reference, and let $\nu$ be a probability on the momentum mark space ${\mathbb{R}}^d$. Use the marked configuration space ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ and local restrictions ${\xi }_{\mathrm{\Delta }}$ of Marked periodic continuation and local restriction. Define translations by $T_{y}q=\{x+y:x\in q\}$ and $T_y^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}\xi =\{(x+y,p):(x,p)\in \xi \}$. Then there is a unique probability law $P^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ on ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ whose positional projection is $P$ and whose marks, conditional on the entire positional configuration, are independent with common law $\nu$. If $P$ is translation invariant, meaning $(T_y{)}_{\mathrm{\#}}P=P$ for every $y\in {\mathbb{R}}^d$, then $(T_y^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}{)}_{\mathrm{\#}}{P}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}=P^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ for every $y$. For the Gaussian ${\nu }_b$ of Interacting marked phase-space and reference measures, this is the independent Gaussian-marked positional lift. This is a mathematical marking construction, not a physical preparation or a convergence result.

Assume AC, the ultrafilter extension principle, DC, and HB. Use the model and dimensionless energy ${\mathcal{H}}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$ of Interacting marked phase-space and reference measures. Let $L_n\to \mathrm{\infty }$, $V_n=L_n^d$, and integers $N_n$ satisfy $N_n\mathrm{/}{V}_n\to \rho >0$, where $\rho$ is an interior attainable density for the positional canonical rate. Fix $b>0$. Assume $F_b(\rho )$ is the periodic exact-particle log-partition rate supplied by Periodic fixed-particle canonical limit for every $b^{\mathrm{\prime }}$ in a neighborhood of $b$, and is differentiable at $b$. Write $C_n^{\mathrm{p}\mathrm{h}}$ for the full fixed-$N_n$ canonical law, and define $e_b(\rho )=-{\mathrm{\partial }}_b{F}_b(\rho )+\frac{d\rho }{2b}\mathrm{.}$ Then, for each fixed $\epsilon >0$, there exist $c_{\epsilon }>0$ and $n_0$ such that $C_n^{\mathrm{p}\mathrm{h}}\text{\hspace{0.17em}⁣}(\mid {\mathcal{H}}_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{V}_n-e_b(\rho )\mid >\epsilon )\le 2e^{-c_{\epsilon }{V}_n}(n\ge n_0)\mathrm{.}$ Consequently there is a nonincreasing sequence ${\delta }_n\downarrow 0$ for which the full phase band $A_n=\{N=N_n,\mathrm{\mid }{\mathcal{H}}_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{V}_n-e_b(\rho )\mathrm{\mid }\le {\delta }_n\}$ has $C_n^{\mathrm{p}\mathrm{h}}(A_n)\to 1$. It has positive finite phase-reference measure for all sufficiently large $n$. This constructs a controlled shrinking window; it makes no claim about every prescribed shell width or an exact energy surface.

Fix the model, physical reference conventions, mass, potential and energy window specified by Interacting marked phase-space and reference measures and Total-energy concentration and controlled canonical windows. This is a single-species model on the flat torus of side ${\mathrm{\ell }}_{0}L$, represented by ${\mathrm{\Lambda }}_L=[0,L{)}^d$. Identify the action reference $a_0$ of Classical particle counting and physical observables with $h_{\mathrm{r}\mathrm{e}\mathrm{f}}={\mathrm{\ell }}_0{p}_{\ast }$ from the marked model, and take the controls $\lambda =(L,bc)$ in Canonical thermal preparation. Its physical Hamiltonian is $H_{N,\lambda }=H_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=E_0{\mathcal{H}}_{L,bc}$ from Interacting marked phase-space and reference measures, so ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{N,\lambda }=b{\mathcal{H}}_{L,bc}$. In the corresponding canonical coordinates, the exact-$N$ reference $R_{L,N}$ of that mathematical model is $e^{-V_L}{\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, where ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is the one-species quotient reference of Classical particle counting and physical observables. The factor $e^{-V_L}$ is independent of $N$ and cancels when normalizing a fixed-$N$ canonical law, a fixed-$N$ band restriction, or the full grand-sector law. Adopt the following alternative preparations whenever their normalizations are finite and positive:

1. At fixed exact particle number $N$, controls and $T>0$, use the canonical density proportional to $e^{-\beta H}$ relative to the declared phase reference, as in Canonical thermal preparation.
2. For a sequence satisfying all hypotheses of Total-energy concentration and controlled canonical windows—including AC, the ultrafilter extension principle, DC, HB, an interior attainable density, the periodic exact-particle log-rate hypothesis near $b$, and differentiability at $b$—at fixed exact $N_n$ use the uniform full-phase reference law restricted to its constructed band $A_n$. “Uniform” means normalized restriction of that finite reference measure; it is an energy-window preparation, not a zero-width surface measure. This postulate makes no existence claim for that specific band when those theorem hypotheses fail.
3. For particle exchange, use the countable-sector grand preparation of Grand canonical particle-exchange preparation. Its full dimensionless activity is the fugacity $z_{\mathrm{p}\mathrm{h}}:=e^{{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mu }>0$, where ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=1\mathrm{/}(k_{B}T)$ in J${}^{-1}$ and the chemical potential $\mu$ is in J per particle. When matching positional activity $z=e^{\alpha }$ at inverse temperature $b$, choose $z_{\mathrm{p}\mathrm{h}}=z\mathrm{/}{\kappa }_b$, as in Interacting marked phase-space and reference measures.

These laws are adopted statistical preparations. They do not follow from Hamiltonian energy conservation, do not assert that any isolated dynamics reaches these distributions, and do not encode an experimental observation. The band sequence is the particular controlled shrinking window constructed conditionally by Total-energy concentration and controlled canonical windows.

Assume AC and the hypotheses and notation of Total-energy concentration and controlled canonical windows. Let $R_n=R_{L_n,N_n}$ be the finite phase reference of Interacting marked phase-space and reference measures, let $A_n$ be the constructed band, and set $M_n^{\mathrm{p}\mathrm{h}}(E)=R_n(E\cap A_n)\mathrm{/}{R}_n(A_n)$. Let $C_n^{\mathrm{p}\mathrm{h}}$ be the full fixed-$N_n$ canonical probability. Then $D_n=D(M_n^{\mathrm{p}\mathrm{h}}\mathrm{\parallel }{C}_n^{\mathrm{p}\mathrm{h}})$ satisfies $D_n\mathrm{/}{V}_n\to 0$. This is a mathematical statement about a normalized measure restriction; it does not assert $D_n\to 0$ or global total-variation convergence.

Assume AC, the ultrafilter extension principle, DC and HB, the exact superstable regular pair interaction and periodic-image convention of Interacting marked phase-space and reference measures, and integer side lengths $L_n\to \mathrm{\infty }$ with exact particle numbers $N_n\mathrm{/}{V}_n\to \rho >0$ at an interior attainable density. Fix $b>0$. Assume the positional canonical log rate is $F_b(\rho )$, and choose finite $\alpha$ with $P_{\mathrm{L}\mathrm{e}\mathrm{b}}(\alpha ,b)=F_b(\rho )+\alpha \rho ,$ where $P_{\mathrm{L}\mathrm{e}\mathrm{b}}$ is the grand pressure per numerical volume with Lebesgue $1\mathrm{/}N!$ normalization. Let the positional grand variational minimizer of Superstable regular continuum grand pressure and variational formula at activity $\exp (\alpha )$ and inverse temperature $b$ be the singleton $\{P_{\ast }\}$. Let $R_n=R_{L_n,N_n}$ be the phase reference of Interacting marked phase-space and reference measures, let $A_n$ be the band of Total-energy concentration and controlled canonical windows, set $M_n^{\mathrm{p}\mathrm{h}}(E)=R_n(E\cap A_n)\mathrm{/}{R}_n(A_n)$, and let $B_n$ be its positional marginal. For any local-tame test $f$, write $B_n[f]=E_{B_n}f({\omega }^{(L_n)})$ and $C_n^{\mathrm{p}\mathrm{o}\mathrm{s}}[f]=E_{C_n^{\mathrm{p}\mathrm{o}\mathrm{s}}}f({\omega }^{(L_n)})$, where ${\omega }^{(L_n)}$ is the periodic continuation. Let ${\bar{B}}_n$ be the barycenter of $R_{L_n,\omega }$ under $B_n$. Then ${\bar{B}}_n$ converges to $P_{\ast }$ in ${\tau }_L$, equivalently $B_n[f]\to P_{\ast }(f)$ for every local-tame $f$; also $C_n^{\mathrm{p}\mathrm{o}\mathrm{s}}[f]\to P_{\ast }(f)$ for every such $f$. The periodized empirical fields of the exact-$N_n$ canonical laws converge in probability in ${\tau }_L$ to $P_{\ast }$ by Periodic fixed-particle canonical limit, and $P_{\ast }$ has intensity $\rho$. Here $P_{\ast }$ is an infinite-volume positional variational state; no finite-volume grand-canonical local convergence or DLR identification is asserted.