Canonical free-cube limit at every interior attainable density

Stability and [F3] give $Z(L,N;\beta )\le e^{\beta BN}{L}^{dN}\mathrm{/}N!$. If $N\mathrm{/}{L}^d\to \rho >0$, the factorial integral bounds [F4] imply $\mathrm{lim\hspace{0.17em}sup}{L}^{-d}\log Z(L,N;\beta )\le \rho (1+\beta B-\log \rho ),$ so $b(\rho )$ and $F(\rho )$ are bounded above on every compact positive-density interval. The vacuum term gives $b(0)=0$: $Z(L,0)=1$ is a lower bound, and the same stability estimate with $N\mathrm{/}{L}^d\to 0$ gives a nonpositive upper limit.

The effective domain of $F$ has nonempty interior. Fix a core cube of side $s>0$ and a gap $g>2R_0$; place one particle in each selected core of a lattice of spacing $s+g$ inside a large box. The absolute interaction from any one occupied core to all others is bounded by $C{\sum }_{m\ge 1}{m}^{d-1}\psi (mg)<\mathrm{\infty }$ by [F1]–[F2], with the gap satisfying the stated hypothesis of the diagonal-gap estimate. Restricting each particle to a subcube of volume $v_0>0$ therefore gives $Z(L,N;\beta )\ge v_0^N{e}^{-C_{0}N}$ for every density strictly below $(s+g{)}^{-d}$, after omitting unused lattice cells. Thus $F$ is finite on a nonempty interval of positive densities; by its definition as a finite-mixture envelope it is concave.

Fix a finite list of core preparations $(s_i,n_i)$ with $0<Z(s_i,n_i;\beta )<\mathrm{\infty }$. For each $i$, choose a gap $g_i=g(s_i)$ from [F2] along a sequence $s_i\to \mathrm{\infty }$ so that $g_i\mathrm{/}{s}_i\to 0$ and the positive interaction between distinct cores in the same lattice packing, divided by core volume, tends to zero. Set tile side $a_i=s_i+g_i$ and effective density $r_i=n_i\mathrm{/}{a}_i^d$. If $M_i$ such tiles carry exactly $n_i$ particles in their central $s_i$-cubes, the same shell count as in the proof of Free-cube grand-pressure limit for decreasing integrable tails gives cross-energy penalty per occupied volume at most $C(n_i\mathrm{/}{s}_i^d{)}^2(s_i\mathrm{/}{g}_i{)}^d\eta (g_i)$, which tends to zero by [F2]. Exact factorization over disjoint core cubes follows because assigning the labeled coordinates to their cores contributes $N!\mathrm{/}{\prod }_i(n_i!{)}^{M_i}$, cancelling the global $N!$.

For any finite convex combination in the definition of $F(\rho )$, choose core preparations with densities tending to its ${\rho }_i$ and rates tending to $b({\rho }_i)$. If their effective densities do not strictly bracket the target $\rho$, add arbitrarily small weights at one attainable density below and one above $\rho$; such densities exist because $\rho$ is interior. Their mixture coefficients can then be perturbed by quantities tending to zero so the effective mean is exactly $\rho$. Place the fixed core types in finitely many macroscopic slabs with these volume fractions, inserting fixed empty layers wider than $R_0$ between slabs. For fixed core sizes and gaps, the distinct-slab cross energy divided by total volume tends to zero: for interaction distances at most $R$, only $O(R{L}^{d-1})$ particles lie in the interface strips, while the remaining contribution is bounded by a constant times $L^d{\sum }_{\mathrm{\mid }j{\mathrm{\mid }}_{\mathrm{\infty }}>R}{\mathrm{\Psi }}_j$; take $L\to \mathrm{\infty }$ and then $R\to \mathrm{\infty }$.

First reserve an outer layer of fixed width $W$. For each large $L$, put ${\rho }_L=N_L\mathrm{/}{L}^d$ and choose the slab fractions in the remaining volume to match the count $N_L$ at density ${\rho }_L{L}^d\mathrm{/}(L^d-O(W{L}^{d-1}))$. This target tends to $\rho$, so the strict bracketing in step 3.1 applies uniformly for large $L$ and absorbs the reserved layer without assuming its one-particle capacity replaces a bulk density. Rounding the finitely many slab and core counts then leaves at most $C_1{L}^{d-1}$ particles unplaced, with $C_1$ depending on the fixed preparations but not on $W$; the error comes only from the finitely many slab faces and incomplete boundary tiles. Choose $W$ large enough to contain at least $C_1{L}^{d-1}$ small cells separated from each other and every core by more than $R_0$. Put each missing particle in one such cell. Its interaction with the fixed preparations is bounded by the same summable tail, so the correction changes log weight by only $O(L^{d-1})$. This constructs exactly $N_L$ particles for every sequence $N_L\mathrm{/}{L}^d\to \rho$, and gives ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{L}^{-d}\log Z(L,N_L;\beta )\ge {\sum }_i{\theta }_i\frac{\log Z(s_i,n_i;\beta )-{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}_i}{a_i^d},$ where ${\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}_i\mathrm{/}{a}_i^d\to 0$ by step 2.1. Taking preparations approaching the selected convex combination and then the supremum gives $\mathrm{lim\hspace{0.17em}inf}\ge F(\rho )$.

By the definition of $b$, every sequence with density tending $\rho$ has limsup rate at most $b(\rho )\le F(\rho )$. Together with step 4.1 this proves the asserted limit. A finite concave function on an open interval is continuous: fix $a<\rho <b$ in the interval; the chord slopes from $a$ and $b$ bound the difference quotients on a smaller interval around $\rho$, yielding a finite local Lipschitz bound. This establishes finiteness and continuity of $F$ on its effective interior while leaving excluded endpoints undecided.