Superquadratic cell coercivity from a divergent or hard core

Write $N_i=N_{C_i}(\omega )$. The negative part of each pair potential is bounded both by $2B$, where $B$ is the stability constant of $\varphi$, and, at large separation, by the lower-regular envelope. Set ${\mathrm{\Psi }}_j$ to a finite constant on the finitely many near-cell offsets and to the envelope evaluated at $\mathrm{\mid }j{\mathrm{\mid }}_{\mathrm{\infty }}-1$ on the remaining offsets. The shell count $\mathrm{\mid }\{j:\mathrm{\mid }j{\mathrm{\mid }}_{\mathrm{\infty }}=m\}\mathrm{\mid }\le c_d{m}^{d-1}$ and the cell-envelope conclusion of [F2] give ${\sum }_j{\mathrm{\Psi }}_j<\mathrm{\infty }$. Since $N_i{N}_j\le (N_i^2+N_j^2)\mathrm{/}2$, the total negative pair contribution is at most $C{\sum }_i{N}_i^2$ for a finite $C$.

In case (i), choose an integer $k_0\ge 2$ with $1\mathrm{/}(k_0-1)\le r_1$. Then $\chi (1\mathrm{/}k)$ and $\chi (1\mathrm{/}(k-1))$ are both defined for every $k\ge k_0$. Partition one unit cell into $k^d$ half-open cubes of side $1\mathrm{/}k$, put ${\chi }_k=\chi (1\mathrm{/}k)$, and $\mathrm{\Delta }{\chi }_k={\chi }_k-{\chi }_{k-1}\ge 0$. If the cell contains $N$ points, the identity ${\sum }_j{n}_j^2-N=2{\sum }_j\left(\genfrac{}{}{0px}{}{n_j}{2}\right)$ and nonnegativity of ${\sum }_j(n_j-N\mathrm{/}{k}^d{)}^2$ give at least $\frac{1}{2}(N^2\mathrm{/}{k}^d-N)$ same-subcube pairs at scale $k$. A fixed pair counted at scale $k$ has separation $s<1\mathrm{/}k$, hence $k<1\mathrm{/}s$; summing its nonnegative weights over all scales where it is counted is at most ${\sum }_{k=k_0}^{\lfloor 1\mathrm{/}s\rfloor }\mathrm{\Delta }{\chi }_k\le \chi (s)$. Thus the positive core energy in the cell dominates the sum of these weighted pair counts. With $K_N=\lfloor (N\mathrm{/}2{)}^{1\mathrm{/}d}\rfloor$, each $k\le K_N$ satisfies $N^2\mathrm{/}{k}^d-N\ge N^2\mathrm{/}(2k^d)$, so the core energy is at least $\frac{N^2}{4}{\sum }_{k=k_0}^{K_N}\frac{\mathrm{\Delta }{\chi }_k}{k^d}\mathrm{.}$ Summation by parts writes the coefficient as an initial constant plus ${\sum }_{k=k_0}^{K_N-1}{\chi }_k(k^{-d}-(k+1{)}^{-d})$; and $k^{-d}-(k+1{)}^{-d}\ge d\mathrm{/}(2^{d+1}{k}^{d+1})$. The series ${\sum }_k\chi (1\mathrm{/}k)k^{-d-1}$ diverges: on dyadic intervals $[2^{-j-1},2^{-j}]$, monotonicity bounds the integral above by $C_d{2}^{-jd}\chi (2^{-j-1})$, while the corresponding block $2^{j+1}\le k<2^{j+2}$ contributes at least $c_d{2}^{-jd}\chi (2^{-j-1})$ to that series. The divergent core integral therefore forces the displayed coefficient to tend to infinity.

In case (i), sum the cell core bound from step 1.2 and subtract the negative-pair bound from step 1.1. The resulting coefficient of $N_i^2$ tends to infinity with $N_i$. Above a finite threshold it dominates $2C{N}_i^2$; below that threshold its possible negative remainder is bounded by a constant times $N_i$. Taking $h(N)$ to be half the positive cell lower bound above the threshold and zero below it gives a nondecreasing extended-valued function with $h(N)\mathrm{/}{N}^2\to \mathrm{\infty }$ and $H_{\varphi }\ge {\sum }_{i}h(N_i)-bN$.

In case (ii), choose $k$ with $1\mathrm{/}k<r_h$ and partition every unit cell into $k^d$ subcubes. A finite-energy configuration has at most one point in each subcube, so $N_i\le k^d$ for every $i$. Define $h(N)=0$ for $N\le k^d$ and $h(N)=+\mathrm{\infty }$ for $N>k^d$. If any cell exceeds this occupancy, the configuration has a forbidden pair and $H_{\varphi }=+\mathrm{\infty }$; otherwise stability gives $H_{\varphi }\ge -BN={\sum }_{i}h(N_i)-BN$. This proves the hard-core alternative.