Proof. Disjoint balls of radius \(1/3\) about the points lie in \([-1/3,L+1/3]^3\), so \(M_L\le C(L+1)^3\). This also bounds the possible cardinalities; compactness of the configuration space for each cardinality shows that the maximum is attained. The integer lattice gives a positive lower asymptotic density.

Fix \(s>0\) and repeat a maximizing configuration in \([0,s]^3\) by translations in \((s+1)\mathbb Z^3\). Points in distinct translates are separated by at least one: in a coordinate in which the translates differ, their coordinate separation is at least \((s+1)-s=1\). Taking all complete translates contained in \([0,L]^3\) gives \[\liminf_{L\to\infty}\frac{M_L}{L^3} \ge \frac{M_s}{(s+1)^3}.\] Letting \(s\to\infty\) along a sequence realizing the upper asymptotic density makes the right side tend to that upper density. The lower and upper limits therefore agree. ◻

Proof. If five points formed a clique, rescale their distances by \(l_t^{-1}\) and take the four difference vectors from one of them. Their Gram matrix differs entrywise by at most \(2\Delta+\Delta^2\) from the matrix with diagonal one and off-diagonal entries \(1/2\). For a diagonal entry this follows from \(1\le r^2\le(1+\Delta)^2\); for an off-diagonal entry use polarization, \(\left\langle v_i,v_j\right\rangle=(\left|v_i\right|^2+\left|v_j\right|^2-\left|v_i-v_j\right|^2)/2\). The reference matrix has smallest eigenvalue \(1/2\). The perturbation has operator norm at most \(4(2\Delta+\Delta^2)<1/2\), so the Gram matrix would be positive definite and have rank four. Four vectors in \(\mathbb R^3\) cannot have this Gram matrix.

For the edge bound, maximize \(\sum_{\{i,j\}\in E}u_i u_j\) over \(u_i\ge0\), \(\sum_i u_i=1\), choosing a maximizer with the smallest possible support. If two nonadjacent vertices have positive weights, the objective is affine in their two weights with their sum fixed. Putting their combined weight on one of the vertices cannot decrease the objective, contradicting minimality of the support. The support is therefore a clique of some size \(r\le4\). Its objective is \[\frac12\left(1-\sum_i u_i^2\right) \le\frac12(1-1/r)\le\frac38.\] For \(m>0\), apply this bound to equal weights \(1/m\). The empty graph case is immediate. ◻

Proof. The diameter \(\sqrt3 b\) of a small cell is less than one. By 2, among \(m\) particles in one cell at least \(m(m-1)/2-3m^2/8=m^2/8-m/2\) pairs have distance outside \(I_t\). Each of these pairs contributes \(t^2\), while all remaining core contributions are nonnegative. Summing over cells proves the first bound, and also proves it when core terms between cells are deleted. For the attractive contribution, including extra diagonal self terms only enlarges the bound, and symmetry gives \[\sum_{r<s}K(x_r,x_s) \le\frac12\sum_{i,j}J_{ij}n_i n_j \le\frac14\sum_{i,j}J_{ij}(n_i^2+n_j^2) \le\frac M2\sum_i n_i^2.\] This proves (8).

There are at most \((L/b+2)^3\le216V\) nonempty cells. Hence \(\sum_i n_i^2\ge N^2/(216V)\). Under (9), \(D_0-M/2\ge t^2/16\), giving (10) with the stated \(c\). The requirements on \(t\) are compatible: \(M\) grows linearly in \(t\), whereas \(D_0\) grows quadratically. The constants \(c\) and \(c_b\) are independent of \(t\) and of the attraction ranges. ◻

Proof. The lower bound follows by bounding the integrand from above using (10) and using \(n!\ge(n/\mathrm e)^n\) for \(n\ge1\). For the other bound, let \(X_1,\ldots,X_n\) be independent uniform points in \(\Lambda_L\). Since \(U^\varphi\le U^{\psi_t}\) and \(\int_{\mathbb R^3}\psi_t(\left|x\right|)\,\,\mathrm dx\le t^2\left|B(0,2)\right|\), \[\mathbb E U^\varphi \le\frac{n(n-1)}{2L^3}\int_{\mathbb R^3}\psi_t(\left|x\right|)\,\,\mathrm dx \le Ct^2n^2/L^3.\] Jensen’s inequality gives \(\log Z^\varphi_{L,n}\ge n\log L^3-\log n!-\beta Ct^2n^2/L^3\). Now use \(n!\le n^n\). At \(n=0\) both bounds are equalities. ◻

Proof. Enclose \(\mathcal D\) in \(\beta\in[\beta_-,\beta_+]\) and \(\left|h\right|\le H\). Put \(V=L^3\). By (12), the logarithm of the summand at \(n\ge1\) is at most \[-\beta t(ct-\zeta/2)\frac{n^2}{V} +(th+\beta t^2/2+1)n-n\log(n/V).\] The choice (9) implies \(ct-\zeta/2\ge ct/2\). Set \(\kappa=\beta_-ct^2/2\) and \(B=tH+\beta_+t^2/2+1\). For \(n/V\ge1\) the displayed expression is at most \(-\kappa n^2/V+Bn\). Choose \(D\ge\max(1,2B/\kappa)\). For \(n>DV\) this is at most \(-\kappa n^2/(2V)\le-(\kappa D/2)n\). Summing this geometric majorant proves the assertion with \(a_0=\kappa D/2\). ◻

Proof. We first prove (14), without using a pressure limit or an approximation by finite-range potentials. At \(d=0\), both bounds in (12) tend to zero. Fix \(d>0\) and \(\beta>0\). Let \(H_*\) be the lower limit of \(H^\varphi_{L,n}(\beta)\) as \(L\to\infty\) and \(n/L^3\to d\), and choose \(s_j\to\infty\) and integers \(n_j\) with \[n_j/s_j^3\longrightarrow d,\qquad H^\varphi_{s_j,n_j}(\beta)\longrightarrow H_*.\] The finite volume bounds make \(H_*\) finite. Consider any other sequence \(L\to\infty\), \(n/L^3\to d\). For each fixed, sufficiently large \(j\), place \(k=\lfloor L/(s_j+2)\rfloor^3\) cubes of side \(s_j\) in \(\Lambda_L\), separated by gaps of width at least two. For any nonnegative integers \(m_1,\ldots,m_k\) summing to \(n\), restriction to configurations with these box counts gives \[ Z^\varphi_{L,n}(\beta) \ge\prod_{i=1}^k Z^\varphi_{s_j,m_i}(\beta). \tag{17}\] Indeed, interactions between different boxes are nonpositive. There are \(n!/\prod_i m_i!\) disjoint assignments of particle labels to the boxes, so the global factor \(1/n!\) leaves exactly the product of their factors \(1/m_i!\).

We retain the favorable count \(n_j\) in most boxes and correct the total particle number in the remaining boxes. Only a vanishing fraction of boxes will change, and (12) will bound their canonical costs uniformly. To make this exact for every target count, put \[q_j=\lfloor(4d+4)s_j^3\rfloor, \qquad c_d=\min(d/2,1).\] For all sufficiently large \(j\), \(n_j\ge c_ds_j^3\) and \(q_j-n_j\ge c_ds_j^3\). Start with \(n_j\) particles in every box. If \(E=n-kn_j\ge0\), fill boxes successively to \(q_j\), with at most one last box given an intermediate integer count. If \(E<0\), empty boxes successively, again with at most one last intermediate count. This changes at most \(r\) boxes, where \[ \frac rk\le\frac1{c_d} \left|\frac{n}{ks_j^3}-\frac{n_j}{s_j^3}\right|+\frac1k. \tag{18}\] For fixed large \(j\) and then sufficiently large \(L\), the target lies between \(0\) and \(kq_j\): its mean count divided by \(s_j^3\) tends to \(d(1+2/s_j)^3<q_j/s_j^3\). Thus the adjustment never exhausts the available boxes. The quotient and remainder in division of \(E\) by \(q_j-n_j\), or of \(-E\) by \(n_j\), give the asserted integer counts.

All counts used have density in \([0,4d+4]\). By (12), their normalized canonical costs have absolute value at most some \(K<\infty\), uniformly for large \(j\). Taking logarithms in (17) consequently yields \[H^\varphi_{L,n}(\beta) \le\frac{ks_j^3}{L^3} \left(H^\varphi_{s_j,n_j}(\beta)+2K\frac rk\right).\] First let \(L\to\infty\) with \(j\) fixed. The prefactor tends to \((s_j/(s_j+2))^3\), and the upper limit of \(r/k\) is at most \[c_d^{-1}\left|d(1+2/s_j)^3-n_j/s_j^3\right|.\] Then let \(j\to\infty\). This error tends to zero and the preceding upper bound tends to \(H_*\). Every sequence has lower limit at least \(H_*\) by its definition; thus every sequence has the same limit. This proves (14) with the required order of limits.

For density convexity, fix \(d_1,d_2\ge0\) and \(\theta\in[0,1]\). Choose \(s_j\to\infty\), integers \(m_{r,j}\) with \(m_{r,j}/s_j^3\to d_r\) for \(r=1,2\), and integers \(q_j\to\infty\). In a cube of side \(q_j(s_j+2)\), use \(q_j^3\) separated boxes of side \(s_j\), assigning count \(m_{1,j}\) to \(\lfloor\theta q_j^3\rfloor\) of them and count \(m_{2,j}\) to the rest. The total density tends to \(\theta d_1+(1-\theta)d_2\). The product inequality and the already proved all-sequence limit, applied directly to this diagonal sequence, give \[H_\varphi(\beta,\theta d_1+(1-\theta)d_2) \le\theta H_\varphi(\beta,d_1) +(1-\theta)H_\varphi(\beta,d_2).\] This proof does not presuppose continuity in density. A finite convex function is continuous at positive densities: on any interior compact interval its secant slopes are bounded by secants to two exterior points. Continuity at zero follows from (12). Hölder’s inequality makes \(\log Z^\varphi_{L,n}(\beta)\) convex in \(\beta\). Passing to the canonical limit proves the stated concavity.

The grand sums are finite by 4, with \(\zeta=0\). Fix \((\beta,h)\). Choose \(D\) there and write \(V=L^3\). Above \(DV\) the entire sum tends to zero. Among the remaining \(\lfloor DV\rfloor+1\) terms choose a maximizing count \(n_L\); the maximum is at least one because it includes the vacuum. Therefore \[\frac{\log\Xi^\varphi_L}{tV} \le h\frac{n_L}{V}-H^\varphi_{L,n_L}(\beta) +\frac{\log(\lfloor DV\rfloor+2)}{tV}\] for all sufficiently large \(L\). From any subsequence choose a further subsequence with \(n_L/V\to d\in[0,D]\). The canonical limit bounds its upper limit by \(hd-H_\varphi(\beta,d)\). Conversely, for every fixed \(d\ge0\) the single term \(n=\lfloor dV\rfloor\) bounds the lower limit by \(hd-H_\varphi(\beta,d)\). This proves existence of the pressure and the first duality in (16).

The vacuum gives \(Q_\varphi\ge0\). The canonical lower bound and \(d\log d-d\ge-1\) bound the first supremum in (16) by a finite quadratic maximum, uniformly over the family on any fixed parameter compact. Hölder’s inequality on the disjoint union of all configuration spaces makes each \(\log\Xi^\varphi_L\) jointly convex in \((\beta,h)\); its pointwise limit has the same property.

Finally fix \(d>0\). Convexity of the finite function \(H_\varphi(\beta,\cdot)\) gives a supporting slope \(h\) there. Explicitly, one may choose \(h\) between \[\sup_{0\le v<d}\frac{H_\varphi(\beta,d)-H_\varphi(\beta,v)}{d-v} \quad\text{and}\quad \inf_{v>d}\frac{H_\varphi(\beta,v)-H_\varphi(\beta,d)}{v-d}.\] The left endpoint is at most the right endpoint by convexity, and both are finite by comparison with secants through fixed points on either side of \(d\). The resulting supporting inequality holds for all \(v\ge0\), so the first duality gives \(Q_\varphi(\beta,h)=hd-H_\varphi(\beta,d)\). For all other \(h\) the same first duality gives the opposite variational inequality. This proves the second equality in (16) without a differentiability or uniqueness assumption. ◻

Proof. Fix a compact parameter set \(K\subset\Omega\), and choose \(\beta_->0\) and \(H<\infty\) such that \(\beta\ge\beta_-\) and \(\left|h\right|\le H\) on \(K\). Throughout the proof \(V=L^3\), and \(I_t=[l_t,(1+\Delta)l_t]\).

Free configurations and the upper bound. Call a pair bad when its distance is less than one and is not in \(I_t\). A configuration is free if it has no bad pair. In a free configuration, join distinct particles whose distance is less than one. Every edge then has length at most \((1+\Delta)l_t\). The endpoints of a path of two edges are therefore at distance at most \(2(1+\Delta)l_t<1\), and hence are joined. Each connected component is consequently a clique with all pair distances in \(I_t\). By 2, its size is at most four. Choosing one representative from every component gives a set with pairwise distances at least one, so the component count is at most \(M_L\).

Consider the grand integral over free configurations with their energy discarded: \[\Xi_{L,\mathrm{free}}(h) =\sum_{n\ge0}\frac{\mathrm e^{thn}}{n!} \int_{\Lambda_L^n}\mathbf 1_{\mathrm{free}}(x_1,\ldots,x_n) \,\mathrm dx_1\cdots\,\mathrm dx_n.\] For an ordered list of component sizes \(m_1,\ldots,m_k\), with \(m_i\in\{1,2,3,4\}\) and \(\sum_i m_i=n\), the number of ordered partitions of the particle labels into these blocks is \(n!/\prod_i m_i!\). On summing over all ordered size lists, division by \(k!\) counts each unordered family of label blocks once. This remains true when sizes repeat: the label blocks themselves are distinct and have exactly \(k!\) orderings.

In a block of size \(m\), choose its least label as the root. Its position has volume at most \(V\), and each of the other \(m-1\) particles lies in a ball of volume \(C l_t^3\) about it, where \(C=4\pi(1+\Delta)^3/3\). After cancellation of the global factor \(1/n!\), the block therefore contributes at most \(V(C l_t^3)^{m-1}/m!\). Discarding all further restrictions within and between components and summing their sizes gives \[ \Xi_{L,\mathrm{free}}(h) \le\sum_{k=0}^{M_L}\frac{(V\Theta_t(h))^k}{k!}, \qquad \Theta_t(h)=\sum_{m=1}^4 \frac{\mathrm e^{thm}(C l_t^3)^{m-1}}{m!}. \tag{51}\]

To pass from free configurations to all configurations, let \(B(X)\) be the set of labels incident to a bad pair. There are at least \(\left|B(X)\right|/2\) bad pairs, and each has core energy exactly \(t^2\). All other core terms are nonnegative, and \(X\setminus B(X)\) is free. Thus the following pointwise bound holds: \[ \mathrm e^{-\beta U^{\psi_t}(X)} \le\sum_{J\subset\{1,\ldots,n\}} \mathrm e^{-\beta t^2\left|J\right|/2} \mathbf 1_{\mathrm{free}}(X\setminus J). \tag{52}\] Indeed, the summand \(J=B(X)\) already bounds the left side. For \(r=\left|J\right|\) and \(m=n-r\), the label factor is \(\binom{m+r}{r}/(m+r)!=1/(r!m!)\). The removed positions integrate to \(V^r\). Tonelli’s theorem and the grand sum now give the exact factorization of the upper bound \[ \Xi_L^{\psi_t}(\beta,h) \le \exp\!\left(V\mathrm e^{th-\beta t^2/2}\right) \Xi_{L,\mathrm{free}}(h). \tag{53}\] Shell endpoints are included in \(I_t\), although their energy may be positive. Pairs at distance exactly one are not edges of the component graph. Both conventions merely discard nonnegative energy in the upper bound, so these arguments hold pointwise, including at the endpoints.

Set \(a_t(h)=t^{-1}\log\Theta_t(h)\). The four exponential rates in \(\Theta_t\) are \[h,\qquad 2h-3,\qquad 3h-6,\qquad 4h-9.\] The middle two are convex combinations of the first and last. Their positive prefactors in (51) are fixed, so \[ a_t(h)=\max\{h,4h-9\}+O(t^{-1}), \tag{54}\] uniformly for \(\left|h\right|\le H\). For \(k\ge1\), the inequality \(k!\ge(k/\mathrm e)^k\), with \(x=k/V\), gives \[\frac1{tV}\log\frac{(V\Theta_t)^k}{k!} \le x a_t(h)+\frac{x(1-\log x)}t \le \frac{M_L}{V}\max\{0,a_t(h)\}+\frac1t,\] since \(x(1-\log x)\le1\) for \(x>0\). The same final bound covers \(k=0\). There are only \(M_L+1=O(1+V)\) terms in (51). First letting \(L\to\infty\), with \(t\) fixed, and using [f:packing,f:thermodynamics], we obtain \[ q^{(t)}(\beta,h) \le p\max\{0,a_t(h)\}+\frac1t +\frac1t\mathrm e^{th-\beta t^2/2}. \tag{55}\] The final term is at most \(t^{-1}\mathrm e^{tH-\beta_-t^2/2}\) on \(K\). Thus (54) proves the required uniform upper bound.

Zero-energy configurations and the lower bound. Fix \(s>0\) and \(0<\nu\le1\). Dilate a maximizing packing in a side-\(s\) cube by \(1+\nu\), and center it in a cube of side \[P=(s+2)(1+\nu).\] Repeat this pattern in adjacent period cubes. Centers in one period have separation at least \(1+\nu\); centers in different periods have at least this separation as well. Every center has distance at least \(1+\nu\) from the boundary of its period cube. A cube of side \(mP\) contains exactly \(k=m^3M_s\) centers.

Choose \(r\in(0,\nu/16)\) and let \(b_0=4\pi r^3/3\). At every center \(z\), put the root particle in \(B(z,r)\). There are two constructions, with cluster size \(j=1\) or \(j=4\). For \(j=4\), choose vectors \(a_2,a_3,a_4\in\mathbb R^3\) such that \(0,a_2,a_3,a_4\) are the vertices of a regular tetrahedron of side \(1+\Delta/2\). Fix \(\delta\in(0,\Delta/24)\) and require \[x_i-x_1\in B(l_t a_i,\delta l_t),\qquad i=2,3,4.\] A root-to-vertex distance differs from \((1+\Delta/2)l_t\) by at most \(\delta l_t\), and another edge differs by at most \(2\delta l_t\). All six distances consequently lie strictly inside \[((1+\Delta/3)l_t,(1+2\Delta/3)l_t),\] where the core vanishes. Write \(b_1=4\pi\delta^3/3>0\). The change from positions to the root and the three relative vectors has Jacobian one. For one prescribed label ordering the integration volume at each center is therefore \(b_0(b_1l_t^3)^3\), with positive relative volume \(b_1^3l_t^9\). For \(j=1\), the corresponding volume is \(b_0\).

Every particle lies within \(r+2l_t\) of its center. For all sufficiently large \(t\), depending on \(\nu,r\) but not on \(m,\beta,h\), these center neighborhoods are disjoint and lie inside their period cubes, and intercluster distances are greater than \(1+\nu/2>1+1/t\). Hence all configurations just described have exactly zero core energy.

The centers are distinguishable. Assign an unordered set of \(j\) particle labels to each of the \(k\) centers, and retain just one deterministic ordering of each assigned set, for example the increasing order. There are \((jk)!/(j!)^k\) such assignments. Different assignments send some label to different disjoint center neighborhoods, so their integration events are disjoint. Possible alternative choices of the root within the same set are never summed. Dividing by the canonical factorial \((jk)!\) gives \[ Z^{\psi_t}_{mP,jk}(\beta) \ge\left(\frac{b_0(b_1l_t^3)^{j-1}}{j!}\right)^k, \qquad j\in\{1,4\}. \tag{56}\] Using this term in the grand sum and first letting \(m\to\infty\), we find \[ q^{(t)}(\beta,h) \ge c_{s,\nu}\left(jh-3(j-1) +\frac1t\log\frac{b_0b_1^{j-1}}{j!}\right), \qquad c_{s,\nu}=\frac{M_s}{((s+2)(1+\nu))^3}. \tag{57}\] The vacuum also gives \(q^{(t)}\ge0\). The constants in (57) have no dependence on \(\beta,h\), and \(c_{s,\nu}\) can be made arbitrarily close to \(p\) by taking \(s\) large and \(\nu\) small. More explicitly, on \(\left|h\right|\le H\) the two functions \(jh-3(j-1)\), \(j=1,4\), have a common finite absolute bound. Given any error tolerance, first choose \(s,\nu\) so that replacing \(c_{s,\nu}\) by \(p\) changes both branches by less than that tolerance, and then take \(t\) large enough to make the two logarithmic terms small. Taking the maximum of these branches and zero preserves the uniform error, even where a nonvacuum branch is negative. Together with (55), this proves (50) on \(K\). ◻

Proof. Apply the uniform-in-\(t\) bound in 4 to the bare core \(\psi_t\), mass \(A\), and parameter compact \(K\). For all sufficiently large \(t\), every active shift lies in one fixed interval \([0,U]\). Enlarge \(U\), if necessary, so that \(4p\le U\).

Write \(d_1=0,d_2=p,d_3=4p\). Completing the square gives the identity \[ F_*(\lambda,u) =\max_{1\le i\le3} \left\{L_i(\lambda)-\frac{\beta A}{2}(u-d_i)^2\right\}. \tag{60}\] Its supremum over \(u\) is \(P_*(\lambda)\), and all its maximizing values belong to \(D_*\). The same identity proves (59).

The shifted points \[K'=\{(\beta,h+\beta Au):(\beta,h)\in K,\ 0\le u\le U\}\] form a compact subset of \(\Omega\). Put \(E_t=\sup_{K'}\left|q^{(t)}-q_*\right|\), which tends to zero by 6. Both transforms can be maximized over \([0,U]\), so \[\sup_K\left|P_1^t-P_*\right|\le E_t.\] If \(u\) is active for \(P_1^t(\lambda)\), approximation of its objective and of the maximum shows \(F_*(\lambda,u)\ge P_*(\lambda)-2E_t\). By (59), \[ \mathop{\mathrm{dist}}(u,D_*) \le 2\sqrt{\frac{E_t}{\beta_-A}}. \tag{61}\] This estimate is uniform on \(K\) and requires no strict inequality between the three affine branches. ◻

Proof. Because \(g\) is an interior point of the nondegenerate triangle, the two numbers \[\eta=\min_{\left\lVert z\right\rVert=1}\max_i\left\langle v_i-g,z\right\rangle>0, \qquad \zeta=\min_{i<j}\mathop{\mathrm{dist}}\bigl(g,[v_i,v_j]\bigr)>0\] are positive. For the first assertion, one may also see positivity directly: if all three inner products were nonpositive, their sum would be zero, so all would vanish, contradicting noncollinearity. Choose \[0<\tau<\min\{p/16,\eta/4,\zeta/2\}, \qquad \gamma=\eta/2.\] For any unit vector \(z\) and any allowed vertices \(w_i\), \[\max_i\left\langle w_i-g,z\right\rangle\ge\eta-\tau>\gamma.\] The support-function characterization of inclusion of compact convex sets proves (63). Every point of the hull of two radius-\(\tau\) balls lies within distance \(\tau\) of the segment joining their centers. Since \(\tau<\zeta\), this also proves (64). Finally, all three planes agree at \(\lambda_*\), and hence for \(z=\lambda-\lambda_*\), \[P_*(\lambda)-P_*(\lambda_*)-\left\langle g,z\right\rangle =\max_i\left\langle v_i-g,z\right\rangle\ge\eta\left\lVert z\right\rVert\ge\gamma\left\lVert z\right\rVert.\] ◻

Proof. Use the uniform optimizer bound \([0,U]\) from 8 with \(K=W_1\). All shifted points \(x=(\beta,h+\beta Au)\) lie in the fixed compact \(K'\) from that proof. Choose one \(\sigma>0\) such that \[K''=\{x+(s,0):x\in K',\ \left|s\right|\le\sigma\}\subset\Omega,\] and set \(E'_t=\sup_{K''}\left|q^{(t)}-q_*\right|\to0\). For every \((e,d)\in\partial q^{(t)}(x)\), its two supporting inequalities at \(x+(\sigma,0)\) and \(x-(\sigma,0)\), together with the independence of \(q_*\) from \(\beta\), imply \[ -\frac{2E'_t}{\sigma}\le e\le\frac{2E'_t}{\sigma}. \tag{69}\] These are fixed steps on an enlarged compact, so the estimate also covers outer parameters on the boundary of \(W_1\).

At a shifted point corresponding to an active \(u\), 5 gives \(d=u\) for every underlying support. By 5, the actual active generators of \(\partial P_1^t(\lambda)\) are precisely the vectors in the union \[G_t(\lambda)= \bigcup_{u\ \mathrm{active}} \left\{(e+Au^2/2,u):(e,u)\in \partial q^{(t)}(\beta,h+\beta Au)\right\}, \qquad \partial P_1^t(\lambda)=\mathop{\mathrm{conv}}G_t(\lambda).\] For a nearest \(d_i\in D_*\), every such generator \(w\) satisfies \[\left\lVert w-v_i\right\rVert \le\frac{2E'_t}{\sigma} +\left(1+\frac A2(U+4p)\right)\left|u-d_i\right|.\] Equations (61) and (69) show that this tends uniformly to zero over all outer parameters, all active optimizers, and all underlying supports. Thus, for all sufficiently large \(t\), \[G_t(\lambda)\subset \partial P_1^t(\lambda)\cap \bigcup_i\overline B(v_i,\tau/2) \qquad(\lambda\in W_1).\] Taking convex hulls proves one inclusion in (68); the other follows from convexity of the subdifferential. Finally, (67) follows from 8. All earlier size conditions are lower bounds on \(t\), so they can be imposed at the same time. ◻

Proof. Fix \(\lambda=(\beta,h)\in W(s-S_*a)\), and choose an active shift \(u\) for \(T_aq\) at \(\lambda\). Let \(x=(\beta,h+\beta au)\). The distance from \(\lambda\) to the boundary of \(W(s)\) is at least \(S_*a\). Since \(\left|x-\lambda\right|\le\beta_{\max}U_*a\), every point \[y,\quad y+(0,\pm ka),\quad x+(0,\pm ka), \qquad \left|y-x\right|\le la,\] lies in \(W(s)\). The surplus \(a\) leaves these test points in the interior of \(W(s)\).

A common density band. For \((e,d)\in\partial P(y)\), the supporting inequality at \(y\), the Lipschitz bound on \(P\), and the bound \(\left\lVert P-q\right\rVert_{W_1}\le\delta\) give \[\begin{align*} \pm ka\,d &\le P(y+(0,\pm ka))-P(y)\\ &\le q(x+(0,\pm ka))-q(x) +2\delta+2C_*la\\ &\le \pm ka\,u+\frac{(ka)^2}{2\beta a} +2\delta+2C_*la. \end{align*}\] The last step is the active-point parabola of 5. Using both signs, (70), and the hypothesis on \(\delta/a\), we obtain \[ \left|d-u\right|\le \frac{k}{2\beta_{\min}}+\frac{2C_*l}{k} +\frac{2\delta}{ka}<\frac p8. \tag{73}\] The same \(u\) works for every \(y\) in this neighborhood.

One slope group near the shifted point. The set in the right side of (71) is nonempty: the full subdifferential is nonempty, and the assumed hull identity identifies it with the convex hull of that set. If a generator belongs to \(\overline B(v_i,r)\), then \[\left|u-d_i\right|<\frac p8+\tau<\frac{3p}{16}.\] Since the distinct \(d_i\) are separated by at least \(p\), at most one index \(i\) can satisfy this condition. It is therefore one common index for the entire neighborhood. By convexity of the ball and the assumed hull identity, \[ \partial P(y)\subset\overline B(v_i,r) \qquad(\left|y-x\right|\le la). \tag{74}\]

Transfer to the physical pressure and its transform. Take \(v\in\partial q(x)\) and a unit vector \(z\in\mathbb R^2\). At \(y=x+la z\), choose any support of \(P\). Using (74) to bound this support and the approximation at \(x,y\), we get \[la\,v\cdot z \le q(y)-q(x) \le P(y)-P(x)+2\delta \le la\,(v_i\cdot z+r)+2\delta.\] Testing every unit \(z\) proves \[\left|v-v_i\right|\le r+\frac{2\delta}{la}.\] By 5, the generating supports of the active branch of \(T_aq\) are obtained by adding \((au^2/2,0)\). They all lie in \(\overline B(v_i,r^+)\) by the definition of \(r^+\) and \(\left|u\right|\le U_*\). The exact convex-hull formula over all active branches gives (72). ◻

Proof. The case \(n=1\) is 7. Each \(q_n\) belongs to the family, and each \(P_n\) has the common Lipschitz bound \(C_*\). Apply 10 inductively with \(P=P_n\), \(q=q_n\), \(a=a_{n+1}\), \(\delta=\delta_n\), \(r=r_n\), and \(s\) equal to the radius of \(W_n\). The first condition in (81) keeps the parameter radius above \(1/8\), while the second gives \(r_n<\tau\). The other hypotheses are (77) and the first condition in (82). The new domain and slope radius are exactly \(W_{n+1}\) and \(r_{n+1}\). ◻

Proof. Minimize \(P_n(\lambda)-g\cdot\lambda\) on the compact ball \(W(1/8)\). For a boundary point, 9 and (83) give \[\begin{align*} &[P_n(\lambda)-g\cdot\lambda] -[P_n(\lambda_*)-g\cdot\lambda_*]\\ &\hspace{2em}> \frac{\gamma}{8}-\frac{2\gamma}{32} =\frac{\gamma}{16}>0. \end{align*}\] Thus a minimizer \(\lambda_n\) is interior, and \(g\in\partial P_n(\lambda_n)\). By 8, this subdifferential is generated by its intersections with the three balls \(\overline B(v_i,r_n)\). The vector \(g\) lies outside the convex hull of any two of the larger balls \(\overline B(v_i,\tau)\). Each of the three intersections must therefore be nonempty. Choose one actual support from each. Their triangle contains \(\overline B(g,\gamma)\) by 9, and their convex hull is contained in \(\partial P_n(\lambda_n)\). ◻

Proof. Since \(\sum_na_n<A+1\) and \(R_n\ge1\), the attraction series converges uniformly. The core and every summand are bounded, continuous, and radial. The common cell estimate of 1 applies to the sum and gives \(U^\Phi\ge-B_0N=-(t^2/2)N\).

Set \(\varepsilon=1/32\). A nonzero summand at radius \(r\) must have \(R_n\ge r\), and then \[r^{3+\varepsilon}R_n^{-3}\le R_n^\varepsilon.\] The core vanishes for \(r\ge1+1/t<2\) and is at most \(t^2\). For every \(r\ge1\), therefore, \[r^{3+\varepsilon}\left|\phi(r)\right| \le 2^{3+\varepsilon}t^2 +t\left\lVert w\right\rVert_\infty\sum_{n:R_n\ge r}a_nR_n^\varepsilon \le C.\] Finally, \[\sum_{n\ge1}a_nR_n^{1/32} =A R_1^{1/32}+\sum_{n\ge2}R_1^{-2^{n-1}/32}<\infty\] by the geometric-series bound following (78). ◻

Proof. The field part of either support inequality says that \(\rho\) supports \(Q(\beta_c,\cdot)\) at \(h_c\). Density duality therefore gives \(H(\beta_c,\rho)=h_c\rho-Q(\beta_c,h_c)\). The full support inequality, with \(s=s_i\), reads \[Q(\beta,h)\ge Q(\beta_c,h_c) +s(\beta-\beta_c)+\rho(h-h_c).\] Rearrange and take the supremum over \(h\). The reverse duality gives \[H(\beta,\rho)\le H(\beta_c,\rho)-s(\beta-\beta_c).\] The function \(H(\,\cdot\,,\rho)\) is finite and concave on \((0,\infty)\), so its one-sided derivatives at an interior point exist and are finite. Dividing the last inequality by positive and negative increments yields the stated ordering. ◻

Proof of 1. Take the potential (86) and the constants in 10. Write \(g=(g_1,\rho_0)\), where \(\rho_0=5p/3>0\), and set \[\eta=\min\{\gamma/2,\rho_0/2\},\qquad I=(\rho_0-\eta,\rho_0+\eta)\subset(0,\infty).\] For every \(\rho\in I\), the two points \[(g_1-\gamma/2,\rho),\qquad(g_1+\gamma/2,\rho)\] lie in \(\overline B(g,\gamma)\), since their squared distances from \(g\) are \((\gamma/2)^2+(\rho-\rho_0)^2<\gamma^2/2\). By (88), both support the same pressure at the same \((\beta_c,h_c)\), independently of \(\rho\). Lemma 11 therefore gives finite one-sided canonical slopes whose difference is at least \(\gamma\) for every \(\rho\in I\). The all-sequence canonical limit of 2 applies in particular to \(N_L=\lfloor\rho L^3\rfloor\) for every \(\beta>0\). The specified free energy equals \[f(\beta,\rho)=\frac{t}{\beta}H_\Phi(\beta,\rho).\] For each \(\rho\in I\), it is finite for every \(\beta>0\). The finite concave function \(H_\Phi(\,\cdot\,,\rho)\) is continuous at \(\beta_c\); hence the common derivative term from \(1/\beta\) cancels in the difference, giving \[f'_-(\beta_c,\rho)-f'_+(\beta_c,\rho) =\frac{t}{\beta_c} \bigl(H'_{\Phi,-}(\beta_c,\rho) -H'_{\Phi,+}(\beta_c,\rho)\bigr) \ge\frac{t\gamma}{\beta_c}>0.\] The potential and the density interval are fixed before the limiting critical parameter is selected, so this one \(\beta_c\) works throughout \(I\). ◻