Proof. Inverting the comparison. We extract a density approximation from the inverse comparison (13). Fix an integer \(q\ge2\) and choose \(h_h>C_{\mathrm{cap}}+1\). At a point in its comparison band, write \[v=|y|^4H_j(y),\qquad t=\left(\frac{|y|^6\mu_j(y)}{k}\right)^{2/3}\ge0.\] Choose \(0<h_l<h_h\) and \(0<\xi<\min\{h_l/16,1\}\), to be decreased below. The cap and the high implication rule out \(t\ge h_h\): taking heights increasing to \(h_h\) would give \(v\ge h_h-\xi>C_{\mathrm{cap}}\). If \(h_l<t<h_h\), heights approaching \(t\) from below and above give \(|v-t|\le\xi\), and hence \(|v_+-t|\le\xi\). If \(t\le h_l\), heights decreasing to \(h_l\) in the low implication give \(v\le h_l+\xi\). This last case includes \(t=h_l\) and allows arbitrarily negative \(v\); both \(t\) and \(v_+\) still lie in \([0,h_l+\xi]\). Consequently \[k\bigl|t^{3/2}-(v_+)^{3/2}\bigr| \le \begin{cases} \dfrac32 k\sqrt{h_h}\,\xi,& h_l<t<h_h,\\[2pt] k(h_l+\xi)^{3/2},& t\le h_l. \end{cases}\] We may therefore choose \(h_l=h_l(q)\) and \(\xi=\xi(q)\) so that the right-hand side is at most \(q^{-8}\). These choices are fixed before the comparison’s smallness threshold for \(s\) is imposed.

Choosing the observation scale. For this fixed \(q\), take \(L_1=q^2\). The admissibility condition gives \[\frac{r_0}{s_\ell} =\varepsilon\bigl(Z_\ell s_\ell^3\bigr)^{-1/3}\longrightarrow0.\] Thus, for all sufficiently large \(\ell\), the largest dyadic radius not exceeding \(s_\ell/q\) has an index \(j<J_\ell\) and satisfies \[\frac{s_\ell}{2q}\le r_j\le\frac{s_\ell}{q}.\] Here \(j<J_\ell\) follows from \(r_{J_\ell}>s_\ell/2\) and \(q\ge2\). Moreover \(4L_1r_j\ge2qs_\ell\), so the comparison covers the band \(r_j\le|y|\le qs_\ell\). The rescaled comparison band thus extends from a radius in \([1/(2q),1/q]\) to \(q\). On its event we have \[ H_j(y)\le C_{\mathrm{cap}}|y|^{-4},\qquad \bigl|\mu_j(y)-k(H_j(y)_+)^{3/2}\bigr| \le q^{-8}|y|^{-6} \quad(r_j\le|y|\le qs_\ell). \tag{23}\] The exceptional probability is at most \(C_qs_\ell^{25}\). The observations and master kernels in (10)–(11) are fixed independently of \(L_1,h_l,h_h,\xi\). Thus this new application of the comparison concerns exactly the same \(\mu_j,H_j\) as the fixed-parameter barrier construction.

Choosing the stages and retained data. Choose strictly increasing stage indices \(\ell_q\) so that, for every \(\ell\ge\ell_q\), all the preceding requirements for that fixed \(q\) hold, including \(C_qs_\ell^{25}\le1/q\). On \(\ell_q\le\ell<\ell_{q+1}\) set \(q_\ell=q\), and choose the deterministic index \(j_\ell\) just described. Then \(q_\ell\to\infty\), (16) holds, and the comparison events have probability tending to one. No bound on the growth of \(C_q\), or on the smallness thresholds as functions of \(q\), is needed.

For each system, first choose the conditional-integral versions of the fields. At each index \(j\), fix an \(\mathcal F_j\)-measurable full-measure set on which all its imported identities, regularity properties, and barrier inequalities hold. Spatial continuity and a countable dense family of distributional tests supply the continuum assertions; the finitely many such sets are fixed before any data are selected. Intersect the comparison event at \(j_\ell\) with its corresponding valid-data set and with \[\left\{\int_{\mathbb R^3}p_{j_\ell}(y)\,\,\mathrm dy\le1\right\}.\] Call the resulting event \(G_\ell\). The intermediate error estimate (14) and Markov’s inequality give \[\mathbb P(G_\ell^c) \le\frac1{q_\ell}+C\bigl(s_\ell^{32}+s_\ell^\gamma\bigr) \longrightarrow0.\] In particular the retained-data assertion is pointwise on specified valid versions, not an assertion left to an exceptional set after a datum is selected.

Rescaling. Each conditional density has mass \(Z_\ell\), and the Coulomb kernel has degree \(-1\), so (17) gives (18) exactly. The comparison (23) gives (19). Since \(f(s^4u)=s^6f(u)\) for \(s>0\), rescaling the imported distributional subsolution inequality gives (20), with the barriers (21). The support scales to the ball of radius \(a_\ell\), and on \(G_\ell\) \[\int_{\mathbb R^3}p_\ell(x)\,\,\mathrm dx =s_\ell^3\int_{\mathbb R^3}p_{j_\ell}(y)\,\,\mathrm dy\le s_\ell^3,\] which proves (22). The finite-system regularity follows from the imported barrier regularity and the positive lower bound on packet widths. For the field remainder, a bounded integrable density has a locally Lipschitz Coulomb potential: near the singularity the kernels \(|z|^{-1}\) and \(|z|^{-2}\) are integrable in three dimensions, while away from it both kernels are bounded. This proves the remaining assertions. ◻

Proof. Newton’s spherical-average identity is \[\frac1{4\pi}\int_{\mathbb S^2}\frac{\,\mathrm d\omega}{|d\omega-z|} =\frac1{\max(d,|z|)}.\] It follows either by direct integration in the angle with \(z\), or by the mean-value property inside and outside the sphere. Applying it to (18) and using the exact equality of the nuclear coefficient and total density mass gives \[ \overline F_\ell(d):= \frac1{4\pi}\int_{\mathbb S^2}F_\ell(d\omega)\,\,\mathrm d\omega =\int_{|z|>d}\left(\frac1d-\frac1{|z|}\right) \sigma_\ell(z)\,\,\mathrm dz\ge0. \tag{26}\] All these integrals are finite for each \(\ell\). No upper bound on \(\lambda_\ell\) is needed.

Fix an annulus \(\alpha\le |x|\le\beta\) with \(0<\alpha<\beta<\infty\). For large \(\ell\), it lies in the band of (19). The cap and (26) imply, for every radius in this annulus, \[ \int_{\mathbb S^2}|F_\ell(d\omega)|\,\,\mathrm d\omega =2\int_{\mathbb S^2}(F_\ell(d\omega))_+\,\,\mathrm d\omega -\int_{\mathbb S^2}F_\ell(d\omega)\,\,\mathrm d\omega \le 8\pi C_{\rm cap}d^{-4}. \tag{27}\] Radial integration therefore bounds the full \(L^1\) norm of \(F_\ell\) on the annulus. Equation (19) also bounds \(\sigma_\ell\) uniformly there.

Here is the local estimate we shall use twice. Take a ball \(Q\) whose closure stays away from zero, and a larger fixed annulus containing its closure. The localized potential \[P_\ell=K*(\sigma_\ell\mathbf 1_Q)\] has uniformly bounded values and first derivatives: this follows from the density bound and the integrability of \(|z|^{-1}\) and \(|z|^{-2}\) on bounded subsets of \(\mathbb R^3\). Since \(\Delta F_\ell=4\pi\sigma_\ell\) on \(Q\), the function \(h_\ell=F_\ell+P_\ell\) is distributionally harmonic there. Its \(L^1(Q)\) norm is bounded by (27) and the bound on \(P_\ell\). On every smaller concentric ball, harmonic interior estimates bound \(h_\ell\) and its gradient by this \(L^1\) norm. For completeness, convolution with a fixed smooth radial averaging kernel reproduces a harmonic function in the interior; differentiating that convolution gives these bounds. The same argument applies to continuous distributionally harmonic functions, by first mollifying on interior sets. Subtracting \(P_\ell\) gives uniform local value and Lipschitz bounds for \(F_\ell\).

A covering by such balls, followed by a diagonal Arzelà–Ascoli argument, gives local uniform precompactness and a locally Lipschitz limit. The error in (19) tends uniformly to zero on every fixed compact annulus, so \(\sigma_\ell\to k(F_+)^{3/2}\) there. Passing to distributions, to the cap, and to the spherical averages proves (24).

To compare with \(U_\ell\) on large spheres, we next strengthen the one-sided cap to an absolute bound for \(F\). For \(D>0\) set \(F^{(D)}(x)=D^4F(Dx)\). The equation, cap, and nonnegative spherical means in (24) are invariant under this dilation. In particular, on a fixed annulus the same argument as in (27) bounds the \(L^1\) norm of every \(F^{(D)}\), while the source \(k(F^{(D)}_+)^{3/2}\) is uniformly bounded by the cap. The local estimate just proved is therefore uniform in \(D\). It yields both precompactness of the dilates and a uniform bound \(|F^{(D)}(\omega)|\le C_*\) for \(|\omega|=1\). Writing \(x=D\omega\) proves (25). ◻

Proof. We compare the barrier and the screened field on a fixed ball after subtracting a Coulomb correction that vanishes away from the origin. Work along the subsequence converging to \(F\). Fix \(S>0\); eventually \(a_\ell<S<q_\ell\). Define \[ e_\ell(x)=q_\ell^{-8}|x|^{-6} \mathbf 1_{\{a_\ell\le|x|\le S\}}, \qquad L_\ell=K*(p_\ell+e_\ell)\ge0. \tag{29}\] Subtracting the Poisson equation for \(F_\ell\) from (20) cancels the nucleus and the entire inner density source. Since \(4\pi\sigma_\ell\le f(F_\ell)+4\pi e_\ell\) on \(a_\ell\le|x|<S\) and \(\Delta(-L_\ell)=4\pi(p_\ell+e_\ell)\), the result is \[ \Delta(U_\ell-F_\ell-L_\ell) \ge \mathbf 1_{\{|x|\ge a_\ell\}} \bigl(f(U_\ell)-f(F_\ell)\bigr) \quad\hbox{in }B(0,S). \tag{30}\] This is an inequality on the whole ball, including the origin and the sphere \(|x|=a_\ell\); no restriction or differentiation across an interface has been made.

Put \[c_S=(C_{\rm inv}+C_*+1)S^{-4}.\] Uniform convergence on \(|x|=S\), the bound (25), the upper barrier in (21), and \(L_\ell\ge0\) give, for large \(\ell\), \[U_\ell-F_\ell-L_\ell<c_S\qquad\hbox{on }|x|=S.\] We justify the comparison throughout the ball. The difference \[W_\ell=U_\ell-F_\ell-L_\ell-c_S =(U_\ell-\lambda_\ell K)+K*\sigma_\ell-L_\ell-c_S\] extends continuously and locally in \(H^1\) across zero. Indeed, the first term is locally Lipschitz by hypothesis. The other potentials have bounded integrable densities for each fixed \(\ell\), and thus locally bounded gradients. In the case of \(L_\ell\) the density is also compactly supported. These bounds need not be uniform in \(\ell\).

The strict boundary inequality implies that \((W_\ell)_+\) has compact support in \(B(0,S)\) and belongs to \(H^1_0(B(0,S))\). It may therefore be used in (30), by approximation with nonnegative smooth tests in \(H^1\). The right side is bounded on this fixed ball for this fixed \(\ell\), since its reaction is cut off at the positive radius \(a_\ell\). On the positive set of \(W_\ell\) outside that radius, we have \(U_\ell>F_\ell\), because \(L_\ell+c_S>0\). Monotonicity of \(f\) consequently gives \[-\int_{B(0,S)}|\nabla(W_\ell)_+|^2\,\,\mathrm dx \ge\int_{B(0,S)}\mathbf 1_{\{|x|\ge a_\ell\}} \bigl(f(U_\ell)-f(F_\ell)\bigr)(W_\ell)_+\,\,\mathrm dx \ge0.\] It follows that \((W_\ell)_+=0\), or equivalently \[ U_\ell\le F_\ell+L_\ell+c_S\qquad\hbox{in }B(0,S). \tag{31}\]

For each fixed \(x\) with \(0<|x|<S\), we next show that \(L_\ell(x)\to0\). For large \(\ell\), the support and mass bound in (22) imply \[(K*p_\ell)(x)\le \frac{2s_\ell^3}{|x|}\longrightarrow0.\] The other source has small total mass: \[\int e_\ell\,\,\mathrm dx =\frac{4\pi}{3}q_\ell^{-8}(a_\ell^{-3}-S^{-3}) \le\frac{32\pi}{3}q_\ell^{-5}\longrightarrow0.\] On the fixed ball \(B(x,|x|/4)\) its density is at most \(C_xq_\ell^{-8}\). Integrability of the Coulomb kernel bounds its potential contribution there by \(C_xq_\ell^{-8}\); on the complement the kernel is at most \(4/|x|\), so the contribution is at most \((4/|x|)\int e_\ell\). Hence \(L_\ell(x)\to0\).

Now (31) and the lower barrier give \[F(x)+c_S\ge B|x|^{-4}.\] The argument holds for every fixed \(S>|x|\) along the same convergent subsequence. Letting \(S\to\infty\) proves the lower bound in (28); the upper bound was already known. ◻

Proof. The local potential and harmonic estimates in Lemma 6 apply uniformly to the dilates \(F^{(D)}(x)=D^4F(Dx)\): their values and their sources are bounded on every fixed compact annulus. Thus the dilates are locally uniformly precompact, with locally Lipschitz limits satisfying the same equation.

Write \[b=\sup_{x\ne0}|x|^4F(x),\qquad h=\inf_{x\ne0}|x|^4F(x), \qquad 0<h\le b<\infty.\] Choose \(x_n\) with \(|x_n|^4F(x_n)\to b\), and let \(D_n=|x_n|\), \(\omega_n=x_n/|x_n|\). After passing to a subsequence, \(F^{(D_n)}\to v\) locally uniformly and \(\omega_n\to\omega\in\mathbb S^2\). Local equicontinuity gives \[v(x)\le b|x|^{-4}\quad(x\ne0),\qquad v(\omega)=b.\] The equation passes to the limit because its reaction is continuous and locally uniformly bounded. Since \(\Delta |x|^{-4}=12|x|^{-6}\) in three dimensions, this contact implies \[ 4\pi k b^{3/2}\le12b. \tag{33}\] Here is a weak proof of the contact assertion. If the inequality failed, the continuous function \(g=v-b|x|^{-4}\) would satisfy \(g\le0\), \(g(\omega)=0\), and \(\Delta g\ge\delta>0\) on a sufficiently small ball about \(\omega\). For \(0<\eta<\delta/6\), the function \(g-\eta|x-\omega|^2\) is weakly subharmonic there, equals zero at the center, and is strictly negative on the boundary. Choose a level strictly between its boundary maximum and zero. Testing its Laplacian with the positive part above that level, as in the proof of Lemma 7, contradicts the value at the center. This proves (33) without a classical second derivative assumption.

Dilating instead along points approaching the infimum gives a limit \(v\ge h|x|^{-4}\) with equality at a unit point. If \(4\pi k h^{3/2}<12h\), the same argument applies to \(h|x|^{-4}-v\), whose Laplacian is then strictly positive near its zero maximum. Consequently \[4\pi k h^{3/2}\ge12h.\] Together with (33), this gives \(b\le A_*\le h\). Since \(h\le b\), both extrema equal \(A_*\), proving (32). The argument allows \(D_n\) to tend to zero, to infinity, or to neither; no extremum of the original field had to be attained. ◻

Proof. Lemma 6 gives a locally uniformly convergent subsequence of every subsequence. Its limit satisfies the positive bounds of Lemma 7, so Lemma 8 identifies it as \(A_*|x|^{-4}\). Uniqueness of every subsequential limit proves convergence of the full sequence. The density convergence follows from (19). ◻

Proof. If either assertion failed, there would be a positive discrepancy along a subsequence and a choice of one datum from each corresponding \(G_\ell\) realizing at least half that discrepancy. Those selected fields satisfy all the same deterministic hypotheses, contradicting Proposition 9. This argument uses no measurable selection: uniformity is a deterministic assertion about all the retained data, after the conditional versions have been fixed. ◻

Proof. Abbreviate \(s=s_\ell\) and \(r_0=\varepsilon Z_\ell^{-1/3}\), and set \(\delta_s=c_1s^w\to0\). If a packet centered at \(z\) has \(|z|\ge r_0\), then its width obeys \(t_z\le\delta_s|z|\). Consequently \[ \mathcal K_z(y)\ne0 \quad\Longrightarrow\quad (1-\delta_s)|z|\le |y|\le(1+\delta_s)|z|. \tag{37}\] This estimate holds also for arbitrarily distant centers. If instead \(|z|<r_0\), its packet is contained in the ball of radius \[r_0+c_1r_0^{1+w}=o(s),\] because \(r_0/s=\varepsilon(Z_\ell s_\ell^3)^{-1/3}\to0\). Such packets therefore cannot meet any fixed annulus \(s\mathcal A(a,b)\) for large \(\ell\). For the remaining packets, (37) shows that a packet meeting this annulus has center in \(s\mathcal A(a/2,2b)\) once \(\delta_s\) is small. Each packet has mass one, proving (35).

The same inequalities show that every packet meeting the contracted annulus \(s\mathcal A(a+\eta,b-\eta)\) has its center in \(s\mathcal A(a,b)\). Conversely, every packet whose center lies in \(s\mathcal A(a,b)\) is entirely contained in the enlarged annulus \(s\mathcal A(a-\eta,b+\eta)\). For example, \(\delta_s<\eta/b\) suffices for these inclusions once the clamped packets have been excluded. Summing the mass-one packets proves both inequalities in (36). ◻

Proof. Put \(Y_\ell=\int_{\mathcal A(a,b)}\sigma_\ell\) and \(X_\ell=N_\ell(a/2,2b)\). Equations (34) and (35), followed by conditional Jensen, give \[0\le Y_\ell\le s_\ell^3\mathbb E[X_\ell\mid\mathcal F_{j_\ell}], \qquad \|Y_\ell\|_2\le s_\ell^3\|X_\ell\|_2.\] The annular count bound (9), with the common offset \(D_0\) and \(s_\ell<1\), therefore yields \[ \|Y_\ell\|_2 \le C\bigl(1+\sqrt{D_0}\,s_\ell^{7/2}\bigr)\le C'. \tag{40}\] In particular, for the actual complement of the retained event, \[\mathbb E\bigl[Y_\ell\mathbf 1_{G_\ell^c}\bigr] \le C'\mathbb P(G_\ell^c)^{1/2}\longrightarrow0.\] No independence of \(G_\ell\) from the conditional fields is required. On \(G_\ell\), Corollary 10 implies that \(Y_\ell\) converges to \(\int_{\mathcal A(a,b)}\rho_*\) uniformly over all retained data. Since \(\mathbb P(G_\ell)\to1\), this proves (38).

Taking expectations in (34) gives \(\mathbb EY_\ell=s_\ell^3\mathbb EP_\ell(a,b)\). Apply (38) to the contracted and enlarged annuli in (36). It follows that \[\begin{align*} \int_{\mathcal A(a+\eta,b-\eta)}\rho_* &\le \liminf_{\ell\to\infty}s_\ell^3\mathbb EN_\ell(a,b)\\ &\le \limsup_{\ell\to\infty}s_\ell^3\mathbb EN_\ell(a,b) \le \int_{\mathcal A(a-\eta,b+\eta)}\rho_*. \end{align*}\] Letting \(\eta\downarrow0\) proves (39), because \(\rho_*\) is continuous and integrable on a neighborhood of the fixed closed annulus. ◻

Proof. For fixed \(M\), eventually \(Ms_\ell<1\). Cover the region between \(Ms_\ell\) and \(1\) by shells \(u<|x|<2u\), with \(u=2^pMs_\ell<1\) and \(p=0,1,\ldots\). The annular bound (9), applied with the fixed shape \((1,2)\), gives \[\mathbb E\#\{i:u<|x_i|<2u\} \le C\bigl(u^{-3}+\sqrt{D_0u}\bigr)\le C_Du^{-3} \qquad(0<u<1).\] The expected raw mass outside radius \(1\) is bounded by the other estimate in (9). Summing the geometric series, and allowing the last shell to overlap this exterior region, gives \[s_\ell^3\mathbb E\#\{i:|x_i|>Ms_\ell\} \le C_D M^{-3}\sum_{p=0}^{\infty}2^{-3p}+Cs_\ell^3.\] The constants depend only on the fixed count estimates and \(D_0\); they do not depend on \(M\). Taking the upper limit proves the claim. ◻

Proof. Fix \(M>a\). Lemma 12 gives convergence on \(\mathcal A(a,M)\), whereas Lemma 13 bounds the remaining exterior mass after rescaling by \(CM^{-3}\). Thus, with \(T_\ell(r)=\int_{|y|>r}\rho_{\Psi_\ell}(y)\,\mathrm dy\), \[\begin{align*} \liminf_{\ell\to\infty}s_\ell^3T_\ell(as_\ell) &\ge \int_{\mathcal A(a,M)}\rho_*,\\ \limsup_{\ell\to\infty}s_\ell^3T_\ell(as_\ell) &\le \int_{\mathcal A(a,M)}\rho_*+CM^{-3}. \end{align*}\] Let \(M\to\infty\). The limiting density is integrable outside every positive radius, and \[\int_{|x|>a}\rho_*(x)\,\mathrm dx =\frac{4\pi kA_*^{3/2}}{3a^3} =\frac{4A_*}{a^3} =\frac{b_{\mathrm{TF}}^3}{a^3},\] using \(4\pi kA_*^{1/2}=12\) and \(4A_*=81\pi^2/2\). This proves (42). ◻

Proof of Theorem 1. For each sufficiently large integer \(m\), put \(s=m^{-1/3}\) and choose a finite integer \[Z_{\min}(m)\ge \max\{m+1,\,Z_{\mathrm{bar}}(m^{-1/3}),\,m^{4/3}\}.\] The barrier threshold is uniform over normalized ground states. Consequently every sequence with \(m\to\infty\) and \(Z\ge Z_{\min}(m)\) is admissible: in particular, \(Zs^3\ge m^{1/3}\to\infty\).

Fix \(0<a_-<b_{\mathrm{TF}}<a_+\). We claim that, for all sufficiently large integers \(m\), all integers \(Z\ge Z_{\min}(m)\), and every normalized neutral ground state \(\Psi\) of charge \(Z\), \[ \int_{|y|>a_-m^{-1/3}}\rho_\Psi(y)\,\mathrm dy>m, \qquad \int_{|y|>a_+m^{-1/3}}\rho_\Psi(y)\,\mathrm dy<m. \tag{43}\] Otherwise choose strictly increasing violating integers \(m_\ell\), charges \(Z_\ell\ge Z_{\min}(m_\ell)\), and corresponding ground states \(\Psi_\ell\). With \(s_\ell=m_\ell^{-1/3}\) this is an admissible sequence. Theorem 14, applied separately at \(a_-\) and \(a_+\), gives normalized tail limits \(b_{\mathrm{TF}}^3/a_-^3>1\) and \(b_{\mathrm{TF}}^3/a_+^3<1\). Hence both inequalities in (43) eventually hold on this sequence, a contradiction. This argument includes arbitrary choices of the ground states at the selected charges.

The tail mass is nonincreasing in its radius and tends to zero, so the set defining \(R_m(\Psi)\) is nonempty. The first inequality in (43) excludes every radius at most \(a_-m^{-1/3}\) from that set; the second places \(a_+m^{-1/3}\) in the set. Thus \[ a_-\le m^{1/3}R_m(\Psi)\le a_+ \tag{44}\] for all the states and charges just specified.

Now fix any family \((\Psi_Z)_{Z\ge1}\) of normalized neutral ground states. For each fixed sufficiently large integer \(m\), removing the finite prefix \(Z<Z_{\min}(m)\) changes neither the inner upper limit nor the inner lower limit. Both \[m^{1/3}\limsup_{\substack{Z\to\infty\\Z>m}}R_m(\Psi_Z) \quad\hbox{and}\quad m^{1/3}\liminf_{\substack{Z\to\infty\\Z>m}}R_m(\Psi_Z)\] therefore lie between \(a_-\) and \(a_+\). Finally let \(m\to\infty\) and squeeze with arbitrary \(a_-<b_{\mathrm{TF}}<a_+\). This proves both asserted limits. No convergence in \(Z\) at fixed \(m\) is used. ◻