The theorem and its normalization

In atomic units, let \(Z\) be a positive integer and let \[ H_{Z,N} =\sum_{i=1}^N\left(-\frac12\Delta_i-\frac{Z}{|x_i|}\right) +\sum_{1\le i<l\le N}\frac1{|x_i-x_l|} \tag{1}\] act on \(\bigwedge^N L^2(\mathbb R^3;\mathbb C^2)\). Write \(E_Z(N)=\inf\sigma(H_{Z,N})\) and \(E_Z(0)=0\). For integers \(Z>m\ge1\), define \[I_m(Z)=E_Z(Z-m)-E_Z(Z).\] The kinetic coefficient of the corresponding Thomas–Fermi functional is \(T=\frac3{10}(3\pi^2)^{2/3}\). More explicitly, put \[\mathcal T_Z(\rho) =T\int_{\mathbb R^3}\rho^{5/3} -Z\int_{\mathbb R^3}\frac{\rho(x)}{|x|}\,dx +\frac12\iint_{\mathbb R^3\times\mathbb R^3} \frac{\rho(x)\rho(y)}{|x-y|}\,dx\,dy,\] and let \(E_Z^{\mathrm{TF}}(N)\) be its infimum over nonnegative densities of mass \(N\) and finite displayed terms. Set \(I_m^{\mathrm{TF}}(Z)=E_Z^{\mathrm{TF}}(Z-m)-E_Z^{\mathrm{TF}}(Z)\). The constant in the conjecture is characterized by the classical Thomas–Fermi weak-ionization law (Bénilan and Brezis 2004, sec. 6, Corollary 5): \[ \lim_{Z\to\infty}I_m^{\mathrm{TF}}(Z)=a_{\mathrm{TF}}m^{7/3},\qquad m>0. \tag{2}\] Our identification of the coefficient in 6 recovers this characterization in the present normalization.

The limits in [eq:iterated-upper,eq:iterated-lower] are precisely the energy part of the generalized ionization conjecture formulated by Solovej (Solovej 2016, sec. 3, Equation (1)). The joint assertion [eq:joint-limit] strengthens that order of limits. For historical comparison, Solovej established uniform first-ionization upper bounds and sharp generalized radius estimates in unrestricted Hartree–Fock theory (Solovej 2003, Theorems 3.8 and 1.5); the generalized conjecture and its Hartree–Fock history are discussed in (Solovej 2016, sec. 3). Here the energy is the exact spectral infimum of [eq:hamiltonian], with no restriction to Slater determinants.

For this same full two-spin Coulomb model, the companion (OpenAI 2026, Corollary 1.3) also proves \(0<c\le E_Z(Z-1)-E_Z(Z)\le C<\infty\) uniformly for every integer \(Z\ge1\). This first-ionization bound does not assert convergence at fixed \(m\) and is distinct from the large-\(m\) limits above.

A price per electron

The proof first allows the particle number to vary. For fixed \(Z\), abbreviate \(E_n=E_Z(n)\), and introduce the unconstrained energy and the priced minimum \[E=\inf_{n\ge0}E_n,\qquad P(\lambda)=\min_{n\ge0}\bigl(E_n+\lambda n\bigr),\qquad\lambda>0.\] The minimum exists, as verified in 2. Its advantage is local: replacing the exterior electrons by a trial state need not preserve their number. The price accounts for the change exactly. At the length scale \(s=\lambda^{-1/4}\), the removed charge is of order \(s^{-3}\). The main intermediate result identifies its coefficient for every minimizing sector, including at prices where several sectors minimize.

Analytic precedents and proof structure

The local kinetic comparison combines two classical constructions. Coherent wave packets provide the upper semiclassical energy (Lieb 1981, sec. V.A.1); see also (Solovej 2003, Lemma 8.2). Neumann cube eigenvalues and their lattice count supply a lower comparison with the same leading coefficient (Lieb and Simon 1977, Theorems III.10–III.13). Their remainders are controlled by local particle counts. This is different from the coarse Lieb–Thirring kinetic inequality (Lieb and Thirring 1975), whose positive universal constant is sufficient for preliminary density bounds; the spectral splitting proof used in the companion follows Rumin (Rumin 2011, Theorem 1.4 and Section 3.1).

Screened exterior potentials and outward Thomas–Fermi comparisons are central to Solovej’s Hartree–Fock argument (Solovej 2003, secs. 6, 10, and 12). Our use of conditional states requires a separate transfer from local quantum energies to conditional fields, while accounting for changes in particle number. The eventual singular profile belongs to the classical Sommerfeld comparison theory (Lieb and Simon 1977, sec. IV), including the nonspherical comparisons and chemical-potential estimates of (Solovej 2003, Lemma 4.4, Remark 4.5, Theorem 4.6, and Corollary 4.7). The issue here is to obtain that profile from the correlated quantum state and preserve its positive singularity through conditioning.

We use the localization, screening, and Thomas–Fermi estimates proved in the companion Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms (OpenAI 2026). In particular, its uniform neutral-sector bound \(E_Z(Z)-E\le C\) fixes the zero-price integration constant, and its local-field estimates retain explicit control of an arbitrary energy offset (OpenAI 2026, Proposition 12.3 and Proposition 4.5). The required positive-price conditional comparisons and barrier propagation are proved in [sec:comparison,sec:barriers]. Their offset dependence is essential: a minimizing priced state has energy at most \(E+Cs^{-7}\), so the allowed offset grows as \(s\to0\). A fixed-offset conclusion from the neutral-atom theory cannot by itself yield 1.

2 obtains the rough deficit bound and proves that the rescaled exterior electron mass vanishes beyond a fixed radius. In 3, noisy observations of the electrons permit local conditional comparisons while controlling the cost of conditioning. A Thomas–Fermi trial state in a small ball is compared with the priced many-body state. A fresh spatial cut records the local particles and leaves a core outside the patch, whose conditional screened potential is harmonic on the patch. Matching this core comparison to the noisy-observation law yields simultaneous upper and lower implications between local density and local potential. Two posterior laws occur in this argument; keeping them distinct is necessary for the Coulomb comparison.

The remaining difficulty is to retain the singular inner charge through successive averaging. 4 constructs a positive subsolution at the innermost scale and propagates it through the observation scales. The construction pays for exceptional conditional data with an error density of vanishing expected mass. The weak differential inequality is proved on open neighborhoods of the retained branches before taking their maximum, so no derivative is restricted to an activity interface. 5 then selects conditional data, rescales by \(s\), and obtains a decaying solution of the unit-price Thomas–Fermi equation. The barrier prevents loss of the singularity. A maximum-principle argument identifies its Sommerfeld coefficient and proves uniqueness, which forces the same deficit for every minimizing sector.

Finally, 6 integrates the slopes of the priced minimum and brackets the desired particle number between two minimizing sectors. Only monotonicity of \(E_Z(N)\) is used in this step; convexity in \(N\) is not needed. The analogous Thomas–Fermi calculation identifies the coefficient with [eq:tf-constant]. A sequential argument then gives the specified order of limits in [eq:iterated-upper,eq:iterated-lower].

Imported estimates and their hypotheses

A one-particle square cut consists of nonnegative bounded-gradient Lipschitz functions \(o,c\) with \(o^2+c^2=1\). Conditional on the positions, assign the particles independently to the out and core groups with probabilities \(o^2\) and \(c^2\). The cut data record the out labels, positions, and spins. For a finite sequence of cuts, stage labels are recorded as well. Conditional core states are obtained by the corresponding products of cut factors, so antisymmetry is preserved within each group. This is the localization convention of (OpenAI 2026, Lemma 3.4).

The local estimate needed for the later patch comparison has a quantitative condition on these cuts. Fix, once and for all, \[L=10^5,\qquad A=40,\qquad a_y=|y|/L,\qquad B_y=B(y,a_y) \quad(y\ne0).\] For an unshifted \(\delta\)-state put \[\begin{gathered} m_y=\max(a_y^{-3},1)+\sqrt{\delta a_y},\qquad J_y=V(y)-\sum_{|x_i-y|\ge Aa_y}\frac1{|y-x_i|},\\ P_* =\max\left(1,\sup_{y\ne0} \frac{\mathbb E\bigl[\#\{i:x_i\in B_y\}^2\bigr]}{m_y^2}\right). \end{gathered}\] The number \(P_*\) is finite for each finite system, since the count is at most the sector particle number and \(m_y\ge1\).

The rough charge deficit

A cutoff at one fixed exterior scale

Nested observations and their energy cost

Fix \(0<\varepsilon<1\), to be chosen in 4, and define \[ r_0=\varepsilon Z^{-1/3},\qquad r_j=2^j r_0,\qquad J_* =\max\{j\geq0:r_j\leq s\},\qquad \ell_j=r_j^{1.01}\quad(0\leq j<J_*). \tag{19}\] Eventually \(2r_0\leq s\). At each scale \(j<J_*\), observe the unordered array of points \(x_i+\ell_jU_{ji}\). All coordinates of all \(U_{ji}\) are independent, also independently of the raw configuration, with density \[\zeta(u)=c_\zeta\exp\!\left(-\frac1{1-u^2}\right) \mathbf 1_{\{|u|<1\}}.\] Labels are forgotten separately in each array. Let \(\mathcal F_j\) be the information in arrays \(j,\ldots,J_*-1\), and let \(\mathcal F_{J_*}\) be trivial. These sigma-fields decrease with \(j\).

Use the constants \(L=10^5\), \(A=40\) and \(a_y=|y|/L\) from 4. Fix a real radial function \(g\in C_c^\infty(B(0,1))\) with \(\int g^2=1\), set \(g_b(x)=b^{-3/2}g(x/b)\), and choose \[w=10^{-5},\qquad 0<c_1<(10L)^{-1}.\] Writing \(d_x=|x|\), define the master packets and conditional fields by \[ \begin{aligned} D_x^{\mathrm b}&=\max(d_x,r_0),& t_x&=c_1D_x^{\mathrm b}\min(D_x^{\mathrm b},s)^w,\\ \mathcal K_x(y)&=g_{t_x}(y-x)^2,& \mu_j(y)&=\mathbb E\!\left[\sum_i\mathcal K_{x_i}(y) \,\middle|\,\mathcal F_j\right],\\ H'_j(y)&=V(y)-\lambda-(K*\mu_j)(y). \end{aligned} \tag{20}\] The superscript \(\mathrm b\) indicates the lower clamp, not a power. The width function is globally Lipschitz, with \(\operatorname{Lip}(t)\leq c_1(1+w)s^w\), and \(t_x\geq c_1r_0^{1+w}\). Thus, for each fixed system, \(\mu_j\) is smooth in its evaluation variable, has mass \(n\), and has deterministic bounds on all spatial derivatives. Its Coulomb potential and first derivatives are also continuous and deterministically bounded. These bounds at a fixed system will justify conditional integration; they are not claimed to be uniform in \(Z\).

We use versions obtained by integrating against the conditional law of the raw configuration. Such versions can, for example, be specified by the finite-array likelihoods in (OpenAI 2026, Lemma 5.1). They give, simultaneously in space, \[ \mathbb E[\mu_j\mid\mathcal F_{j+1}]=\mu_{j+1},\qquad \mathbb E[H'_j\mid\mathcal F_{j+1}]=H'_{j+1},\qquad \Delta H'_j=-4\pi\nu+4\pi\mu_j. \tag{21}\] Continuity extends identities first obtained on a countable dense set. In particular, \(H'_j\) is subharmonic off the nucleus.

Accordingly, the unshifted offset of the tilted state is at most \(D_0+C r_j^{-2.02}\log^5(e/p)\), while its shifted energy is at most \[ P(\lambda)+C r_j^{-2.02}\log^5(e/p). \tag{23}\] This distinction is what permits the local comparison below for \(D_0\) up to order \(s^{-7}\).

Statement and preliminary bounds

The two implications compare the field with each positive density height from opposite sides. They will yield an approximate Thomas–Fermi relation in 5, while permitting the field itself to be arbitrarily negative where the density is small.

We prove the proposition in the rest of this section. The proof uses the arbitrary-offset screening estimates from 2 and the energy cost in 8; it does not apply the fixed-offset inverse comparison of (OpenAI 2026, Proposition 10.1) with a growing \(D_0\).

Fix \(j<J_*\) and put \(r=r_j\). On the enlarged band \(r/2\leq d_y\leq8Mr\), write \[a=a_y,\qquad t=t_y,\qquad \tau=t/a.\] Uniformly there, for fixed \(M\), \[ a\asymp_M r,\qquad c_Ma^w\leq\tau\leq Cs^w. \tag{25}\] If \(\mathcal K_x(y)\ne0\), the Lipschitz bound for the widths gives \(|x-y|\leq2t\) and \(t_x/t=1+O(s^w)\). Changing variables at scale \(t\) therefore gives \[ \int\mathcal K_z(y)\,dz=1+O(s^w). \tag{26}\] On the local support, the kernel is bounded by \(Ct^{-3}\) and is Lipschitz, in either its center or its evaluation variable, with constant \(Ct^{-4}\).

A fine patch comparison with the price included

Fix a point \(y\) in the original band and a height \(h_l/2\leq h\leq2h_h\). Suppose that one of the following events, intersected with \(\mathcal C_j\), has probability \(p\geq r^{42}\): \[ \begin{array}{ll} d_y^6\mu_j(y)>kh^{3/2},&d_y^4H'_j(y)<h-\xi/4,\\[2pt] d_y^6\mu_j(y)<kh^{3/2},&d_y^4H'_j(y)>h+\xi/4. \end{array} \tag{31}\] Write \(Q\) for the original joint law conditioned on the event. When spatial cuts are introduced, enlarge this law by fresh labels with their ordinary conditional probabilities given the raw positions. Its raw marginal is the state from 8. Set \[\delta'=Cr^{-2.02}\log^5(e/r^{42}),\qquad \delta=D_0+\delta'.\] Then the raw state has unshifted offset at most \(\delta\), shifted offset at most \(\delta'\), and \[ m_y\leq C_Ma^{-3},\qquad \delta'=o(a^{-6.98}). \tag{32}\] The constants below may depend on the fixed band parameters.

Put \(\beta=a^{0.2}\) and \(b=\beta a\), with \(s\) small enough that \(\beta<1/100\). For each fixed band parameter \(M\), the scales satisfy \[b\ll\ell_j\ll t\ll a.\] Here each ratio of consecutive scales tends to zero uniformly in \(j\), also when \(r/s\to0\), by [eq:packet-scale-ratios]. The fine width \(b\) makes the semiclassical errors small; the broader master width \(t\) absorbs both this smearing and the observation displacement \(\ell_j\), while \(t/a\to0\) permits a local field comparison. The patch construction below has three outputs. Its energy comparison controls the Coulomb discrepancy and a negative-field penalty. The former, together with noisy-array matching, recovers the conditional density; the latter permits the high-density implication to pass to expectation. Finally, the two conditional-expectation identities recover the mean potential. We establish these transfers in that order.

Choose a square cut, using the sine and cosine of a smooth angle, that is full out on \(B(y,t_0)\), supported out in \(B(y,t_0+b)\), and has gradients bounded by \(C/b\), where \(5a\leq t_0\leq6a\) is selected below. Retain the out particles satisfying \(|x_i-y|<t_0-7b\), and call the other out particles deleted. Let \(n_o\) denote the total number of out particles and \(n_{\mathrm{del}}\) the number of deleted particles; thus the retained count is \(n_o-n_{\mathrm{del}}\). Then \(t_0\) may be chosen so that \[ \mathbb E_Q n_o^2\leq C_Ma^{-6},\qquad \mathbb E_Q n_{\mathrm{del}}^2\leq C_M\beta a^{-6}. \tag{33}\] For every candidate \(t_0\), the total out count is bounded by the raw count in \(B(y,7a)\), which lies in a fixed annulus about the nucleus with radius comparable to \(a\). The annular count estimate [eq:annular-count], with [eq:patch-offsets], bounds the expected square of this encompassing count by \(C_Ma^{-6}\), uniformly in \(t_0\). The deleted particles lie in the raw strip \(t_0-7b\leq|x_i-y|\leq t_0+b\). Integrating its squared count over \(t_0\in[5a,6a]\), each ordered pair contributes for a set of lengths at most \(8b\). Averaging selects \(t_0\) for the second bound in [eq:patch-counts], while preserving the first bound.

Recovering the conditional density from the patch

By [eq:patch-euler], the patch field satisfies \(\Delta W=4\pi kW_+^{3/2}\) on \(B(y,4a)\). We will use the bounds \[ W\leq Ca^{-4}\text{ on }B(y,3a),\qquad \sup_{B(y,2t)}|W-W(y)| \leq C\tau\bigl(a^{-4}+W(y)_-\bigr). \tag{42}\] For the first, compare on the ball of radius \(R=4a\) with \[U(x)=\frac{CR^4}{(R^2-|x-y|^2)^4}.\] For large universal \(C\), \(\Delta U\leq4\pi kU^{3/2}\); it diverges at the boundary. Testing with the positive part of \(W-U\) proves the bound. For the second, the Coulomb potential of \(\rho\mathbf 1_{B(y,2a)}\) is \(O(a^{-4})\) with gradient \(O(a^{-5})\) on that ball, by the first bound and direct integration of the Coulomb kernel. Adding this potential to \(W\) leaves a harmonic function bounded above by \(Ca^{-4}\). Apply the interior gradient estimate to its nonnegative harmonic distance from this upper bound. Its center value is at most \(Ca^{-4}+W(y)_-\), which proves the oscillation estimate. These comparisons are classical or weak positive-part tests: compact \(L^{5/3}\) densities have continuous potentials, so \(W\) is continuous.

Define the local averaging operator \[\mathcal R f(y)=\int\mathcal K_z(y)f(z)\,dz.\] On every compatible path under \(Q\), including its fresh cut labels, \[ |\mathcal R\sigma(y)-\mu_j(y)| \leq C_Mr^{-3}(\ell_j+b)t^{-4}=o(a^{-6}). \tag{43}\] Here is the reason this comparison uses the original observation posterior correctly. Match any compatible raw configuration to the \(j\)th observed multiset. Each matched displacement is at most \(\sqrt3\ell_j\), so the raw kernel sum is within \(C_Mr^{-3}\ell_jt^{-4}\) of the sum evaluated at that multiset. The same is true of its posterior average \(\mu_j\) at these data. Only \(C_Mr^{-3}\) entries can contribute, by \(\mathcal C_j\). All relevant raw centers lie in the fully retained region of the fresh cut, since \(t/a\to0\), \(b/a\to0\), and \(\ell_j/a\to0\). Replacing each such center by its fine smear changes the sum by at most \(C_Mr^{-3}bt^{-4}\). This proves [eq:noisy-array-sandwich] without identifying the two posterior laws. The ratios of these errors to \(a^{-6}\) are bounded by \[C_M\bigl(a^{.01-4w}+a^{.2-4w}\bigr)=o(1).\]

Let \[G=\{D(\sigma-\rho)\leq a^{-7+.01}\}.\] By [eq:patch-comparison], \(Q(G^c)\leq C_Ma^{.01}\). The function \(z\mapsto\mathcal K_z(y)\) is a compactly supported Lipschitz test with gradient norm at most \(Ct^{-5/2}\). Coulomb duality (OpenAI 2026, Lemma 3.1) therefore gives, on \(G\), \[ |\mathcal R\rho(y)-\mathcal R\sigma(y)| \leq Ca^{-7/2+.005}t^{-5/2}=o(a^{-6}). \tag{44}\] Indeed the relative gain is at least \(a^{.005-(5/2)w}\). All the small errors here are uniform for fixed parameters along [eq:scaling-regime], including scales with arbitrarily small \(r/s\).

On \(G\), the high-density antecedent in [eq:fixed-point-failures] forces \[ W(y)\geq(h-\xi/16)d_y^{-4}. \tag{45}\] Otherwise [eq:patch-field-oscillation] would imply \(W(z)\leq(h-\xi/32)d_y^{-4}\) on the support of the averaging kernel. This remains true for arbitrarily negative \(W(y)\): its upper estimate contains the nonpositive term \(-(1-C\tau)W(y)_-\). The Euler relation, [eq:packet-approximate-mass], and [eq:noisy-array-sandwich,eq:local-coulomb-duality] would then contradict the high-density antecedent. In the same way the low-density antecedent forces, on \(G\), \[ W(y)\leq(h+\xi/16)d_y^{-4}. \tag{46}\] The upper bound in [eq:patch-field-oscillation] lets [eq:low-patch-field] pass to expectation with an \(o(a^{-4})\) error.

The high case needs a bound on the negative field on \(G^c\). The high antecedent and [eq:noisy-array-sandwich] hold on all paths under \(Q\), and give \[\int_{B(y,2t)}\sigma\geq ca^{-6}t^3.\] Using [eq:patch-field-oscillation] on this ball and absorbing \(C\tau W(y)_-\) yields \[ W(y)_-\leq C(a^{-6}t^3)^{-1}\int W_-\sigma+C\tau a^{-4}. \tag{47}\] Consequently [eq:patch-comparison] implies \[\mathbb E_Q[\mathbf 1_{G^c}W(y)_-]=o(a^{-4}),\] since the ratio of its bound to \(a^{-4}\) is at most \(C_Ma^{.02}\tau^{-3}+C\tau \leq C_Ma^{.02-3w}+Cs^w=o(1)\). Thus [eq:high-patch-field] also passes to expectation with an \(o(a^{-4})\) error. This step explicitly controls the negative part; no lower cap on the conditional field is assumed.

Matching the potentials of the two conditional laws

We adapt the averaged field transfer of (OpenAI 2026, Lemma 5.3) to the present priced fields. The preceding patch estimates supply its required small errors; we give the two-law decomposition and estimates explicitly. We prove \[ \left|\mathbb E_Q\bigl(H'_j-W\bigr)(y)\right|=o(a^{-4}). \tag{48}\] Write \(P^{\mathrm{raw}}\) and \(P^{\mathrm{ms}}\) for the raw and master-smeared electron potentials at \(y\), and \(P^{\mathrm{ret}},P^{\mathrm{del}}\) for the raw potentials of retained and deleted out particles. Because the event defining \(Q\) is in \(\mathcal F_j\), the original observation tower gives \(\mathbb E_Q(K*\mu_j)(y)=\mathbb E_QP^{\mathrm{ms}}\). The core in \(W\) instead uses the fresh-cut tower in the tilted raw law. Combining these two identities, and cancelling both nucleus and price, expresses \(\mathbb E_Q(H'_j-W)(y)\) as the expectation of \[ (P^{\mathrm{raw}}-P^{\mathrm{ms}})-P^{\mathrm{del}} -(P^{\mathrm{ret}}-K*\sigma)(y)+K*(\rho-\sigma)(y). \tag{49}\]

By Hölder’s inequality and the ordinary density bound in [eq:patch-comparison], for \(0<u\leq4a\) the expected raw potential from \(B(y,u)\) is at most \[ \int_{B(y,u)}\frac{\rho_{\mathrm{tilt}}(z)}{|z-y|}\,dz \leq C_Ma^{-4}(u/a)^{1/5}. \tag{50}\] Radial smearing never increases the potential of a point charge and leaves it unchanged outside the smear radius. The master-smearing difference in [eq:two-posterior-potential-identity] can therefore come only from centers within \(2t\) of \(y\); the fine-smearing difference can come only from centers within \(\sqrt3b\). [eq:raw-short-range-potential] bounds both by \(o(a^{-4})\). Deleted centers have distance comparable to \(a\), so [eq:patch-counts] gives \(\mathbb E_QP^{\mathrm{del}}\leq C_Ma^{-4}\sqrt\beta\).

For the remaining signed potential, truncate the kernel at height \(t^{-1}\). The test \(z\mapsto\min(|z-y|^{-1},t^{-1})\) has homogeneous Sobolev gradient norm \(O(t^{-1/2})\). Coulomb duality, or its compactly supported approximation, and [eq:patch-comparison] give expected absolute truncated error at most \[C_Mt^{-1/2}a^{-7/2+.01}=o(a^{-4});\] the relative gain is \(a^{.01}\tau^{-1/2}\leq C_Ma^{.01-w/2}\). The truncation loss for \(\rho\) is at most \(Ca^{-6}t^2\) by [eq:patch-field-oscillation]. For \(\sigma\), radial smearing and [eq:raw-short-range-potential] bound it by \[C_Ma^{-4}\bigl((t+\sqrt3b)/a\bigr)^{1/5}.\] These also vanish relative to \(a^{-4}\) by [eq:packet-scale-ratios]. This proves [eq:potential-matching].

In the high case, [eq:high-patch-field,eq:negative-patch-control] give \(\mathbb E_QW(y)\geq(h-\xi/16)d_y^{-4}-o(a^{-4})\), whereas the failed comparison in [eq:fixed-point-failures] gives \(\mathbb E_QH'_j(y)\leq(h-\xi/4)d_y^{-4}\). [eq:potential-matching] contradicts these inequalities eventually. The low case follows in the same way from [eq:low-patch-field] and the upper bound on \(W\). We conclude that each fixed-point failure in [eq:fixed-point-failures], intersected with \(\mathcal C_j\), has probability less than \(r^{42}\).

Spatial and height nets

Take a spatial net of mesh at most \(r^2\) in the original band, with \(O_M(r^{-3})\) points, and a fixed finite height net in \([h_l/2,2h_h]\) of mesh at most \(\xi/64\). Exclude the fixed-point failures at all net pairs, the complement of \(\mathcal C_j\), and the enlarged-cap failure. Their total probability is \(O_M(r^{39})+O_M(r^{40})\), in particular at most \(Cr^{25}\). All these events are \(\mathcal F_j\)-measurable.

The normalized density \(d_y^6\mu_j(y)\) changes by at most \(C_Mr^{1-4w}=o(1)\) between a point and its nearest net point, by [eq:conditional-density-bounds]. For the field, work on balls of radius a small fixed multiple of \(r\) within the enlarged cap band. Adding the Coulomb potential of the local source \(\mu_j\) makes \(H'_j\) harmonic there and bounds that harmonic function above by \(C_Mr^{-4-3w}\), using [eq:conditional-density-bounds,eq:enlarged-field-cap]. The positive-harmonic gradient argument used in [eq:patch-field-oscillation] shows that \(v(y)=d_y^4H'_j(y)\) varies over a displacement \(O(r^2)\) by at most \[ C_Mr^{1-3w}+C_Mr\,v(y)_-. \tag{51}\]

Suppose a high-density failure with tolerance \(\xi\) occurs at \((y,h)\). If \(v(y)\geq0\), [eq:net-field-variation] puts the nearby net value below \(h-3\xi/4\) for small \(s\). If \(v(y)<0\), the same upper estimate is bounded by \((1-C_Mr)v(y)+C_Mr^{1-3w}\), and gives the identical conclusion since \(h\geq h_l>16\xi\). Choose a height-net value between \(h-\xi/4\) and \(h-\xi/8\). The small normalized-density variation preserves the strict high antecedent at this lower height, while the field value violates its comparison with tolerance \(\xi/4\). This is an excluded net failure.

A low-density failure has \(v(y)>h+\xi>0\). Its nearest net value is above \(h+3\xi/4\) by [eq:net-field-variation]. A height-net value between \(h+\xi/8\) and \(h+\xi/4\) preserves the low antecedent and again yields an excluded failure with tolerance \(\xi/4\). Thus both comparisons hold everywhere on the band for every \(h_l\leq h\leq h_h\). Together with [eq:enlarged-field-cap], this proves 9.

Selecting conditional data

Take an arbitrary sequence in [eq:scaling-regime], with an arbitrary minimizing index \(n\) in each system. By [eq:rough-deficit], after passing to a subsequence we may suppose \[q_s:=s^3(Z-n)\longrightarrow q_*.\] Choose integers \(l\to\infty\) sufficiently slowly and a scale \(j<J_*\) such that \[ \frac1{2l}\le a_*:=\frac{r_j}{s}\le\frac1l. \tag{62}\] This is possible because \(r_0/s\to0\) and the radii are dyadic. In addition to the fixed parameters used in the barrier construction, apply 9 at this scale with \(M=l^2\), a fixed upper height greater than \(C_{\mathrm{cap}}+1\), and lower height and tolerance sufficiently small in terms of \(l\). On its good event, \[ \left|\mu_j(y)-k\bigl(H'_j(y)_+\bigr)^{3/2}\right| \le l^{-8}|y|^{-6} \qquad(r_j\le|y|\le ls). \tag{63}\] Indeed, let \(u=|y|^4H'_j(y)\) and \(v=(|y|^6\mu_j(y)/k)^{2/3}\). The cap on \(u\) excludes \(v\) at or above the upper height: applying the high implication at heights approaching that endpoint would contradict the cap. Above the lower endpoint, the two implications at heights approaching \(v\) squeeze \(u\) to within the chosen tolerance of \(v\). At or below the lower endpoint they bound \(u\) above by that endpoint plus the tolerance. Thus \(|u_+-v|\le h_l+\xi\), with both \(u_+\) and \(v\) in \([0,h_h+1]\); this uses no lower bound on \(u\). Uniform continuity of \(t\mapsto t^{3/2}\) on that interval proves the assertion after choosing \(h_l\) and \(\xi\) sufficiently small. The spatial range is covered because \(4Mr_j\ge2ls\).

For each fixed \(l\) these are fixed parameters of 9; its thresholds hold eventually and its exceptional probability tends to zero. A slow diagonal choice therefore makes [eq:local-tf-approximation] hold outside a set of probability tending to zero while also ensuring [eq:intermediate-scale].

We can select data on which, in addition, \[ \int p_j\le1, \qquad s^3\int_{|y|>2R_0s}\mu_j(y)\,dy\le\eta_s, \qquad\eta_s\longrightarrow0. \tag{64}\] The first excluded probability tends to zero by [eq:barrier-error-mass]. For the second, a master packet centered in \(|x|\le R_0s\) is contained in \(|y|\le2R_0s\) for all sufficiently far systems. The conditional tower and [eq:exterior-cutoff] show that the expectation of its left side tends to zero, uniformly in \(j\). Markov’s inequality supplies a deterministic threshold \(\eta_s\to0\) with excluded probability tending to zero. These two events and the good comparison event have positive joint probability. Choose a point in their intersection where all the conditional identities and barrier inequalities hold. Only this finite-system selection is needed; every selected density still has total mass exactly \(n\).

Compactness and the exterior boundary condition

On the selected data, suppress the sequence index and put \[F_s(x)=s^4\bigl(V-K*\mu_j\bigr)(sx), \qquad \rho_s(x)=s^6\mu_j(sx).\] Then \[\Delta F_s=-4\pi Zs^3\delta_0+4\pi\rho_s, \qquad s^4H'_j(sx)=F_s(x)-1.\] The total rescaled electron mass need not be bounded. We instead control the screened field through its spherical mean. Newton’s averaging identity and \(\int\mu_j=n\) give \[ \overline F_s(d):=\frac1{4\pi}\int_{\mathbb S^2}F_s(d\omega)\,d\omega =\frac{q_s}{d} +\int_{|x|>d}\left(\frac1d-\frac1{|x|}\right)\rho_s(x)\,dx. \tag{65}\] In particular \(\overline F_s(d)\ge q_s/d\).

On every compact annulus, 9 gives the uniform pointwise upper bound \(F_s(x)\le1+C_{\mathrm{cap}}|x|^{-4}\) for sufficiently far systems. Together with [eq:spherical-mean] and boundedness of \(q_s\), this bounds \(F_s\) in local \(L^1\) away from zero: integrate \(|F_s|=2(F_s)_+-F_s\) first over spheres and then over radii. Equation (63) also bounds \(\rho_s\) uniformly on these annuli.

For completeness, choose a ball whose closure avoids the origin, and add to \(F_s\) the Coulomb potential of \(\rho_s\) restricted to a slightly larger ball. The sum is harmonic on the larger ball. The added potential and its gradient are uniformly bounded by the integrals of \(|x-y|^{-1}\) and \(|x-y|^{-2}\) against a bounded density. Interior harmonic estimates, using the local \(L^1\) bound, give uniform bounds and equicontinuity on the smaller ball. A diagonal Arzelà–Ascoli argument now gives a subsequence converging locally uniformly on \(\mathbb R^3\setminus\{0\}\) to a field \(F_\infty\). The scaled error in [eq:local-tf-approximation] tends to zero locally uniformly, so the densities converge there to \(k((F_\infty-1)_+)^{3/2}\). Hence \[ \Delta F_\infty=4\pi k\bigl((F_\infty-1)_+\bigr)^{3/2}, \qquad F_\infty(x)\le1+C_{\mathrm{cap}}|x|^{-4}. \tag{66}\] The equation holds classically off zero. For example, bounded local sources first give local Hölder regularity of the field; the reaction is then locally Hölder, and interior Poisson regularity gives the asserted conclusion.

The selected tail bound says precisely \[\int_{|x|>2R_0}\rho_s(x)\,dx\longrightarrow0.\] Local uniform density convergence therefore makes \(F_\infty\) harmonic on \(|x|>2R_0\). For \(d>2R_0\), the last term of [eq:spherical-mean] is nonnegative and at most \(d^{-1}\) times this tail mass. Consequently \[ \overline F_\infty(d)=q_*/d\qquad(d>2R_0). \tag{67}\]

These facts imply the uniform boundary condition \[ F_\infty(x)\longrightarrow0\qquad(|x|\to\infty). \tag{68}\] Here no positivity has yet been assumed. On large annuli rescaled to a fixed annulus, the upper cap bounds the positive part, and [eq:limit-exterior-mean] bounds the integral of the negative part. Interior harmonic estimates thus give a uniform absolute bound for \(F_\infty\) throughout the exterior. Given any sequence \(R_i\to\infty\), its dilates \(F_\infty(R_i x)\) have a subsequence converging locally on punctured space to a bounded harmonic function, with the same absolute bound on every punctured compact set. Its singularity at zero is removable, and Liouville’s theorem makes it constant. Its spherical mean is zero by [eq:limit-exterior-mean], so the constant is zero. If [eq:limit-decay] failed, dilating at a sequence of offending radii and taking a convergent subsequence of the corresponding unit directions would contradict this conclusion. This proves uniform decay.

Transferring the positive barrier

We claim that \[ F_\infty(x)\ge B|x|^{-4}\qquad(x\ne0). \tag{69}\] The field in this assertion is the field without the subtracted price. This distinction permits the reaction comparison below.

Fix an outer radius \(S>0\) and write \(U_s(x)=s^4u_j(sx)\). Inside \(a_*\), the distributional inequality for \(U_s\) and the equation for \(F_s\) contain the same point-nucleus and density \(\rho_s\) terms; the inequality also contains the rescaled error from \(p_j\). Outside \(a_*\), [eq:local-tf-approximation] gives, once \(l>S\), \[\rho_s(x)\le k\bigl((F_s(x)-1)_+\bigr)^{3/2} +l^{-8}|x|^{-6}.\] Define the nonnegative correction potential \[ L_s=K*\left(s^6p_j(s\,\cdot) +l^{-8}|\cdot|^{-6}\mathbf 1_{\{a_*\le|\cdot|\le S\}}\right). \tag{70}\] Scaling [eq:barrier-invariant], subtracting the equation for \(F_s\), and using \(\Delta K=-4\pi\delta_0\) gives on \(B(0,S)\) \[ \Delta(U_s-F_s-L_s) \ge\mathbf 1_{\{|x|\ge a_*\}}\bigl[f(U_s)-f(F_s-1)\bigr]. \tag{71}\] In particular, subtracting \(L_s\) cancels both the accumulated source error and the approximation error with the required sign.

Choose any \[c>\left(C_{\mathrm{inv}}S^{-4} -\min_{|x|=S}F_\infty(x)\right)_+.\] For sufficiently far systems, local uniform convergence on this sphere, the upper bound for \(U_s\), and \(L_s\ge0\) give \(U_s-F_s-L_s\le c\) on the boundary. The same inequality holds throughout the ball. Indeed, test [eq:barrier-transfer-inequality] with \((U_s-F_s-L_s-c)_+\). On its positive set, \(U_s>F_s+L_s+c>F_s-1\), so the reaction difference is nonnegative. The test yields \[\int_{B(0,S)}\left|\nabla(U_s-F_s-L_s-c)_+\right|^2\le0.\] It is legitimate on the full ball: the nuclear singularities in \(U_s-F_s\) cancel, the remaining offsets are continuous and locally \(H^1\), and the correction is the potential of a bounded compactly supported density for each fixed system. After cancellation all right-hand sides are locally bounded, including near zero, because the reaction term vanishes on \(|x|<a_*\). Approximation therefore allows the compactly supported nonnegative \(H^1\) test.

For each fixed \(x\ne0\), \(L_s(x)\to0\). The first density in [eq:barrier-correction] has mass at most \(s^3\) by [eq:selected-data] and is supported in \(|x|\le a_*\). The second has total mass at most \[C l^{-8}a_*^{-3}\le C l^{-5},\] by [eq:intermediate-scale]; its density near a fixed nonzero evaluation point is \(O_x(l^{-8})\). Splitting its potential integral into a small ball around that point and its complement proves the claim. Finally, for fixed \(0<|x|<S\) and sufficiently far systems, \[U_s(x)\ge B|x|^{-4}\left(1-\frac{a_*}{8|x|}\right).\] Pass to the limit in \(U_s\le F_s+L_s+c\), then decrease \(c\) to its permitted endpoint. Letting \(S\to\infty\) and using [eq:limit-decay] proves [eq:limit-positive-barrier].

The singular profile and uniqueness

The dilation argument below is a direct form of the Sommerfeld comparison principle. The homogeneous zero-price profiles are covered by (Solovej 2003, Lemma 4.4 and Remark 4.5), including nonspherical fields. Here dilation removes the unit price near the singularity; a separate comparison then proves uniqueness for the priced equation.

Integrating the price and bracketing the sector

The same charge in Thomas–Fermi theory

The classical weak-ionization law was established by Bénilan and Brezis (Bénilan and Brezis 2004, sec. 6, Corollary 5), following the bounds and chemical-potential integration argument of Lieb and Simon (Lieb and Simon 1977, Theorems IV.11–IV.12). We give the argument in the present normalization to identify its coefficient with the charge obtained from the conditional quantum limit.

We allow real nuclear charges and real electron masses in this subsection. For clarity, the admissible finite-mass class can be written as \[\mathcal X=\{\rho\ge0:\rho\in L^1(\mathbb R^3)\cap L^{5/3}(\mathbb R^3)\}.\] All terms of \(\mathcal T_Z\) are finite on this class. For the direct term, interpolation gives \(\rho\in L^{6/5}\), and the Sobolev inequality gives the embedding \(L^{6/5}\subset(\dot H^1)^*\) and the finite Coulomb energy. For the nuclear term, use \(|x|^{-1}\in L^{5/2}_{\mathrm{loc}}\) near zero and the finite mass away from zero. Thus this is precisely the finite-mass class in the definition of \(E_Z^{\mathrm{TF}}(M)\).

The specified order of limits