Proof. Approximate an \(n\)-electron trial state by one with compact support. Wedge it with a normalized spin orbital in a disjoint distant ball. Dilate the orbital to make its kinetic energy small and separate the supports far enough to make the added repulsion small. Disjoint spatial supports make the energy of the normalized wedge the sum of the group energies and their direct cross repulsion. Dropping the added nuclear attraction proves \(E_{n+1}\leq E_n\). A normalized orbital dilated about one nucleus has kinetic energy \(O(s^{-2})\) and attraction at most \(-c z_m s^{-1}\), proving \(E_1<0\).

Suppose \(\Delta=E_{n-1}-E_n>0\), and take a normalized minimizing sequence \(\psi_j\). The form bound above gives a uniform \(H^1\) bound. Choose nonnegative Lipschitz functions \(g_S,h_S\) with \[g_S^2+h_S^2=1,\quad g_S=1\text{ on }B(0,S),\quad g_S=0\text{ outside }B(0,2S),\quad |\nabla g_S|+|\nabla h_S|\leq C/S.\] They may be constructed as the cosine and sine of a cutoff angle. For each subset \(I\) of the particle labels, multiply \(\psi_j\) by \(\prod_{i\in I}g_S(x_i)\prod_{i\notin I}h_S(x_i)\). The squared norms of these localized states sum to one. The product partition identity for the kinetic energy adds at most \(Cn/S^2\) to the sum of their forms.

The all-\(g_S\) component has energy at least \(E_n\) times its squared norm. If \(k<n\) coordinates carry \(g_S\), slice in the remaining coordinates. Antisymmetry in the \(k\) retained coordinates gives their energy lower bound \(E_k\geq E_{n-1}\). Drop the other kinetic terms and all pair terms involving an \(h_S\) coordinate. For \(S>2\max_m|R_m|\), their total nuclear attraction is bounded below by \(-2Zn/S\) times the component’s squared norm. If \(p_j(S)\) is the total squared mass of components other than all-\(g_S\), it follows that \[(\Delta-2Zn/S)p_j(S) \leq q_n[\psi_j]-E_n+Cn/S^2.\] The probability that any \(|x_i|\geq2S\) is at most \(p_j(S)\), because the all-\(g_S\) product vanishes there. Taking first \(j\to\infty\), then \(S\to\infty\), proves tightness.

Local Rellich compactness and tightness give, along a subsequence, strong \(L^2\) convergence to a normalized antisymmetric \(\Psi\), weak \(H^1\) convergence, and almost-everywhere convergence. The attraction converges: the infinitesimal form bound applied to \(\psi_j-\Psi\), with its relative-bound parameter subsequently tending to zero, gives \[\sum_i\int V(x_i)|\psi_j-\Psi|^2\longrightarrow0.\] Cauchy–Schwarz with this weight handles the difference of the attractive expectations. The kinetic energy is weakly lower semicontinuous, and the nonnegative repulsion is lower semicontinuous by Fatou’s lemma. Hence \(\Psi\) minimizes. Variation in the form domain gives the weak eigen-equation at \(E_n\). ◻

Proof. Let \(\Psi\) be the minimizer from Lemma 4. For \(l,\varepsilon>0\) put \[V_l(x)=\sum_m z_m(|x-R_m|^2+l^2)^{-1/2},\qquad U=V_l+\varepsilon,\qquad w=U^{-1}.\] For these fixed parameters, \(w\) and its first derivatives are bounded. Test the weak eigen-equation with \((\sum_iw(x_i))\Psi\) and take real parts. This summed multiplier is symmetric; after taking this admissible test we may expand its individual summands in the ambient form.

In the \(i\)th summand, the Hamiltonian on the other \(n-1\) coordinates, minus \(E_n\), has nonnegative pairing after weighting by \(w(x_i)\). This follows by slicing in \(x_i\) and its spin and using \(E_{n-1}\geq E_n\). The remaining kinetic pairing is also nonnegative. Indeed, with \(\phi=w(x_i)\Psi\), \[\begin{align*} \operatorname{Re}\int\nabla_i\Psi\cdot \overline{\nabla_i(w(x_i)\Psi)} &=\int U(x_i)|\nabla_i\phi|^2 -\frac12\int(\Delta V_l)(x_i)|\phi|^2\geq0,\\ \Delta V_l(x)&=-3l^2\sum_m z_m(|x-R_m|^2+l^2)^{-5/2}\leq0. \end{align*}\] The integration by parts is justified by large-radius cutoffs and \(H^1\) approximation; all displayed coefficients and their needed derivatives are bounded for fixed \(l,\varepsilon\). We obtain \[ \mathbb E_\Psi\sum_{i<j}\frac{w(x_i)+w(x_j)}{|x_i-x_j|} \leq\mathbb E_\Psi\sum_iw(x_i)V(x_i). \tag{5}\]

Let \(l\downarrow0\) at fixed \(\varepsilon\). Dominated convergence applies, using \(V/\varepsilon\) for attraction and \(2K/\varepsilon\) for a pair. These functions have finite expectations on the form domain. Hence (5) holds with \(w=(V+\varepsilon)^{-1}\), and its right side is at most \(n\).

For a configuration outside the null set of nuclear collisions, set \[q_{im}=\frac{w(x_i)z_m}{|x_i-R_m|},\qquad s_i=\sum_mq_{im}=\frac{V(x_i)}{V(x_i)+\varepsilon}\leq1.\] The triangle inequality gives \[\frac{w(x_i)+w(x_j)}{|x_i-x_j|} \geq\sum_m\frac{q_{im}q_{jm}}{z_m}.\] In fact, multiplying the right side by \(|x_i-x_j|\) bounds it above by \(w(x_i)w(x_j)(V(x_i)+V(x_j))\leq w(x_i)+w(x_j)\). Weighted Cauchy–Schwarz and \(z_m\geq1\) now imply \[\begin{align*} \sum_{i<j}\sum_m\frac{q_{im}q_{jm}}{z_m} &=\frac12\sum_m\frac{(\sum_iq_{im})^2-\sum_iq_{im}^2}{z_m}\\ &\geq\frac{(\sum_i s_i)^2}{2Z}-\frac n2. \end{align*}\] Here \(\sum_mq_{im}^2/z_m\leq\sum_mq_{im}^2\leq s_i^2\leq1\). As \(\varepsilon\downarrow0\), each \(s_i\) tends to one almost surely, and the squared sum is bounded by \(n^2\). Taking expectations in the last inequality and using (5) gives \(n\geq n^2/(2Z)-n/2\), as asserted. ◻

Proof. The set of strict sectors is nonempty since \(E_1<0\), and finite by Lemma 5. After its largest member there are no strict drops. Monotonicity therefore gives \(E_n=E_N\) for \(n>N\), whereas \(E_n\geq E_N\) for \(n<N\). Existence of \(\Psi\) follows from Lemma 4. Every other strict sector has particle number at most \(N\). ◻

Proof. Both \(F\Psi\) and \(F^2\Psi\) belong to \(\mathcal Q_n\). Test the weak eigen-equation with \(F^2\Psi\) and expand the gradients. Subtraction from \(q_n[F\Psi]\) cancels all multiplication potentials and mixed-gradient terms, leaving (7). ◻

Proof. Strict binding follows inductively beginning with the vacuum. Suppose \(\Phi\) is a ground state in sector \(n-1\). With the radial core cutoff \(g_S\) used in the compactness proof, put \(F_S=\prod_i g_S(x_i)\). Its norm tends to one, and the multiplier identity gives normalized energy \(E_{n-1}+O_Z(S^{-2})\). Wedge this state with a normalized radial spatial orbital of fixed spin, supported in \(3S<|x|<4S\), with kinetic energy \(O(S^{-2})\). The supports are separated. Newton’s spherical mean identity is \[ \frac1{4\pi}\int_{\mathbb S^2}|s\omega-x|^{-1}\,d\omega =\frac1{\max(s,|x|)}. \tag{8}\] It follows directly by integrating \((s^2+|x|^2-2s|x|t)^{-1/2}/2\) over \(-1<t<1\). Every core radius is at most \(2S\), so the added electron’s attraction plus direct repulsion is at most \(-(Z-n+1)/(4S)\). For \(n\le Z\) this strictly dominates the kinetic and cutoff errors when \(S\) is large. Lemma 4 supplies the next ground state. Its tightness argument gives the asserted compactness for every minimizing sequence, completing the induction. ◻

Proof. Let \(\Phi\) be a ground state in a strict sector, supplied by Lemma 4. For \(l,\varepsilon>0\) set \[U(x)=(|x|^2+l^2)^{-1/2}+\varepsilon,\qquad w(x)=U(x)^{-1}.\] For fixed parameters \(w\) and its first derivatives are bounded. Test the weak eigen-equation by \((\sum_iw(x_i))\Phi\) and take real parts. This is a symmetric admissible multiplier. In its \(i\)th summand, the form on the other \(n-1\) coordinates is at least \(E_{n-1}\mathbb E_\Phi w(x_i)\), by slicing in \(x_i\) and its spin. The selected-coordinate kinetic pairing is nonnegative. Indeed, writing \(\phi=w(x_i)\Phi\) and hence \(\Phi=U(x_i)\phi\), integration by parts gives \[\begin{align*} \operatorname{Re}\int\nabla_i\Phi\cdot \overline{\nabla_i(w(x_i)\Phi)} &=\int U(x_i)|\nabla_i\phi|^2 -\frac12\int\Delta U(x_i)|\phi|^2\geq0,\\ \Delta U(x)&=-3l^2(|x|^2+l^2)^{-5/2}\leq0. \end{align*}\] The identity follows first for smooth compactly supported functions; the bounded coefficients for fixed \(l,\varepsilon\), approximation in \(H^1\), and large-radius cutoffs give it in the present domain. Since \(E_{n-1}-E_n>0\), discarding that additional nonnegative term leaves \[ \mathbb E_\Phi\sum_{i<j}\frac{w(x_i)+w(x_j)}{|x_i-x_j|} \leq \mathbb E_\Phi\sum_i w(x_i)\frac Z{|x_i|}. \tag{10}\] Let \(l\downarrow0\) at fixed \(\varepsilon\). Domination by \(2K(x_i-x_j)/\varepsilon\) for pairs and \(V(x_i)/\varepsilon\) for attraction, which are integrable on the form domain, justifies the limit. Then \(w(x)=|x|/(1+\varepsilon|x|)\). As \(\varepsilon\downarrow0\), both sides increase and the right side tends to \(nZ\). The left side tends to the expectation of \(\sum_{i<j}(|x_i|+|x_j|)/|x_i-x_j|\), which is at least \(n(n-1)/2\) by the triangle inequality. Thus \(n\leq2Z+1\).

The strict indices form a nonempty finite set by Lemma 8. After its largest element \(N_*\) there can be no further strict drop; monotonicity implies \(E_n=E_{N_*}\) for every \(n>N_*\). For \(n<N_*\) monotonicity gives \(E_n\geq E_{N_*}\). This proves (9), without any assumption on possible plateaus before \(N_*\). Attainment follows once more from Lemma 4. ◻

Proof. For compactly supported \(f\in L^{5/3}\) we also have \(f\in L^{6/5}\). Riesz representation on \(\dot H^1\) gives a unique solution \(u\) of \(-\Delta u=4\pi f\). The convolution \(K*f\) is continuous: near the singularity use \(K\in L^{5/2}_{\mathrm{loc}}\) and continuity of translations in that space; away from it use dominated convergence. It is bounded locally and decays like \(O(|x|^{-1})\) at infinity, hence belongs to \(L^6\). It has the same distributional Laplacian as \(u\). A harmonic \(L^6\) function is zero, by mollification and the mean-value property on arbitrarily large balls. Thus \(u=K*f\). Testing its weak equation with \(u\) gives the first identity in (11); testing with \(v\) and using Cauchy–Schwarz gives the second.

For the noncompact nonnegative case, positivity first gives \[\int f|\varphi|\leq\|f\|_{(\dot H^1)^*}\|\nabla\varphi\|_2 \qquad(\varphi\in C_c^\infty).\] To obtain this estimate, approximate the absolute value by smooth convex truncations with derivative bounded by one. Completion and an almost-everywhere convergent subsequence extend the estimate to \(\dot H^1\) and identify its dual pairing with actual integration. Let \(u\) be the Riesz solution, and let \(u_j=K*(\mathbf1_{B(0,j)}f)\). Testing the difference equation with \((u_j-u)_+\) gives \(u_j\leq u\). These positive-part tests belong to \(\dot H^1\): an \(L^6\) function with gradient in \(L^2\) is in the completion by large-radius cutoff and mollification. For the cutoff error, Hölder’s inequality bounds the gradient of the cutoff times the function by its \(L^6\) norm on the escaping annulus. Now \(u_j\uparrow K*f\) and \(u_j\leq u\in L^6\), so dominated convergence in \(L^6\), the distributional equation, and harmonic uniqueness give \(K*f=u\). Its energy identity follows by testing with \(u\). The convolution is continuous also in this case. Choose a point \(x_0\) where its value is finite. On any fixed bounded set of evaluation points, the sufficiently distant kernel tails are dominated by a constant times \(K(x_0-\cdot)f\), which is integrable. Dominated convergence handles that tail, and the local \(L^{5/3}\)–\(L^{5/2}\) argument handles the remaining compact part.

The spherical average of \(K\) at distance \(d\) from a sphere of radius \(s\) is \(1/\max(d,s)\), by direct angular integration. Decomposing a radial probability into spherical averages proves the smearing assertion. The difference of two independent radial displacements is again radial; applying the same assertion to its distribution and integrating against \(\rho(x)\rho(y)\) proves the energy inequality. ◻

Proof. For a unit spin orbital \(v\), the number operator \(\sum_{i=1}^k|v\rangle\langle v|_i\) is a projection on the antisymmetric space: in a Slater basis containing \(v\), its eigenvalue is zero or one. Its expectation is at most \(\|f\|_2^2\). This proves the occupation assertion and its versions with auxiliary integration and mixtures.

For the kinetic bound, take an eigen-expansion \(\gamma=\sum_\alpha\lambda_\alpha |u_\alpha\rangle\langle u_\alpha|\), and let \(P_s\) be Fourier projection onto \(|p|^2/2\leq s\). The density of its low-frequency parts is at most \(Cs^{3/2}\): Bessel’s inequality bounds each spin evaluation against the corresponding Fourier vector, and there are two spins. The triangle inequality in the weighted orbital-index and spin norm gives, almost everywhere, \[\sum_{\alpha,\sigma}\lambda_\alpha |((1-P_s)u_\alpha)(x,\sigma)|^2 \geq\big(\sqrt{\rho_\gamma(x)}-Cs^{3/4}\big)_+^2.\] Integrating the left side in \(x\) and then in \(s>0\) gives the kinetic trace by Plancherel and the layer-cake formula. The integral of the right side in \(s\) is a positive constant times \(\rho_\gamma(x)^{5/3}\). Truncation proves the assertion also when the kinetic trace is infinite. ◻

Proof. Drop the nonnegative pair repulsion, use \(E\le0\), and split attraction at \(R=Z^{-1/3}\). Outside that ball it is at most \(Zn/R\le3Z^{7/3}\). Inside, Young’s inequality gives \[\int_{|x|<R}\frac Z{|x|}\rho_\psi \le\frac c2\int\rho_\psi^{5/3} +C\int_{|x|<R}(Z/|x|)^{5/2} \le\frac c2\int\rho_\psi^{5/3}+CZ^{7/3},\] where \(c\) is the constant in Lemma 11. Combine this with that kinetic lower bound and the energy bound \(q_n[\psi]\le E+\delta\le\delta\). ◻

Proof. Successive squared-gradient splitting gives \[\sum_h|\nabla p_h|^2+|\nabla c|^2 =\sum_h\Big(\prod_{l<h}c_l^2\Big) (|\nabla s_h|^2+|\nabla c_h|^2).\] The norm-map inequality bounds \(|\nabla o|^2\) by \(\sum_h|\nabla p_h|^2\). Expanding the product partition in the kinetic form, the mixed derivatives cancel because \(o\nabla o+c\nabla c=0\). This proves the stated IMS bound. Multiplication potentials split exactly by the squared partition.

Fix the recorded subset \(I\), and write \(u\) for its out positions and spins and \(\xi\) for the remaining positions and spins. Put \[\kappa_I(u)=\int|\psi_I(u,\xi)|^2\,\mathrm d\xi, \qquad \chi_X(\xi)=\frac{\psi_I(u,\xi)}{\sqrt{\kappa_I(u)}} \quad\text{when }\kappa_I(u)>0,\] where the integral includes the core spin sum. Thus \(\chi_X\) is the normalized core vector, with its phase retained. The stage decoration depends only on \(u\) and does not change this normalized slice. Fubini gives its \(H^1\) regularity almost surely; permutations of core variables preserve the multiplying factors and therefore its antisymmetry. The conditional core–out repulsion is exactly \[\sum_{i\ \mathrm{out}}\int K(x_i-x)\rho_X^c(x)\,\mathrm dx.\] Grouping the core form, out kinetic form, and the remaining multiplication potentials proves (16). All absolute Coulomb terms and kinetic terms are integrable in this calculation by the form bounds and the finite partition. The slice ratios and their integrals give the stated measurable versions; assign arbitrary values on null slices.

For each \(I\), apply the occupation assertion of Lemma 11 to the out group, integrating the other variables. Its one-body matrix \(\gamma_I^o\) obeys \(\gamma_I^o\leq\|\psi_I\|_2^2\). Thus \(\gamma_o=\sum_I\gamma_I^o\leq1\), even though the out particle number varies with \(I\). Its kinetic trace is \(T_o\), and its density is \(o^2\rho_\psi\), by summing the squared partition with one label required to be out. Equation (12) proves the kinetic assertion.

Finally, the entire construction has a single sampling description. Conditional on the raw configuration, independently label each coordinate by its first-out stage or by final core, with probabilities \(p_h^2\) and \(c^2\). An appended cut only splits the previous core probability. Forgetting its new-stage labels and their variables recovers the old observation. This is the required nesting and proves the tower assertion. ◻

Proof. For each center \(y\), momentum \(p\), and spin \(\sigma\), use the orbital \[v_{y,p,\sigma}(x,\tau) =g_b(x-y)e^{ip\cdot x}\mathbf1_{\{\tau=\sigma\}}.\] Integrate its projector over both spins and \(|p|\leq(3\pi^2\rho(y))^{1/3}\) with measure \(\mathrm dy\,\mathrm dp/(2\pi)^3\), obtaining an operator \(\gamma\). The unrestricted packet resolution is the identity: apply Plancherel in \(p\) and then integrate \(g_b^2\) in \(y\). Restriction therefore gives \(0\leq\gamma\leq1\). Momentum-ball integration gives \[\begin{align*} \operatorname{Tr}\gamma&=\int\rho, &\rho_\gamma&=g_b^2*\rho,\\ \operatorname{Tr}(-\Delta/2)\gamma &=c_{\mathrm{TF}}\int\rho^{5/3} +\frac12\|\nabla g\|_2^2b^{-2}\int\rho. \tag{18}\end{align*}\] The reality of \(g\) eliminates the kinetic cross term. These traces are finite. Every positive-eigenvalue orbital of \(\gamma\) belongs to \(H^1\) and is supported in the packet region.

Take a finite eigen-truncation of \(\gamma\). Occupy each retained orbital independently with Bernoulli probability equal to its eigenvalue. Each realization is a normalized Slater determinant, including the possible vacuum. The average one-body energy is the trace energy. Its average repulsion is the direct energy of the truncated density minus \[\frac12\sum_{\sigma,\tau}\iint K(x-y)|\gamma(x,\sigma;y,\tau)|^2\,\mathrm dx\,\mathrm dy,\] where here \(\gamma\) denotes the finite truncation. In particular the average repulsion is at most the direct energy. Equal-orbital direct and exchange contributions cancel, as required by the Bernoulli occupation rule.

Wedge each realization with the normalized core slice. Because their spatial supports are separated, the terms assigning the two groups to different sets of particle labels have disjoint supports. Normalized antisymmetrization consequently preserves the sum of the group energies and adds exactly their direct cross repulsion. Each resulting vector belongs to its own antisymmetric particle-number sector and has energy at least \(E\). Averaging gives the desired inequality with the finite-truncation density and kinetic energy.

For completeness, passage to the full operator needs no selection of infinite Slater states. The truncated densities increase almost everywhere to \(g_b^2*\rho\), and their kinetic energies increase to (18). The full density is bounded and compactly supported, so nuclear integrals converge. The core potential is bounded on the packet region by support separation, so its integrals converge as well. Direct energies converge monotonically. Lemma 10 gives \(D(g_b^2*\rho)\leq D(\rho)\), proving (17). The same argument with no inserted orbitals proves \(E\leq e_X\). ◻

Proof. On a cube of side \(b\), the spin-two Neumann eigenvalues are \(\pi^2|m|^2/(2b^2)\), for \(m\in\mathbb N_0^3\), each with two spin states. The number of lattice points of radius at most \(s\), with this multiplicity, is bounded by \[\frac\pi3(s+\sqrt3)^3:\] attach a disjoint unit cube to each octant lattice point and enclose their union in the octant ball of radius \(s+\sqrt3\). Thus the \(j\)th ordered radius is at least \(((3j/\pi)^{1/3}-\sqrt3)_+\). Summing their squared radii and using \(\sum_{j\leq n}j^{2/3}\geq(3/5)n^{5/3}\) gives the lower bound \[ b^{-2}\big(c_{\mathrm{TF}}n^{5/3}-C(n^{4/3}+n)\big) \tag{22}\] for the sum of the first \(n\) eigenvalues. The leading coefficient is exactly \(c_{\mathrm{TF}}\).

Partition the out coordinates into assignments to the cubes of a grid. Restriction to a cube is admissible for its Neumann form; no zero extension across its boundary is involved. Within each cube the corresponding variables remain antisymmetric. The occupation bound, now in the Neumann basis, gives (22) after slicing all auxiliary variables. If \(n_C\) denotes the out count in a cube, the leading term is \[c_{\mathrm{TF}}\int\left(\sum_C\frac{n_C}{b^3}\mathbf1_C(x)\right)^{5/3} \mathrm dx.\] The error is at most \(Cb^{-2}(n_o^{4/3}+n_o)\), since \(\sum_Cn_C^{4/3}\leq n_o^{4/3}\).

Average the grid over a uniform translation in a fundamental cube and over rotations. At a fixed out configuration, its averaged cell density is \(\sum_{i\ \mathrm{out}}\vartheta_b(x-x_i)\). Before rotation, the probability that two points are in the same translated cube gives precisely the triangular product in (19). Jensen’s inequality bounds the power integral from below by the power of this averaged density. Dropping any chosen particles can only decrease that density and its nonnegative power. This proves (20).

For the pair bound, first discard all pairs not both retained. Smearing each retained point by \(\vartheta_b\) decreases every distinct pair interaction by Lemma 10, because the difference of two independent radial displacements is radial. Each self energy is bounded by \(C/b\), using the kernel’s support and \(L^\infty\) bound. Subtracting these self energies from \(D(\sigma)\) proves (21). ◻

Proof. The set of admissible \(s\) in (23) is downward closed. It contains \(Z^{-1/3}\) and is bounded above, since every sufficiently large ball contains all nuclei. Increasing open balls show that the condition also holds at the supremum. The radius \(2a_y\) is not admissible, proving the third inequality in (24).

If \(s=a_x-|x-y|/L>0\), then \(B(y,Ls)\subset B(x,La_x)\), and hence \(\nu(B(y,Ls))\le a_x^{-3}\le s^{-3}\). Thus \(a_y\ge s\). Interchanging \(x\) and \(y\) proves the Lipschitz bound. If \(d_y>La_y\), a slightly larger ball in the defining family would still contain no nucleus, a contradiction. If \(d_y/L>1\), every radius below \(d_y/L\) is admissible, whereas every strictly larger radius contains a charge at least one and has \(s^{-3}<1\). This proves the asserted equality. Finally, the Lipschitz bound makes all scales in \(B(y,ca_y)\) comparable to \(a_y\); a mesh of spacing a sufficiently small multiple of \(a_y\) gives the stated cover. ◻

Proof. We first give the upper trial at a target \(y\). Append a cut that is full out on \(B(y,Aa_y)\), supported in \(B(y,2Aa_y)\), and has gradients bounded by \(C/a_y\). Write \(X'\) for the enlarged data, \(a=a_y\), and \(h=\mathbb E[J_y\mid X']\). The sliced core vanishes on \(B(y,Aa)\). Take uniform packet centers of total mass \(M_{\rm tr}\ge0\) in \(B(y,a)\), with packet width \(a\). The packet support lies in \(B(y,2a)\) and is separated from the core. The radial mean of the core field and of each far nuclear field is its value at \(y\). Near nuclei contribute positively, and \(J_y\) subtracts far out electrons as well as the core. Therefore Lemma 14 gives \[ E\le e_{X'}-M_{\rm tr}h+C\left(\frac{M_{\rm tr}^{5/3}}{a^2} +\frac{M_{\rm tr}}{a^2}+\frac{M_{\rm tr}^2}{a}\right). \tag{29}\] Here \(M_{\rm tr}\) is the expected occupation of the packet trial; it need not be an integer. This use of an arbitrary trial mass is justified by \(E=\inf_k E_k\).

For \(ah>C_1m_{0,y}\) choose \(M_{\rm tr}=\lambda ah\). The three positive terms in (29), divided by \(M_{\rm tr}h\), are at most \[\frac{C\lambda^{2/3}}{a(ah)^{1/3}},\qquad \frac{C}{a(ah)},\qquad C\lambda.\] The inequalities \(am_{0,y}^{1/3}\ge1\) and \(am_{0,y}\ge1\) allow us to choose \(\lambda>0\) small and then \(C_1\) large so that their sum is less than a fixed number below one. In the other case use the vacuum trial. For all data this proves \[ E\le e_{X'}-ca h_+^2+C\frac{m_{0,y}^2}{a}. \tag{30}\]

The insertion estimate makes a large positive field expensive. We bound the energy of the removed electrons in terms of positive fields at nearby centers. If the present field exceeded a universal multiple of its proposed bound, this comparison would force a larger normalized field after one further deterministic cut. Repetition gives doubly exponential growth, whereas the elementary nuclear bound and the Lipschitz scale permit only exponential growth. We now quantify this argument, retaining the original \(e_*\) throughout.

Start with target \(v_0=y\) and the given data \(X_0=X\). At stage \(j\), append the cut just constructed at the current deterministic target \(v_j\); denote the enlarged data by \(X_{j+1}\). The upper estimate (30) applies with \(y=v_j\) and \(X'=X_{j+1}\). Cover the supports of all cuts, including this new cut, by at most \(C(j+1)\) local cells, and let \(\mathcal U_j\) be the union of their doubled balls \(2B_z\). These are the possible next targets: we estimate the removed energy using their fields under the enlarged data, writing \(X'=X_{j+1}\) in this calculation. Each newly added scale is comparable to a preceding covering scale by the common local scale properties above; the original target and initial covering scales need not be comparable to one another. Let \(a_{\min}>0\) be the minimum scale of the original target and initial covering cells. The Lipschitz property and the bounded local meshes give, for every later target or covering cell at iteration \(j\), \[ a_v\ge C^{-(j+1)}a_{\min},\qquad e_v\le C^{j+1}e_*. \tag{31}\] Indeed each new scale is comparable to a preceding scale; the quantities \(m(a)=\max(a^{-3},1)+\sqrt{\delta a}\) and \(e(a)=m(a)^2/a\) change by a bounded factor when \(a\) does, uniformly in \(\delta\). The total out count obeys \[ \|n_o\|_2\le\sum_{\text{covering cells }z}\|n_z\|_2 \le\sqrt P\sum_zm_z. \tag{32}\] The localization error is at most \(C^{j+1}Pe_*\): for an initial cell this follows from \[D_z^2a_z^{-2}\mathbb En_z \le\sqrt P\,\frac{D_z^2}{a_zm_z}e_z\le C_0Pe_z,\] and for a later cell it follows from \(a_zm_z\ge1\) and \(\mathbb En_z\le\sqrt Pm_z\).

For a covering cell \(B_z\), let \(V_z\) be the potential of nuclei in \(B(z,4a_z)\). Let \(h_z(v)\) be the conditional raw signed field, given the present \(X'\), of all sources at distance at least \(4a_z\) from \(z\). It has a jointly measurable harmonic version on \(3B_z\): all sources are separated from compact subballs, so their kernel derivatives are bounded for the fixed system and may be integrated against the conditional law. On \(B_z\) the field of the conditional core satisfies \[ \Phi_{X'}\le V_z+h_z+\text{the far out-electron potential}. \tag{33}\] For \(v\in2B_z\), the exclusion in \(J_v\) contains \(B(z,4a_z)\), since \(A=40\) and nearby scales are comparable. Extra electrons in \(h_z\) can only lower the field. Extra nuclei have total charge at most \(a_z^{-3}\) and distance at least \(2a_z\) from \(v\). Hence \[ h_z(v)\le\mathbb E[J_v\mid X']+Ca_z^{-4}. \tag{34}\] Set \[G(v,X')=\sqrt{\frac{a_v}{e_*}} \big\|(\mathbb E[J_v\mid X'])_+\big\|_2.\] Positive parts of harmonic functions are subharmonic. Their ball submean inequality, a fixed covering of \(B_z\), and Minkowski’s inequality give \[\begin{align*} \left\|\sup_{B_z}(h_z)_+\right\|_2 &\le Ca_z^{-3}\int_{2B_z}\|(h_z(v))_+\|_2\,dv\tag{35}\\ &\le C^{j+1}\sqrt{\frac{e_*}{a_z}} \left(1+\sup_vG(v,X')\right), \end{align*}\] where the supremum is over \(v\in\mathcal U_j\). The term \(a_z^{-4}\) was absorbed using \(a_z^{-7}\le e_z\le C^{j+1}e_*\).

The out addback in (33) is at most \[C\sum_l\frac{n_l}{\max(a_l,a_z)}.\] Indeed the far exclusion gives distance at least \(3a_z\); if \(a_l\gg a_z\), Lipschitzness of the scale forces every point of \(B_l\) to be a distance at least a fixed multiple of \(a_l\) from \(B_z\). Using \(\|n_l\|_2\le\sqrt{Pe_la_l}\) and \(\sqrt{a_l}/\max(a_l,a_z)\le a_z^{-1/2}\) bounds this addback in \(L^2\) by \(C^{j+1}\sqrt{Pe_*/a_z}\).

For completeness, the nuclear singularities have the required energy scale. If \(Q_z=\nu(B(z,4a_z))\le a_z^{-3}\), weighted convexity gives \[ \int_{B_z}V_z^{5/2} \le Q_z^{3/2}\sum_m z_m \int_{B_z}|v-R_m|^{-5/2}\,dv \le C Q_z^{5/2}a_z^{1/2}\le Ca_z^{-7}. \tag{36}\] In the sum over cells, the power of the number of cells arising from convexity is bounded by \(C^{j+1}\). Lemma 11 and Young’s inequality therefore absorb this attraction at cost \(C^{j+1}e_*\). Multiplying the other bounds by \(\|n_z\|_2\le\sqrt{Pe_za_z}\) and summing cells, the conditional localization identity, with out-pair repulsion dropped, yields \[ q_n[\psi]+\mathrm{IMS} \ge\mathbb Ee_{X'}-C^{j+1}e_* \left(P+\sqrt P\sup_vG(v,X')\right). \tag{37}\] This calculation used the actual list of cuts; it did not assume the conclusion of the lemma for that list.

Combine (30) and (37), use \(\delta\le e_*\), and apply conditional Jensen from the enlarged data back to the preceding data. With \(G(v_j,X_j)\) defined by the same normalization, \[ G(v_j,X_j)^2\le C^{j+1} \left(P+\sqrt P\sup_{v\in\mathcal U_j}G(v,X_{j+1})\right). \tag{38}\] Write \(g_j=G(v_j,X_j)/\sqrt P\). Whenever \(g_j^2\ge2C^{j+1}\), the last inequality permits a deterministic next target \(v_{j+1}\in\mathcal U_j\) with \[g_{j+1}\ge\frac{g_j^2}{4C^{j+1}},\qquad \log g_{j+1}\ge2\log g_j-(j+1)\log C-\log4.\] The series \[S=\sum_{j\ge0}2^{-j-1}\big((j+1)\log C+\log4\big)\] converges. Choose \(M_0>0\) so that \(\exp(2^{j+1}M_0)\ge2C^{j+1}\) for every \(j\). If \(\log g_0>S+M_0\), induction gives \(\log g_j\ge2^jM_0\); thus the condition for selecting each next target always holds. All supremums here are of numerical norms for deterministic cuts; choosing a target close to one of them never chooses a target from sampled data. On the other hand, \(J_v\le Z/(Aa_v)\) pointwise, and (31) gives \[G(v,X')\le\sqrt{\frac{a_v}{e_*}}\frac{Z}{Aa_v} \le C^{j+1}\frac{Z}{A\sqrt{e_*a_{\min}}}.\] For the fixed system and initial cuts the last factor is finite. Its logarithm grows at most linearly in \(j\), contradicting the positive multiple of \(2^j\) forced above. This argument works in both geometries and requires no global lower bound on the scale. It proves (28). ◻

Proof. We divide the proof into the choice of cut, the two energy comparisons, and the deterministic interior bound.

Proof. Fix small \(\epsilon>0\) and \(0<\beta<1/100\), to be chosen in that order. We need only consider \(P\ge\beta^{-4}\), since \(\beta\) will be fixed universally. Then \(\beta^{-2}/(am)\le\sqrt P\) at every target, and the patch of Lemma 18 is available. Every raw particle in \(B_y\) is retained, and its fine smear is contained in \(B(y,1.1a)\) because \(\beta<1/100\). Choose a nonnegative smooth \(\chi\), equal to one there, supported in \(B(y,2a)\), with \(\|\nabla\chi\|_2\le C\sqrt a\). Lemma 10 and (45) give \[n_y\le\int\chi\sigma \le Ca^{-3}+C\sqrt{aD(\sigma-\rho)},\qquad \mathbb En_y^2\le Ca^{-6}+Ca\,\mathbb ED(\sigma-\rho).\] Divide by \(Pm^2\) and use (43). Since \(\delta a\le m^2\), the result is at most \[\begin{align*} C\bigg[&P^{-1}+\epsilon+\sqrt\beta +\frac{\epsilon^{-3/2}\beta^{1/2}}P +\frac{\beta^{-1}}{\sqrt Pm}\tag{59}\\ &+\frac{\beta^{-2}}{P^{1/3}am^{2/3}} +\frac{\beta^{-2}}{\sqrt Pam}\bigg]. \end{align*}\] In the atomic distance geometry, using (46) instead gives the sharper intermediate inequality \[ \frac{\mathbb En_y^2}{Pm_y^2} \le C\left[P^{-1}+\sqrt\beta +\frac{\beta^{-1}}{\sqrt Pm_y} +\frac{\beta^{-2}}{P^{1/3}a_ym_y^{2/3}} +\frac{\beta^{-2}}{\sqrt P a_ym_y}\right]. \tag{60}\] For the common bound it suffices to use (59) in both geometries. First choose \(\epsilon\) small, then \(\beta\) small, and finally a universal threshold \(P_0\) large. The inequalities (26) show that the expression is at most \(1/2\) whenever \(P>P_0\). Enlarge \(P_0\) so that \(P_0\ge\beta^{-4}\); this verifies (39) at every target. If \(P>P_0\), we would obtain \(\mathbb En_y^2/m_y^2\le P/2\) for every \(y\), contradicting the definition of \(P\). Thus \(P\le P_0\). Since \(m_y^2\le2\max(a_y^{-6},1)+2\delta a_y\), this proves (58). ◻

Proof. The count assertion is Lemma 19. Comparable scales have comparable \(m\) and \(e=m^2/a\), uniformly in \(\delta\), so \(e_*\le C(C_0)e_y\) in Lemma 17. Substitute \(P\le C\) into that lemma to obtain the field assertion. ◻

Proof. Take \(\beta=a^{0.2}\) and \(\epsilon=a^{0.02}\). The energy assumption gives \(m\le Ca^{-3}\), while \(\beta^{-2}/(am)\le a^{1.6}\), so the patch hypothesis holds for small \(a\). By Lemma 19, \(P\le C\). The terms on the right of (43) are bounded respectively by constant multiples of \[a^{-6.98},\quad a^{-6.98}+a^{-6.9},\quad a^{-6.93},\quad a^{-4.2},\quad a^{-6.4}+a^{-5.4}.\] All are at most \(Ca^{-7+0.02}\) for \(a<1\), proving (62). Equation (44) gives \(T_o\le Ca^{-7}\). Since the cut is full out on \(B(y,4a)\), the out density equals \(\rho_\psi\) there. Applying Lemma 11 proves (63). ◻

Proof. The expectation is \(\int_{B(y,u)}|x-y|^{-1}\rho_\psi(x)\,dx\). Hölder’s inequality, the raw density bound, and direct radial integration give \[\int_{B(y,u)}\frac{\rho_\psi(x)}{|x-y|}\,dx \le\left(\int_{B(y,4a)}\rho_\psi^{5/3}\right)^{3/5} \left(\int_{|z|<u}|z|^{-5/2}\,dz\right)^{2/5} \le Ca^{-21/5}u^{1/5}.\] This is the stated bound in either geometry. ◻

Proof. By (57) and (47), \(0\le\Delta W\le Ca^{-6}\) on \(B(y,2a)\). Write \[Q(x)=\frac1{4\pi} \int_{B(y,2a)}\frac{\Delta W(z)}{|x-z|}\,\,\mathrm dz.\] For \(x\in B(y,2a)\), direct integration of \(|x-z|^{-1}\) and \(|x-z|^{-2}\) gives \[0\le Q(x)\le Ca^{-4},\qquad |\nabla Q(x)|\le Ca^{-5}.\] Since \(\Delta Q=-\Delta W\) inside that ball, \(H=W+Q\) is harmonic there. By (47), \(H\le C_1a^{-4}\). Choose \(C_2>C_1\) universally and put \(h=C_2a^{-4}-H\), a positive harmonic function on \(B(y,2a)\). The Poisson formula on a sphere of radius \(3a/2\) gives \[\sup_{B(y,a)}|\nabla h|\le Ca^{-1}h(y).\] Indeed the differentiated Poisson kernel on \(B(y,a)\) is bounded by \(C/a\) times its value at the center; integrating the nonnegative boundary values gives precisely this inequality. Also \(h(y)\le Ca^{-4}+W(y)_-\), since \(Q(y)\ge0\). Integrate the gradient along segments in \(B(y,s')\), and add the bound for \(\nabla Q\). This proves (66). ◻

Proof. The compact annulus \(\{\lambda\le|x|\le\Lambda\}\) has a finite cover by cells \(B_z\) whose radii are comparable to one, with constants allowed to depend on \(\lambda,\Lambda,L\). Dilate this cover by \(u\). Because \(a_{uz}=ua_z\), the dilated balls are again cells, now with radii comparable to \(u\). The annular count is bounded by their sum of counts. Minkowski’s inequality and (61) bound its \(L^2\) norm by \(C\sum_z m_z\), which is the right side of (67). Boundary spheres carry zero raw probability, since the one-electron density is absolutely continuous. Thus the same estimate holds with any choice of open or closed endpoints. ◻

Proof. On \(\mathcal C\) the exact likelihood of an array is \[ L_j(x,Y_j)=\sum_{\pi\in S_n}\prod_{i=1}^n\prod_{b=1}^3 \ell_j^{-1}\zeta\!\left( \frac{Y_{j,\pi(i),b}-x_{i,b}}{\ell_j}\right). \tag{70}\] Set it equal to zero outside the chamber. Integration of the permutation sum over \(\mathcal C\) equals integration of the labeled product over all arrays, and is therefore one. Define \[\Lambda_{\ge j}(x,Y)=\prod_{h=j}^{m-1}L_h(x,Y_h),\qquad p_{\ge j}(Y)=\sum_{\boldsymbol\sigma}\int |\Psi(x,\boldsymbol\sigma)|^2\Lambda_{\ge j}(x,Y)\,\mathrm dx,\] with the empty product and \(p_{\ge m}\) both equal to one. At a tuple with \(p_{\ge j}>0\), use the probability density \(|\Psi|^2\Lambda_{\ge j}/p_{\ge j}\) as the conditional raw law. Equivalently, the conditional mean of a nonnegative Borel raw observable \(G\) is the ratio \[\frac{\displaystyle\sum_{\boldsymbol\sigma}\int G(x,\boldsymbol\sigma)|\Psi(x,\boldsymbol\sigma)|^2 \Lambda_{\ge j}(x,Y)\,\mathrm dx} {p_{\ge j}(Y)}.\] On zero-density tuples use the unconditional raw law. All these versions are jointly Borel, including their dependence on an additional spatial variable. At every parent tuple with \(p_{\ge j+1}>0\), the conditional density of the next array is \(p_{\ge j}/p_{\ge j+1}\). Tonelli’s theorem and \(\int L_j\,\mathrm dY_j=1\) show that it integrates to one. Substitution of the ratios gives the tower property, also after nonnegative spatial integration; positive and negative parts give the integrable signed case.

For each multi-index \(\alpha\), \(|\partial_y^\alpha\mathcal K_z(y)| \le C_\alpha t_{\min}^{-3-|\alpha|}\) uniformly in \(z,y\). Differentiation under the conditional integral therefore gives all claimed derivatives and bounds. Tonelli gives the mass \(n\). For the potential and its gradient, split the kernels \(|u|^{-1}\) and \(|u|^{-2}\) at unit distance. The near parts are integrable and are controlled by \(\|\mu_j\|_\infty\); the far parts are controlled by \(n\). Splitting again at a small radius, then using dominated convergence on its complement, proves continuity of the potential and gradient. Finally \(-\Delta K=4\pi\delta_0\) gives (68). In the atomic case this argument does not remove the nuclear singularity of \(V_s\). ◻

Proof. In labeled observation coordinates place the sorting map inside the indicator of \(A\). The translated product density is continuously differentiable in \(L^1\), since \(\zeta\) is smooth and compactly supported. Hence \(p_A\) is continuously differentiable even for a Borel event. The scalar translation score is \[S(u)=-\frac{\zeta'(u)}{\zeta(u)} =\frac{2u}{(1-u^2)^2},\qquad |u|<1.\] Its law is symmetric and centered. It satisfies \(\|S(U)\|_q\le Cq^2\) for \(q\ge2\): indeed, \(T=(1-U^2)^{-1}\) has a finite exponential moment of every order less than one, so integration of its exponential tail gives \(\|T\|_q\le Cq\), while \(|S(U)|\le2T^2\).

For a unit vector \(v\in\mathbb R^{3n}\) the directional derivative of the likelihood pairs the event indicator with \[Q_v=\sum_{h=j}^{m-1}\sum_{i=1}^n\sum_{b=1}^3 \ell_h^{-1}v_{i,b}S(U_{hi,b}).\] Multiplication by independent random signs preserves the joint law of the independent symmetric scores. For deterministic coefficients, \[\prod_\alpha\cosh(sc_\alpha) \le\exp\!\left(\frac{s^2}{2}\sum_\alpha c_\alpha^2\right)\] gives the Gaussian tail estimate and consequently \(\|\sum_\alpha\epsilon_\alpha c_\alpha\|_q \le C\sqrt q(\sum_\alpha c_\alpha^2)^{1/2}\). Conditioning on the scores and then using the triangle inequality in \(L^{q/2}\) yields \[\|Q_v\|_q\le Cq^{5/2} \left(\sum_{h,i,b}\ell_h^{-2}v_{i,b}^2\right)^{1/2} \le Cq^{5/2}\ell_j^{-1}.\] The identity \(\sum_{i,b}v_{i,b}^2=1\) and the geometric square-sum bound are precisely what avoid factors depending on dimension or on the number of observations.

Hölder’s inequality at fixed \(x\) gives \(|D_vp_A(x)|\le Cq^{5/2}\ell_j^{-1}p_A(x)^{1-1/q}\). For \(p_A(x)>0\) choose \(q=\max\{2,\log(e/p_A(x))\}\) and then take the supremum over unit vectors. At a zero the gradient of the nonnegative differentiable function \(p_A\) vanishes. Therefore \[ |\nabla p_A|\le C\ell_j^{-1}p_A\log^{5/2}(e/p_A), \tag{72}\] with right side zero at \(p_A=0\).

Regularize the square root by \(\sqrt{p_A+\eta}\). Since \(\sqrt u\log^{5/2}(e/u)\) is bounded on \([0,1]\), these functions have a common Lipschitz bound. Their uniform limit \(\sqrt{p_A}\) is Lipschitz. Its gradient vanishes almost everywhere on its zero set, and the ordinary chain rule applies elsewhere. Thus \[\left|\nabla\sqrt{p_A/p}\right|^2 \le C\ell_j^{-2}\frac{p_A}{p}\log^5(e/p_A).\] The permutation sum in (70) shows that \(p_A\) is invariant under raw particle permutations and is spin independent. The bounded Lipschitz multiplier therefore preserves antisymmetry and the form domain. Its squared norm and its raw law are as claimed by the definition of \(p_A\).

Lemma 7, applied with eigenvalue \(e_\Psi\), now gives \[q_n[\Psi_A]-e_\Psi \le C\ell_j^{-2}\mathbb E_P[\log^5(e/p_A(x))\mid A].\] Under this conditional law, \[P\{p_A(x)<e^{-u}\mid A\} =p^{-1}\mathbb E_P[p_A\mathbf 1_{\{p_A<e^{-u}\}}] \le\min\{1,e^{-u}/p\}\qquad(u\ge0).\] Integration of this tail, split at \(\log(e/p)\), bounds the fifth moment by \(C\log^5(e/p)\) and proves (71). The proof for a single array is identical with the square-sum constant equal to one. ◻

Proof. Write \[P^{\rm raw}(y)=\sum_iK(y-x_i),\qquad P^{\rm ms}(y)=\sum_i(K*\mathcal K_{x_i})(y),\qquad P^{\rm raw}=P^c+P^{\rm ret}+P^{\rm del}.\] The first identity in (73) follows by multiplying the original \(P\)-conditional expectation defining \(\mu_j\) by \(\mathbf 1_A\), using \(A\in\mathcal F_j\), and dividing by \(P(A)\). The fresh labels do not alter this expectation. For the second identity, condition the raw-plus-label marginal of \(Q\) on \(X\). That marginal is exactly the localization law of \(\Psi_A\); the original observations can be integrated out. Tonelli justifies both identities initially for nonnegative potentials. In particular the first tower is under the original observation law, while the second uses the fresh tilted patch law.

Subtracting the definitions of \(H_j\) and \(h_X\), and then using these towers, gives the exact algebraic identity \[ \begin{split} \mathbb E_Q(H_j-h_X)(y)=\mathbb E_Q\big[& (P^{\rm raw}-P^{\rm ms})-P^{\rm del}\\ &-(P^{\rm ret}-K*\sigma)+K*(\rho_X-\sigma)\big](y). \end{split} \tag{75}\] The estimates below also establish finiteness of all terms, so this subtraction involves no difference of infinite expectations.

For \(0<u\le4a\), the local density bound and Hölder’s inequality give \[ \begin{split} \mathbb E_Q\sum_{|x_i-y|<u}K(y-x_i) &\le\left(\int_{B(y,4a)}\rho_{\Psi_A}^{5/3}\right)^{3/5} \left(\int_{|z|<u}|z|^{-5/2}\,\mathrm dz\right)^{2/5}\\ &\le Ca^{-4}(u/a)^{1/5}. \end{split} \tag{76}\] The complementary raw potential is bounded by \(n/(4a)\). By radial smearing, \(P^{\rm raw}-P^{\rm ms}\ge0\) and only centers with \(|x_i-y|\le t_{x_i}\) can contribute. Lipschitz continuity of the width implies \(|x_i-y|\le t_y+|x_i-y|/2\), hence these centers lie in \(B(y,2t)\). Equation (76) bounds its mean by \(Ca^{-4}\tau^{1/5}\). The same Newton comparison for each fine packet gives \[0\le\mathbb E_Q(P^{\rm ret}-K*\sigma)(y) \le Ca^{-4}\beta^{1/5}.\] The distance and count assumptions for the deleted particles give \(\mathbb E_Q P^{\rm del}(y)\le Ca^{-4}\sqrt\beta\).

For the signed density difference take \(v_t(z)=\min\{|y-z|^{-1},t^{-1}\}\). This is in \(\dot H^1\) and \[\|\nabla v_t\|_2^2=4\pi\int_t^\infty r^{-2}\,\mathrm dr=4\pi/t.\] Lemma 10 and Cauchy–Schwarz in probability give \[\mathbb E_Q\left|\int v_t(\rho_X-\sigma)\right| \le Ct^{-1/2}\sqrt{\mathbb E_QD(\rho_X-\sigma)} \le Ca^{-4}\varepsilon_a\tau^{-1/2}.\] It remains to control the two nonnegative remainders of the truncation inside \(B(y,t)\). In the source-free case, \[\int_{B(y,t)}K(y-z)\rho_X(z)\,\mathrm dz \le Ca^{-6}t^2=Ca^{-4}\tau^2.\] For the nuclear alternative, weighted convexity gives, with \(Q_{\rm loc}=\sum_\alpha q_\alpha\), \[V_{\rm loc}^{3/2} \le Q_{\rm loc}^{1/2}\sum_\alpha q_\alpha|z-R_\alpha|^{-3/2}.\] The assertion is read as zero if \(Q_{\rm loc}=0\). Uniformly in \(R\in\mathbb R^3\), \[\int_{B(y,t)}\frac{\,\mathrm dz}{|y-z|\,|z-R|^{3/2}} \le Ct^{1/2}.\] For completeness, apply Hölder to the second factor with any exponent \(q\in(3/2,2)\) and to the first with conjugate exponent \(q'\in(2,3)\). Integrals of these radial decreasing powers over a ball are no larger than their centered-ball integrals; the powers of \(t\) combine to \(t^{1/2}\). Since \(\rho_X\le C(a^{-6}+V_{\rm loc}^{3/2})\), its remainder is at most \[C\big(a^{-6}t^2+Q_{\rm loc}^{3/2}t^{1/2}\big) \le Ca^{-4}(\tau^2+\tau^{1/2}).\] This proves the asserted nuclear bound even when a nucleus lies at \(y\); only \(V_s(y)\), not the unsmoothed \(V(y)\), was required finite.

For \(\sigma\), only packets centered within \(t+\sqrt3b\) of \(y\) meet the truncation ball. Extend each such packet’s kernel integral to all space and use radial smearing to bound it by its raw point potential. Equation (76) bounds the mean remainder by \(Ca^{-4}(\tau+\sqrt3\beta)^{1/5}\). The truncated \(\sigma\) potential is integrable, being at most \(n/t\). Together with Coulomb duality and the just obtained remainder bound, this also proves integrability of the full \(\rho_X\) potential. The raw and core potentials were already integrable by (76). Hence all the preceding subtractions are valid. Adding the four estimates in (75) proves (74). ◻

Proof. It suffices to consider two functions. For a nonnegative compactly supported smooth test \(\varphi\), choose an increasing Lipschitz function \(\theta_\eta\) that is zero on \((-\infty,0]\) and one on \([\eta,\infty)\). Test the first inequality with \(\theta_\eta(v_1-v_2)\varphi\) and the second with \((1-\theta_\eta(v_1-v_2))\varphi\). These nonnegative compactly supported \(H^1\) tests are legitimate by mollification, since the right sides are bounded on their support. The sum of the left sides is \[\begin{align*} &-\int\bigl[\theta_\eta(v_1-v_2)\nabla v_1 +(1-\theta_\eta(v_1-v_2))\nabla v_2\bigr]\cdot\nabla\varphi\\ &\hspace{8mm} -\int\theta_\eta'(v_1-v_2)|\nabla(v_1-v_2)|^2\varphi. \end{align*}\] The second term is nonpositive. Deleting it and letting \(\eta\downarrow0\) gives the gradient of \(\max(v_1,v_2)\) by the Sobolev positive-part formula. Dominated convergence gives the reaction at the maximum; the reactions agree at ties. Induction proves the finite version.

For the last assertion subtract \(V\) in each inequality. The singular measures cancel, and the reaction for the offsets becomes \(F(y,V(y)+a_i(y))\). It is locally bounded: it vanishes near every nuclear point, and \(V\) is smooth off \(\operatorname{supp}\nu\). Apply the first assertion to the offsets and then add \(V\) back. ◻

Proof. Smear the point charges at any fixed positive widths and leave the smooth charges unchanged. This decomposes \(V_\zeta=V_s+V_c\) with \(V_s\in\dot H^1\) and compactly supported \(V_c\in L^{5/2}\). Thus \[\left|\int V_s\rho\right| \le C\|\nabla V_s\|_2\sqrt{D(\rho)},\qquad \left|\int V_c\rho\right| \le\|V_c\|_{5/2}\|\rho\|_{5/3}.\] For the fixed source and smearing widths, both displayed potential norms are finite. Young’s inequality proves coercivity in both spaces defining \(\mathcal X_+\). A bounded minimizing sequence has weakly convergent subsequences in these two reflexive spaces. Their limits agree as distributions, since both converge against compact smooth tests; nonnegativity passes to this limit. The two linear terms are weakly continuous in their respective spaces, and both positive terms are weakly lower semicontinuous. This proves existence. The strictly convex kinetic term proves uniqueness.

Nonnegative smooth compact variations give \[d:=\tfrac53c_{\mathrm{TF}}\rho^{2/3}-V_\zeta+K*\rho\ge0 \quad\text{almost everywhere}.\] For bounded smooth compactly supported \(\varphi\), the variations \((1+t\varphi)\rho\) are admissible for sufficiently small positive and negative \(t\). Indeed \(|\varphi\rho|\le\|\varphi\|_\infty\rho\), and positivity of the Coulomb kernel bounds its Coulomb energy. The nonnegative pairing property in Lemma 10 therefore justifies differentiating these variations and gives \(\int d\varphi\rho=0\). Hence \(d\rho=0\), proving the Euler equation in (99) and its distributional Laplacian.

To establish finite mass without assuming it, let the source components have centers \(R_m\) and masses \(Q_m\), and put \[U(x)=\sum_m\frac{Q_m}{|x-R_m|},\qquad w(x)=U(x)^{-1}.\] Newton’s theorem gives \(V_\zeta\le U\). Where \(\rho>0\), the Euler equation gives \(K*\rho\le V_\zeta\). For any ball \(B\), multiply this inequality by \(1_Bw\rho\) and restrict the screening integral below to \(B\). Write \(q_m(x)=w(x)Q_m/|x-R_m|\), so that \(\sum_mq_m=1\). The triangle inequality gives \[\frac{w(x)+w(y)}{|x-y|} \ge\sum_m\frac{q_m(x)q_m(y)}{Q_m}.\] The resulting symmetrization and Cauchy–Schwarz yield, with \(m_B=\int_B\rho\), \[m_B\ge\frac12\sum_m\frac{(\int_Bq_m\rho)^2}{Q_m} \ge\frac{m_B^2}{2Q}.\] All integrals in this calculation are local and finite; in particular \(w\) is bounded on \(B\), and the local Coulomb energy is finite. Increasing \(B\) proves \(\int\rho\le2Q\).

For completeness, \(L^1\cap L^{5/3}\) implies continuity and decay of the screening potential. The kernel restricted to a fixed ball belongs to \(L^{5/2}\), so its pairing with translates of \(\rho\) is continuous. The complementary kernel contributes at most \(\|\rho\|_1/R\) when the cutoff radius is \(R\). For fixed \(R\), the local \(L^{5/3}\) norm of \(\rho\) on a ball whose center tends to infinity tends to zero. First letting the center tend to infinity and then \(R\to\infty\) proves decay.

Finally, \((W+\varepsilon)_-\) is compactly supported away from point charges for every \(\varepsilon>0\): \(W\to0\) at infinity, and its positive Coulomb singularities dominate the continuous screening potential. On the support of this test \(\rho=0\), and hence \(\Delta W\le0\). Testing the equation gives \(\int|\nabla(W+\varepsilon)_-|^2\le0\). The test is justified by compact \(H^1\) approximation away from the singularities. Letting \(\varepsilon\downarrow0\) proves \(W\ge0\). ◻

Proof. For a single source, Newton’s theorem and the Euler equation give \(\rho(z)\le k(Q/|z-R|)^{3/2}\). For every \(x,R\) and \(s>0\), \[ \int_{B(x,s)}\frac{dz}{|x-z|\,|z-R|^{3/2}} \le Cs^{1/2}. \tag{102}\] To see the uniformity in \(R\), apply Hölder with exponent \(q_1\in(3/2,2)\) to the second factor and its conjugate \(q_1'\in(2,3)\) to the first. The integrals of both radial powers over \(B(x,s)\) are bounded by their integrals over centered balls of the same radius. Their powers of \(s\) combine to \(s^{1/2}\). Split \(K*\rho(x)\) at \(s=Q^{-1/3}\). The near part is at most \(CQ^{3/2}s^{1/2}\) by (102), and the far part at most \(2Q/s\) by Lemma 31. Both are \(CQ^{4/3}\).

For a point charge this gives \(W\ge Q/(2d)\) for \(d=|x-R|\le cQ^{-1/3}\). Outside that sphere set \(v=\lambda d^{-4}\). For sufficiently small universal \(\lambda>0\), \[\Delta v=12\lambda d^{-6}\ge c_{\mathrm F}v^{3/2},\] and its boundary value is below the preceding lower bound. On the exterior domain, testing the equation for \(v-W\) with \((v-W-\varepsilon)_+\) gives a nonpositive gradient square bounded below by zero. This test has compact support since both fields decay. Thus \(v\le W\). Together with the inner estimate this proves (100).

For the mixture \(F=\sum_i(q_i/q)W_i\), convexity gives \[\Delta F\ge-4\pi\zeta_{\rm truncated}+c_{\mathrm F}F^p \ge-4\pi\zeta+c_{\mathrm F}F^p.\] Compare with the equation for \(W\) and test with \((F-W-\varepsilon)_+\). The support is compact. Near a point charge whose mass was strictly decreased, \(F-W\) tends to \(-\infty\), so the test vanishes there. If the masses agree, the singular terms cancel and the difference is locally \(H^1\). The remaining density terms are locally in \(L^{5/3}\), so compact \(H^1\) approximation is valid. Monotonicity of the power gives \(F\le W\). ◻

Proof. By the scale definition, \(\nu(B(x,2La_x))>(2a_x)^{-3}\). Truncate charges in this ball to total \(q=(2a_x)^{-3}\) and apply Lemma 32. Their distances from \(x\) are at most \(2La_x\), so (100) gives the first bound in (105) with a fixed \(c_0>0\).

If some charges contribute to \(V_c(x)\), denote their total by \(q\le a_x^{-3}\). The screening estimate for each single field of charge \(q\), followed by the mixture comparison, gives \[W_0(x)\ge\sum_{\rm contributors}\frac{z_m}{|x-R_m|}-Cq^{4/3} \ge V_c(x)-Ca_x^{-4}.\] If there are no contributors, the same conclusion follows from \(W_0\ge0\).

For the upper bound, set \(a=a_x\), \(s=4a\), and let \(V_{\rm loc}\) be the potential of nuclei within \(20a\) of \(x\). On \(B(x,s)\) the function \[V_{\rm loc}(z)+\frac{C s^4}{(s^2-|z-x|^2)^4}\] is a supersolution of the field equation when \(C\) is large: the Laplacian of the second term is at most \(80Cs^6/(s^2-|z-x|^2)^6\), whereas its \(p\)th power has the same spatial dependence and coefficient \(C^{3/2}\). The boundary blowup dominates \(W_0-V_{\rm loc}\). The positive-difference test is legitimate because the point singularities cancel. Thus \(W_0(x)\le V_{\rm loc}(x)+Ca^{-4}\). Subtracting \(V_c(x)\) and using the smooth local potential bound proves the upper offset estimate. The expression defining \(h_0\) proves its continuity.

For \(W_s\), the same bump without \(V_{\rm loc}\) is a supersolution after its coefficient is enlarged to absorb the locally bounded source \(4\pi\theta\nu_s\). This proves \(W_s\le Ca^{-4}\). Its Laplacian is bounded by \(Ca^{-6}\) on a local ball. Subtract the Newton potential of this source restricted to that ball. Its gradient is bounded by \(Ca^{-5}\), since the integral of \(|z|^{-2}\) on a ball of radius \(a\) is \(Ca\). The remaining harmonic function is bounded by \(Ca^{-4}\) on a smaller ball; its interior gradient bound proves the stated estimate and local \(C^1\) regularity.

For the lower bound, select and truncate the same charges as in the first paragraph, before multiplying them by \(\theta\). Their smearing widths are at most \(3c_1a\), because their centers are within \(2La\). A single such blob of charge \(\theta(2a)^{-3}\) has bare potential at \(x\) at least \(c_L\theta a^{-4}\): every source point is within \((2L+3c_1)a\) of \(x\). Its screening potential is at most \(C\theta^{4/3}a^{-4}\). Choose \(\theta\) sufficiently small and apply the mixture comparison to obtain \(W_s\ge c_\theta a^{-4}\).

Decrease \(\theta\) further if necessary. The source bound and the lower bound for \(W_0\) imply \[4\pi\theta\nu_s \le c_{\mathrm F}\bigl(1.01^{3/2}-1.01\bigr)W_0^{3/2}.\] Consequently \(1.01W_0\) is a supersolution for the field of \(\theta\nu_s\); its extra negative point sources have the required sign. The positive-difference test for \(W_s-1.01W_0\), supported away from those point singularities, proves the final comparison. Decay follows from Lemma 31. ◻

Proof. In either case the local upper field estimate in (45) or Lemma 33 gives \(h\le Ca^{-4}\) on a fixed ball about \(y\). The patch core is absent there. The Euler equation, the bound on \(\nu_s\), and the weighted power inequality give \[ |\Delta h(z)|\le Ca^{-6} +Ca^{-3/2}\sum_{\rm loc}z_m|z-R_m|^{-3/2}. \tag{108}\] Here all centers in the sum belong to one fixed local ball and their total charge is at most \(a^{-3}\). The same estimate holds for \(h_0\) by Lemma 33.

Take the Newton particular solution for this Laplacian restricted to a ball of radius \(a\). Equation (102) bounds it by \(Ca^{-4}\) on a smaller ball. Its variation over distance \(s\) is at most \(Ca^{-4}(s/a)^{1/2}\). In detail, the part within distance \(Cs\) of either evaluation point is at most \[C\bigl(a^{-6}s^2+a^{-9/2}s^{1/2}\bigr).\] On a dyadic ring of radius \(l\ge Cs\), the difference of the two Coulomb kernels is at most \(Cs/l^2\), and the source mass in that ring is at most \(C(a^{-6}l^3+a^{-9/2}l^{3/2})\). Summing the rings up to radius \(a\) gives the asserted variation.

The harmonic remainder is bounded above by \(Ca^{-4}\). Harnack’s inequality and the harmonic gradient estimate, applied to this upper constant minus the remainder, bound its variation by \(C(s/a)(a^{-4}+h(y)_-)\). Combining the estimates gives (107), as well as continuity. For \(h_0\), Lemma 33 bounds its negative part by \(Ca^{-4}\), which is absorbed by the first term. ◻

Proof. The scale function is \(1/L\)-Lipschitz. The function \(a\mapsto c_1a\min(a,r_*)^w\) has Lipschitz constant \(O(r_*^w)\). Thus \(|t_z-t_y|\le O(r_*^w)|z-y|\). If \(|z-y|\le t_z\), this proves the support and comparability assertions. Replacing \(t_z\) by \(t_y\) changes the kernel by at most \(Cr_*^wt_y^{-3}\) on a ball of radius \(2t_y\), giving the first assertion of (110). The chain rule for a Lipschitz width gives the gradient bounds, first pointwise almost everywhere and then in \(L^2\) by the support bound. Since \(\int\mathcal K_z(y)\,dy=1\), Fubini proves preservation of integrals. Weighted Jensen followed by Fubini, using the bound on \(R1\), proves the \(L^{5/3}\) estimate.

Newton’s theorem gives the nonnegative representation \[q_0(y)=\int\rho_0(z) \bigl[K(y-z)-(K*g_{t_z}^2)(y-z)\bigr]\,dz.\] Only \(|z-y|\le2t_y\) contributes. The local field bound gives \[\rho_0(z)\le Ca_y^{-6} +Ca_y^{-3/2}\sum_{\rm loc}z_m|z-R_m|^{-3/2}.\] Equation (102) therefore bounds the displayed integral by \(C(a_y^{-6}t_y^2+a_y^{-9/2}t_y^{1/2})\), proving (112). Both \(\rho_0\) and \(R\rho_0\) belong to \(L^1\cap L^{5/3}\), hence also to \(L^{6/5}\). Their potentials are continuous and decay by the proof of Lemma 31; the Sobolev inequality and the Coulomb energy identity put their difference in \(\dot H^1\). ◻

Proof. First fix \(\theta\) as in Lemma 33. That lemma implies \[D_*:=\sup_x\frac{|\nabla W_s(x)|^2}{W_s(x)^{5/2}}<\infty\] with a fixed universal bound. Since \(\Delta W_s\le c_{\mathrm F}W_s^{3/2}\), the chain rule gives \[\Delta(W_s^{1+\alpha}) \le(1+\alpha)(c_{\mathrm F}+\alpha D_*)W_s^{\alpha+3/2}.\] Choose \(\alpha>0\) so small that \((1+\alpha)(1+\alpha D_*/c_{\mathrm F})\le1.1\). Consequently \[ \Delta\Gamma_r\le1.1c_{\mathrm F}\sqrt{W_s}\,\Gamma_r. \tag{118}\] All chain rules are local: \(W_s\) is positive and locally \(C^1\) with locally bounded Laplacian, so multiplication in the weak equation justifies them.

Choose \(c_b>0\) smaller than a sufficiently small fixed fraction of \(c_0\). On \(a\ge r\), \(r^{4\alpha}W_s^\alpha\le C^\alpha\). We may therefore choose positive \(\epsilon_0,C'_0\) so small that, throughout this region, \(\Gamma_r\) is a tiny fixed fraction of \(W_s\), is at most \(0.1c_0a^{-4}\), and is at most \((c_b/2)a^{-4}\). At the same time, \[ \Gamma_r\ge\epsilon_0c_\theta a^{-4}. \tag{119}\] These choices precede every smallness requirement on \(r_*\).

The equations for \(h_0\) and \(q_0\) give the exact cancellation \[ \Delta(h_0+q_0)=-4\pi\nu_s+c_{\mathrm F}R(W_0^p). \tag{120}\] Fix \(y\) with \(a_y\ge r\). For a contributing center \(z\), the offset oscillation estimate and (112) imply \[\begin{align*} V_c(z)+b_r(y) &=W_0(z)+h_0(y)-h_0(z)+q_0(y)-\Gamma_r(y)\\ &\le W_0(z)-\Gamma_r(y)+Ca_y^{-4}\tau_y^{1/2}. \end{align*}\] By (119), the final error is \(o(\Gamma_r(y))\) uniformly as \(r_*\to0\). Furthermore, \[W_0(z)\ge\frac{W_s(z)}{1.01} \ge\frac{1-o(1)}{1.01}W_s(y),\] using the gradient and lower bounds for \(W_s\) and \(|z-y|\le2t_y\). The chosen small ratio \(\Gamma_r/W_s\) and the derivative of the power therefore give \[W_0(z)^p-(V_c(z)+b_r(y))_+^p \ge 1.35\sqrt{W_s(y)}\,\Gamma_r(y)\] for sufficiently small damping fractions and then sufficiently small \(r_*\). Indeed the limiting coefficient is \(p/\sqrt{1.01}>1.49\). After integration in \(z\), the factor \(R1(y)=1+o(1)\) leaves the lower bound \(1.3\sqrt{W_s(y)}\Gamma_r(y)\). Combine this with (120) and (118) to prove (115). All terms represent locally integrable densities, so the assertion includes points of a level set of \(a\) up to a Lebesgue-null set; no regularity of those level sets is required.

Because \(q_0\ge0\) and \(\Gamma_r\le0.1W_0\), we have \(B_r\ge0.9W_0\). The upper offset bound follows from \(b_r\le h_0+q_0\). Also \[b_r-h_b=q_0-\Gamma_r+c_ba^{-4}\ge(c_b/2)a^{-4},\] so one may take \(c'=c_b/2\). On the kernel support, \[V_c(z)+h_b(y) =W_0(z)+h_0(y)-h_0(z)-c_ba_y^{-4} \ge ca_y^{-4},\] by the lower bound for \(W_0\), scale comparability, and the offset oscillation estimate, after decreasing \(r_*\) further. This proves (117).

Finally, \(h_0,q_0\) are continuous and locally \(H^1\), and the same is true of \(\Gamma_r\). Their decay proves the remaining assertions. ◻

Proof. The difference is exactly \[b_r-b_{R_+} =C'_0(1-2^{-4\alpha})R_+^{4\alpha}W_s^{1+\alpha}.\] The lower bound for \(W_s\) gives the assertion with \[\gamma=C'_0(1-2^{-4\alpha}) c_\theta^{1+\alpha}(1+\kappa)^{-4(1+\alpha)}>0.\] ◻

Proof. Write \(a=a_y\) and tilt the event in question, if it has positive probability \(P_A\). By the tower property, its average of \(K*\mu_j\) is the tilted raw average of the potential of the master smears. Nuclei within a fixed multiple of \(a\) contribute at most \(Ca^{-4}\) to \(V_s\). More distant electron smears have exactly their bare potential at \(y\): a contributing smear near \(y\) has center within \(2t_y\ll a\). Dropping the remaining negative electron terms therefore gives \[\mathbb E[H_j(y)\mid A]\le Ca^{-4}+\mathbb E_{\psi_A}J_y.\] Lemmas 17 and 19, with no initial cut, bound the last expression by \(Ca^{-4}+C\sqrt{\delta/a}\). Indeed, for small \(a\), their scale quantity is bounded by \(C(a^{-7}+\delta)\). Lemma 26 and \(\ell_j\ge c_{L_1}a^{1+.01}\) imply \[s-C\le C a^{7/2}\ell_j^{-1}\log^{5/2}(e/P_A) \le C_{L_1}a^{2.49}\log^{5/2}(e/P_A).\] Solving for \(P_A\) proves the assertion. The deterministic threshold comes solely from the universal local field estimates. ◻

Proof. Since \(\Delta H_j\ge-Ca_Q^{-6}\) on \(Q'\), adding a quadratic and applying the ball submean inequality gives, for any fixed large \(C'\) and a universal \(C''\), \[\sup_QH_j\le C''a_Q^{-4} +Cr^{-3}\int_{Q'}(H_j-C'a_Q^{-4})_+.\] The same inequality follows for distributional Laplacians by mollification. Choose \(C'\) above the threshold in Lemma 38. Because \(a_z/a_Q\) is bounded and \(a_z\le5L_1r\) on \(Q'\), integration of that tail yields \[\mathbb E(H_j(z)-C'a_Q^{-4})_+^m \le C a_Q^{-4m}r^{A'}\] for arbitrary fixed \(m,A'\). Jensen’s inequality for the spatial average gives the same bound for the integral term. Taking \(C'_{\rm cap}>C''+1\), Markov’s inequality gives the bad-cube probability. On a bad cube its supremum is bounded by the integral term plus \(C''a_Q^{-4}\); the probability just proved controls the latter contribution to every moment. This proves (125).

If a point lies in a bad cube, the corresponding term of \(\Pi\) dominates \((H_j)_+\) there. Otherwise the good-cube bound gives the asserted cap, since \(a_Q\) and the local scale are comparable by a universal factor. At a fixed point only a bounded number of enlarged cubes occur, proving the estimate for \(\mathcal E\). Every participating or enlarged cube lies within distance \(C_{L_1}r\) of some nucleus, by \(d_y\le La_y\). Counting grid cubes in this union proves finiteness and (126). Cube suprema are measurable through a countable dense subset, because \(H_j\) is continuous. The bounds from Lemma 25 also give the last assertion. ◻

Proof. The argument has three steps. Observation matching first relates the conditional density to the particles retained by a fresh patch. The patch comparison then determines the local offset except for a small event, whose negative-field cost must also be controlled. Finally, two separate tower identities transfer the offset comparison back to the master field.

Proof. Conditional Jensen, the master-kernel support, and Lemma 19 give \[\|\mu_j(y)\|_2\le Ct_y^{-3} \|\#\{i:x_i\in B(y,a_y)\}\|_2 \le Ct_y^{-3}a_y^{-3}.\] The tail bound implies \(\|(H_j(y)_+)^{3/2}\|_2\le Ca_y^{-6}\). For the nuclear part of the reaction, weighted convexity and the uniform integral of \(|z-R_m|^{-3/2}\) on a radius-\(2t_y\) ball give \[\int\mathcal K_z(y)V_c(z)^{3/2}\,\mathrm dz \le Ct_y^{-3}a_y^{-9/2}t_y^{3/2} =Ca_y^{-6}\tau_y^{-3/2}.\] This proves (140). Cauchy–Schwarz against \(\mathbb P(F_y)<r^{40}\) contributes \(r^{20}\). Use \(a_y\asymp_{L_1}r\), \(\tau_y\ge c_{L_1}r^w\), and (126) to obtain (141). The arbitrarily high polynomial bounds in Lemma 39 give the other assertions, choosing their exponent larger if necessary. All integrations are licensed by the joint measurable versions already constructed. ◻

Proof. Define two shifted offsets \[u_1=u-\lambda_1R_+^{-4},\qquad u_2=H-\lambda_2R_+^{-4}-\Pi.\] Use the following open cover to define \(u'\): \[ \begin{array}{c|c} \text{open region}&u'\\ \hline a<(1+\kappa)R_+&\max(u_1,u_2)\\ (1+\kappa/2)R_+<a<2L_1R_+&\max(u_1,u_2,b_{R_+})\\ a>L_1R_+&b_{R_+}. \end{array} \tag{146}\] The definitions agree on overlaps. On the lower overlap, Lemma 37 and (144) give \[u_1\ge b_r-\lambda_1R_+^{-4}>b_{R_+}.\] On the upper overlap the shifted cap inequalities, ensured by the choice of \(L_1\), put both \(u_1\) and \(u_2\) below \(b_{R_+}\). Thus the local maxima glue to a continuous locally \(H^1\) function. No assertion about the measure or regularity of a level set of \(a\) is required. The new exterior bounds follow from the same comparisons: for \(R_+\le a<(1+\kappa)R_+\) the lower bound is supplied by \(u_1\), and elsewhere the barrier is an included branch.

We verify the differential inequality locally. At a fixed point in one of the open regions, discard branches more than \(\eta\) below the maximum. There is a neighborhood on which these discarded branches cannot attain the maximum, while every retained branch is within \(2\eta\) of it. We show that each retained branch satisfies the common inequality \[ \Delta v\ge -4\pi\nu_s+4\pi\mathbf 1_{\{a<R_+\}}\mu_j -4\pi\widetilde d+\mathbf 1_{\{a\ge R_+\}}f_y(v) \tag{147}\] on that neighborhood.

For \(u_1\), the old core inequality suffices on \(a<r\), and monotonicity of \(f_y\) gives the required reaction on \(a\ge R_+\). On the gain region \(r\le a<R_+\), the error in [m:eq:outward-error] pays \(\mu_j\) at points of \(\mathcal E\cup\mathcal B^{\mathrm h}\). At every other gain point, \(\Pi=0\) and near activity yields \[H\le u-(\lambda_1-\lambda_2)R_+^{-4}+2\eta.\] Here \(h_b\le u\le h_{\mathrm{top}}\). If \(\mu_j>T_y(u)\), then the high comparison applies and \(I_y(\mu_j)\ge u\), including the case where the inverse is clipped at its upper endpoint. It would follow that \[H\ge u-\delta_0a^{-4}\ge u-16\delta_0R_+^{-4},\] contrary to (144). Thus \(4\pi\mu_j\le f_y(u)\). The old reaction supplies precisely the new core term on the gain region.

For \(u_2\), the identity \(\Delta H=-4\pi\nu_s+4\pi\mu_j\) gives the required core term on all of \(a<R_+\), while the first increment in [m:eq:outward-error] pays the Laplacian of \(\Pi\). Consider the remaining points where \(u_2\) is retained. The lower splice or the included barrier gives \[u_2\ge b_{R_+}-2\eta \ge h_b+c'a^{-4}-2e_0R_+^{-4}>h_b.\] The last inequality uses \(a\le2L_1R_+\) and (144). On \(\mathcal E\cup\mathcal B^{\mathrm l}\) the last error term pays the reaction because \(u_2\le H\) and \(f_y\) is nondecreasing. Otherwise \(\Pi=0\) and the cap gives \(H<h_{\mathrm{top}}\). If \(\mu_j<T_y(u_2)\), the low comparison applies and \(I_y(\mu_j)\le u_2\), also when the inverse is clipped at its lower endpoint. We would obtain \[H\le u_2+\delta_0a^{-4}\le u_2+\delta_0R_+^{-4},\] contradicting \(H=u_2+\lambda_2R_+^{-4}\). Therefore \(4\pi\mu_j\ge f_y(u_2)\), as required.

The barrier branch appears only where \(a>R_+\), and Lemma 36 gives (147) there. The old error density and every additional error are nonnegative, so they can only weaken the required lower bound.

These comparisons cover the exact level sets as well: \(a=r\) belongs to the gain region and uses the old reaction, whereas \(a=R_+\) belongs to the new exterior and uses its reaction. All right-side comparisons hold almost everywhere on the chosen neighborhood and therefore upgrade each retained branch’s distributional inequality to (147). The source \(\nu_s\) is a smooth bounded density for the fixed system, and all branches here are continuous and locally \(H^1\). The reaction is also bounded on bounded ranges of its argument: for a fixed system, \(t_{\min}:=\inf_z t_z>0\) and \(V_c\in L_c^{5/2}\) give, for every \(A>0\), \[\sup_{y,\,|h|\le A} f_y(h) \le C t_{\min}^{-3}\int V_c^{3/2} +CA^{3/2}\sup_y R1(y)<\infty.\] Thus the first, nonsingular part of Lemma 28, followed by a partition of unity, proves the inequality for \(u'\) on all of \(\mathbb R^3\). No vanishing of the reaction near the original nuclei is needed in this step. ◻

Proof. For the fixed system there are finitely many scales and cubes, and all master smoothing widths have a positive minimum. Lemma 25 therefore gives deterministic bounds for the conditional densities, their potentials and their gradients. The offsets and error densities in the assertion are jointly measurable: maxima, clipped inverses and failure flags preserve measurability, and each cube supremum may be taken over a fixed countable dense subset.

Inductively the branches have deterministic continuity moduli and local \(H^1\) bounds on each compact set. Their open definitions agree on overlaps; a finite subcover of a compact set supplies a Lebesgue number and hence a modulus for the glued function. The gradient of a finite maximum is almost everywhere one of the branch gradients. Conditional averaging consequently preserves continuity. It preserves the local \(H^1\) bound by testing weak derivatives against compactly supported vector fields, using Fubini and Cauchy–Schwarz. The error terms have deterministic bounded size and support for the fixed system and step, so the same is true after averaging.

Integrate the distributional inequality of Proposition 42 against a fixed nonnegative compact test and then against the explicit positive-parent conditional law of Lemma 25. The preceding bounds justify Fubini. The tower property gives \(\mathbb E[\mu_j\mid\mathcal F_{j+1}]=\mu_{j+1}\), while convexity gives \[\mathbb E[f_y(u')\mid\mathcal F_{j+1}] \ge f_y\bigl(\mathbb E[u'\mid\mathcal F_{j+1}]\bigr).\] This is the required lower bound for the reaction. The exterior bounds and the exterior equality are deterministic and pass to the average. ◻

Proof. Since \(r_0<Z^{-1/3}\le a\), Lemma 36 gives the initial invariant everywhere; there is no initial core region.

At scale \(r=r_j\), apply Lemma 41 to the three increments in [m:eq:outward-error]. The high-failure estimate pays \(\mu_j\) on the gain region; the low-failure estimate pays \(T_y(H)=f_y(H)/(4\pi)\) on the remaining participating region. The same Lemma pays membership in \(\mathcal E\) and the penalty Laplacian. For \(\mathcal B\in\{\mathcal B^{\mathrm h},\mathcal B^{\mathrm l}\}\), use \(\mathbf 1_{\mathcal E\cup\mathcal B}\le\mathbf 1_{\mathcal E}+\mathbf 1_{\mathcal B}\). All these integrands are nonnegative, so restriction to the indicated subregions only improves their bounds. We obtain \[\mathbb E\int(\widetilde d-d_j)\le C M r^{17-3w}.\]

Conditional averaging preserves the unconditional expected integral of a nonnegative density. Because \(\beta=17-3w>0\), \[\sum_{j=0}^{n-1}r_j^\beta =r_*^\beta\sum_{h=1}^{n}2^{-h\beta} \le\frac{r_*^\beta}{2^\beta-1}.\] At the last step the data are trivial, so \(d_n\) is deterministic. This proves (148), uniformly in \(n\). The argument has used the failure probability separately at each spatial point and then integrated it; it has imposed no simultaneous good-data event in space. ◻

Proof of Theorem 1. Use the contradiction sequence and sector eigenstates of Section 8. Its terminal scale is \(r_*=(B_0M/K_{\mathrm{exc}})^{1/3}\). The universal \(B_0\) will exceed the fixed count constant below. All deterministic parameters and tolerances are chosen before taking the sequence limit. The smallness requirements of the preceding lemmas then hold for all its sufficiently late members, uniformly over their scales.

At the final step put \[U=V_c+u_n,\qquad S=\nu-\mathbf 1_{\{a<r_*\}}\mu_n+d_n.\] The signed measure \(S\) has compact support. In fact \(\{a<r_*\}\) lies in a finite union of balls around the nuclei, the error density has compact support, and the nuclei are finite. Since \(\Delta V_c=-4\pi\nu+4\pi\nu_s\), the final invariant implies \[\Delta(U-K*S)\ge\mathbf 1_{\{a\ge r_*\}}f_y(u_n)\ge0.\] The nuclear singularities cancel in the explicit identity \[ U-K*S =u_n-V_s+K*(\mathbf 1_{\{a<r_*\}}\mu_n)-K*d_n. \tag{149}\] The right-hand side is continuous and locally \(H^1\) on all of \(\mathbb R^3\): the last two potentials have bounded compactly supported densities, and the other terms have the stated regularity from the construction.

Far from the nuclei, (143) gives \(U=B_{r_*}=W_0+q_0-\Gamma_{r_*}\), which tends to zero by the deterministic field estimates. The compact source potential \(K*S\) also tends to zero. For every \(\epsilon>0\), the function \((U-K*S-\epsilon)_+\) consequently has compact support and is an admissible \(H^1\) test in the subharmonic inequality. Testing gives a nonpositive squared gradient norm, so this positive part vanishes. Letting \(\epsilon\downarrow0\) proves \[U\le K*S\qquad\text{away from the nuclear points}.\] On every sufficiently large sphere \(U=B_{r_*}\ge .9W_0>0\). The spherical mean of the potential of a compactly supported signed measure, on a sphere centered at the origin and enclosing its support, equals its total mass divided by the sphere radius. Hence \[ 0<\int\,\mathrm dS =Z-N+\int_{\{a\ge r_*\}}\mu_n+\int d_n. \tag{150}\] Only positivity on a far sphere is used here; no asymptotic charge formula for \(W_0\) is needed.

At the last scale the data have been completely averaged away, so \[\mu_n(y)=\mathbb E_\Psi\sum_i\mathcal K_{x_i}(y).\] If a kernel centered at \(x\) contributes to a point with \(a_y\ge r_*\), then its support and the Lipschitz scale estimate imply \[a_y\le a_x+\frac{|x-y|}{L} \le a_x\left(1+\frac{c_1r_*^w}{L}\right).\] For small \(r_*\) this forces \(a_x\ge r_*/2\). Each kernel has integral one, and Lemma 30 therefore gives \[\int_{\{a\ge r_*\}}\mu_n \le\mathbb E_\Psi\#\{i:a(x_i)\ge r_*/2\} \le C M r_*^{-3}.\] Combining this with (148) and (150) yields \[K_{\mathrm{exc}}<C M r_*^{-3}+o(M) =\frac{C}{B_0}K_{\mathrm{exc}}+o(M).\] Choose \(B_0>2C\), increasing it also to ensure the initial scale choice in Section 8. Since \(K_{\mathrm{exc}}/M\to\infty\), the last inequality is impossible. This proves the existence of one universal finite constant bounding the excess per nucleus in every strictly bound sector.

For an arbitrary strictly bound sector, the largest strictly bound sector of the same system has at least as many electrons, so the same bound applies. For every integer \(N>Z+CM\), strict adjacent binding is therefore impossible. Sector monotonicity gives \(E_N\le E_{N-1}\), and exclusion of the strict inequality gives \(E_N=E_{N-1}\). Taking \(M=1\) proves both atomic assertions for every real nuclear charge \(Z\ge1\). ◻

Proof. We first control the observed count and the positive field throughout the band. At a fixed point and height, a hypothetical failure then defines a tilted state. Its fresh patch compares the observed density with the patch field, including the contribution of rare negative fields; the two expectation towers turn this into a contradiction. Finally, spatial and height nets give the simultaneous statement. Every occurrence of \(o(1)\) below tends to zero as \(s\downarrow0\), uniformly for \(r_0\le r=r_j\le s\) and the stated bands, after the fixed parameters in the proposition have been chosen.

Kernel geometry. Put \(a=a_y=d_y/L\), \(t=t_y\) and \(\tau=t/a\). If \(\mathcal K_z(y)\ne0\), the Lipschitz bound on \(t\) gives \(|z-y|\le t_z\le t_y+Cs^w|z-y|\), hence \(|z-y|\le2t_y\). On this support, \(t_z/t_y=1+O(s^w)\). Comparison with the fixed-width kernel \(g_t(y-z)^2\) therefore gives \[ \int\mathcal K_z(y)\,\,\mathrm dz=1+O(s^w),\qquad |\mathcal K_z(y)|\le Ct^{-3},\qquad |\nabla_y\mathcal K_z(y)|+|\nabla_z\mathcal K_z(y)|\le Ct^{-4}. \tag{156}\] The center derivative is a weak derivative where \(t_z\) has a corner; its stated bound also gives the needed Lipschitz estimate. For \(r/2\le d_y\le8L_1r\) one has \(a\asymp_{L_1}r\) and \[ c_{L_1}a^w\le\tau\le Cs^w. \tag{157}\] For example, when \(d_y\ge r_0\), the formula is \(\tau=c_1L\min(d_y,s)^w\); on the indicated band \(\min(d_y,s)\ge c_{L_1}d_y\). The case \(d_y<r_0\) is comparable, since \(d_y\ge r_0/2\).

An observed-count event. Let \(M_j\) be the number of entries of \(Y_j\) with radii in \([r/4,12L_1r]\). For a sufficiently large fixed \(C_{L_1}\), \[ \mathbb P\{M_j>C_{L_1}r^{-3}\}\le r^{40}. \tag{158}\] Indeed, if the event had probability \(p>r^{40}\), its tilted raw state from Lemma 26 would have offset \[\delta\le D_0+Cr^{-2.02}\log^5(e/r^{40}).\] Each configuration compatible with the event has more than \(C_{L_1}r^{-3}\) raw points in \([r/8,13L_1r]\), because observation displacements are at most \(\sqrt3\ell_j=o(r)\). But (67) bounds its expected count by \(C'_{L_1}(r^{-3}+\sqrt{\delta r})\le2C'_{L_1}r^{-3}\) for small \(s\), a contradiction if \(C_{L_1}>2C'_{L_1}\). Call the complementary event in (158) \(\mathcal C_j\). For data in \(\mathcal C_j\), every compatible configuration has at most \(C_{L_1}r^{-3}\) centers relevant to the master kernels on \(r/2\le d_y\le8L_1r\). Conditional averaging and (156) imply there \[ \mu_j(y)\le Cr^{-6-3w},\qquad |\nabla\mu_j(y)|\le Cr^{-7-4w}. \tag{159}\]

A universal spatial field cap. Fix \(y\) in the same enlarged band and tilt an event \(A'=\{H_j(y)>b\,d_y^{-4}\}\) of positive probability \(p\). The original-law tower gives \[\mathbb E[H_j(y)\mid A'] =V(y)-\mathbb E_{\Psi_{A'}}\sum_i(K*\mathcal K_{x_i})(y) \le\mathbb E_{\Psi_{A'}}J_y.\] To justify the last inequality, a master smear whose potential at \(y\) differs from the point potential has \(|x_i-y|<t_{x_i}\) and hence \(|x_i-y|\le2t_y<Aa_y\) for small \(s\). All terms outside that exclusion therefore have their exact point potential, and dropping the remaining electron terms gives \(J_y\). By (61) without an initial cut and Lemma 26, \[ b\le C_2+C_{L_1}r^{2.49}\log^{5/2}(e/p). \tag{160}\] Here \(C_2\) is universal: the local field term in these units is \(C d_y^4a_y^{-4}=CL^4\), and the contribution \(C d_y^{7/2}\sqrt{D_0}\) is at most one after decreasing \(s\). Solving (160) for \(p\) shows that for \(b\ge2C_2\), \[\mathbb P\{d_y^4H_j(y)>b\} \le e\exp\{-c_{L_1}(b-C_2)^{2/5}r^{-0.996}\}.\] Tail integration consequently gives, for each fixed \(M>0\), \[\mathbb E(d_y^4H_j(y)-2C_2)_+\le C_{M,L_1}r^M.\] Subharmonicity permits a simultaneous conclusion without a \(Z\)-dependent derivative estimate. Apply the ball submean inequality with radius \(d_y/10\) for \(3r/4\le d_y\le6L_1r\). The averaging ball is contained in \(\{r/2\le d_z\le8L_1r\}\) and \(d_z\asymp d_y\) there, so \[d_y^4H_j(y)\le C_3+Cr^{-3} \int_{\{r/2\le d_z\le8L_1r\}} (d_z^4H_j(z)-2C_2)_+\,\,\mathrm dz.\] The constant \(C_3\) is universal. The integral term has expectation \(O_{M,L_1}(r^M)\), since its integration volume is \(O_{L_1}(r^3)\). Markov’s inequality proves that, outside an event of probability \(C_{M,L_1}r^M\), \[ H_j(y)\le C_{\rm cap}d_y^{-4} \quad(3r/4\le d_y\le6L_1r),\qquad C_{\rm cap}=C_3+1. \tag{161}\] This establishes the claimed order of dependence of the cap.

A fixed spatial point and height. Fix now \(r\le d_y\le4L_1r\) and \(h\in[h_l/2,2h_h]\). Consider either failure of (155) with tolerance \(\xi/4\), intersected with \(\mathcal C_j\). If its probability were at least \(r^{42}\), call that event \(A'\) and write \(Q=\mathbb P(\,\cdot\mid A')\). The raw marginal is the state \(\Psi_{A'}\) of Lemma 26, with \[ \delta\le D_0+Cr^{-2.02}\log^5(e/r^{42}) \le a^{-7+.02} \tag{162}\] for small \(s\). Apply the refined patch (64) to this raw state at \(y\), with \(\beta=a^{.2}\) and \(b=\beta a\). Extend \(Q\) by fresh cut labels sampled conditional on the raw positions, exactly as in that patch. Let \(X\) be these patch data, let \(\rho_X^c\) be its conditional core density, and let \(\rho,W,\sigma\) be its minimizing density, screened field, and fine-smear density. Thus \[W=V-K*\rho_X^c-K*\rho,\qquad \rho=kW_+^{3/2}\ \hbox{on the patch},\qquad \mathbb E_Q\left[D(\sigma-\rho)+\int W_-\sigma\right] \le Ca^{-7+.02}.\] The roles of these densities and their laws are distinct. The pair \(\mu_j,H_j\) keeps its original-observation definition in (152). Under the event law \(Q\), the fresh patch gives the conditional core density \(\rho_X^c\) and its field \(\Phi_X=V-K*\rho_X^c\); \(\sigma\) is the fine smear of retained out particles, whereas \(\rho\) is the auxiliary minimizing density and \(W=\Phi_X-K*\rho\). The field comparison is made after taking \(Q\)-expectations in (169) below.

Each compatible raw configuration matches the observed array \(Y_j\) after a permutation, with displacement at most \(\sqrt3\ell_j\) per position. Both configurations and their conditional averages at those same data therefore differ, when tested against \(x\mapsto\mathcal K_x(y)\), by at most \(Cr^{-3}\ell_jt^{-4}\). This follows from the count bound and the center Lipschitz estimate in (156); only entries within \(2t+O(\ell_j)\) of \(y\) can contribute. All such raw particles are retained by the fresh patch. Its radial fine smear moves a retained point by at most \(\sqrt3b\), adding at most \(Cr^{-3}bt^{-4}\) to the error. Consequently, on every compatible path, including paths where the Coulomb error is large, \[ |\mathcal R\sigma(y)-\mu_j(y)| \le Cr^{-3}(\ell_j+b)t^{-4} \le Ca^{-6}\big(a^{.01}\tau^{-4}+a^{.2}\tau^{-4}\big) =o(a^{-6}). \tag{163}\] In the comparison with the conditional average, one may first compare each raw sum with \(\sum_i\mathcal K_{Y_{j,i}}(y)\), then average this same bound using the original posterior law. The array is fixed in both comparisons, so no equality of posterior laws is being used.

Let \(G=\{D(\sigma-\rho)\le a^{-7+.01}\}\). Then \(Q(G^c)\le Ca^{.01}\). The test function \(z\mapsto\mathcal K_z(y)\) has support in \(B(y,2t)\) and gradient norm at most \(Ct^{-5/2}\). Coulomb duality gives on \(G\) \[ |\mathcal R(\sigma-\rho)(y)| \le Ca^{-7/2+.005}t^{-5/2} =Ca^{-6}a^{.005}\tau^{-5/2}=o(a^{-6}). \tag{164}\] The positivity of the two powers used so far is explicit: \(.01-4w=.00996\) and \(.005-(5/2)w=.004975\).

On \(G\), a high-density antecedent forces \[ W(y)\ge(h-\xi/16)d_y^{-4}. \tag{165}\] For if the reverse strict inequality held, the oscillation bound (66) on \(B(y,2t)\) would give \(W(z)\le(h-\xi/32)d_y^{-4}\) there for sufficiently small \(s\). This remains uniform when \(W(y)\) is arbitrarily negative: the upper estimate is \(W(y)+C\tau(a^{-4}+W(y)_-)\), and \(v\mapsto v+C\tau v_-\) is increasing for \(C\tau<1\). Using the patch equation and \(\int\mathcal K_z(y)\,\,\mathrm dz=1+O(s^w)\) would then give \[\mathcal R\rho(y) \le k(h-\xi/32)^{3/2}d_y^{-6}(1+O(s^w)) <kh^{3/2}d_y^{-6}-c_{h_l,h_h,\xi}a^{-6},\] contradicting (163)–(164). Similarly, on \(G\) a low-density antecedent forces \[ W(y)\le(h+\xi/16)d_y^{-4}. \tag{166}\] Indeed the contrary is positive, so the lower oscillation estimate gives \(W(z)\ge(h+\xi/32)d_y^{-4}\) on the contributing ball; its reaction density contradicts the low antecedent with the same strict margin. Constants from \(d_y=La\) are fixed before \(s\) is decreased.

The negative field on the exceptional patch event. The upper bound (166) passes to expectation with an \(o(a^{-4})\) error, by the patch bound \(W(y)\le Ca^{-4}\) and \(Q(G^c)\le Ca^{.01}\). To pass the lower bound to expectation we need more than this small probability. Under the high-density antecedent, (163) holds on all paths and gives \(\mathcal R\sigma(y)\ge ca^{-6}\), since \(h\ge h_l/2>0\). The support and upper bound of the master kernel imply the deterministic mass bound \[M_\sigma:=\int_{B(y,2t)}\sigma\ge ca^{-6}t^3.\] For \(Y=W(y)_-\), (66) gives throughout this ball \(W(z)_-\ge(1-C\tau)Y-C\tau a^{-4}\). Taking \(C\tau\le1/2\), integrating against \(\sigma\), and using the mass bound yields \[Y\le C(a^{-6}t^3)^{-1}\int W_-\sigma+C\tau a^{-4}.\] Multiply by \(\mathbf 1_{G^c}\) and use the refined patch budget under the same law \(Q\). We obtain \[ \mathbb E_Q[\mathbf 1_{G^c}W(y)_-] \le Ca^{-4}\big(a^{.02}\tau^{-3}+\tau\big)=o(a^{-4}), \tag{167}\] because \(.02-3w=.01997>0\). Thus the high and low antecedents give, respectively, \[ \mathbb E_QW(y)\ge(h-\xi/16)d_y^{-4}-o(a^{-4}),\qquad \mathbb E_QW(y)\le(h+\xi/16)d_y^{-4}+o(a^{-4}). \tag{168}\]

Transfer between the two conditioning procedures. Apply Lemma 27 with \(V_s=V\), \(V_c=0\) and \(h_X=W\). Its two towers are taken under the original observation law and the fresh tilted patch law, respectively. The refined patch provides all its inputs: the raw density bound and Coulomb budget are (64), the deleted count bound is (48), and \(\rho\le Ca^{-6}\) on the contributing ball by (47). Therefore \[ |\mathbb E_Q(H_j-W)(y)| \le Ca^{-4}\{\tau^{1/5}+\beta^{1/5}+\sqrt\beta +a^{.01}\tau^{-1/2}+\tau^{1/2}\}=o(a^{-4}). \tag{169}\] Here \(\beta=a^{.2}\) and (157) makes every positive-power error vanish. The source-free potential remainder actually has the sharper power \(\tau^2\), so the common upper bound is sufficient. No equality of posterior laws is used in this transfer.

On a high-failure event, its defining inequality is \(d_y^4H_j(y)<h-\xi/4\), whereas (168) and (169) give \(d_y^4\mathbb E_QH_j(y)\ge h-\xi/16-o(1)\), a contradiction for small \(s\). On a low-failure event the inequalities are reversed and give the same contradiction. Thus each such failure intersected with \(\mathcal C_j\) has probability less than \(r^{42}\). Including (158), the two implications at a fixed point and height, with tolerance \(\xi/4\), fail with probability at most \(Cr^{40}\).

Spatial and height nets. Choose a net of the closed band \(\{r\le d_y\le4L_1r\}\) with spacing at most \(r^2\) and at most \(C_{L_1}r^{-3}\) points. For example, select one band point in each intersecting cube of side \(r^2/2\); the cube count follows from the volume of its \(O(r^2)\) enlargement. Choose a finite height net in \([h_l/2,2h_h]\) with mesh at most \(\xi/64\), including its endpoints. Exclude every point-and-height failure, the count failure, and the cap failure with \(M=40\) in (161). A union bound gives exceptional probability \(Cr^{37}\), and hence at most \(Cr^{25}\) for \(r<1\).

On the retained event, (159) shows that \(d_y^6\mu_j(y)\) varies by at most \(Cr^{1-4w}=o(1)\) between a band point and a net point within \(Cr^2\). To control the field, work in a ball of radius \(cr\) around a band point, contained in the enlarged cap band. On this ball \(B\), let \(Q=K*(\mathbf 1_B\mu_j)\). Then \(H_j+Q\) is harmonic on \(B\). The value and gradient of \(Q\) are bounded by \(Cr^{-4-3w}\) and \(Cr^{-5-3w}\), respectively, so \(H_j+Q\) is bounded above by \(Cr^{-4-3w}\). Its nonnegative distance from that upper bound has, by the Poisson formula on a smaller concentric ball, oscillation bounded by its central value times the relative distance. It follows that \[ |H_j(z)-H_j(y)| \le Cr\big(r^{-4-3w}+H_j(y)_-\big) \quad(|z-y|\le Cr^2). \tag{170}\] This estimate also controls the change of the normalized field \(d_y^4H_j(y)\), since \(d_z/d_y=1+O(r)\). In particular, if its value at \(y\) is below \(h-\xi\), its value at \(z\) is below \(h-3\xi/4\) for small \(s\). Arbitrarily negative values cause no problem: the relevant upper estimate has the form \(v+Crv_-+O(r^{1-3w})\), increasing in \(v\) when \(Cr<1\). If the value at \(y\) exceeds \(h+\xi\), it is positive and the cap and (170) make its value at \(z\) greater than \(h+3\xi/4\).

Suppose now a high-density implication failed at some \((y,h)\). At a nearby spatial net point choose a height-net value \(h'\in[h-\xi/4,h-\xi/8]\). The positive lower height bound and the \(o(1)\) density variation preserve the high antecedent at \(h'\), while the field is below \(h-3\xi/4<h'-\xi/4\). This is an excluded net failure. For a low-density failure choose \(h'\in[h+\xi/8,h+\xi/4]\); its antecedent is preserved and the field exceeds \(h+3\xi/4>h'+\xi/4\). The height intervals lie inside \([h_l/2,2h_h]\) by the stipulated smallness of \(\xi\). This proves simultaneity and finishes the proposition. ◻