% Ionization energies for full Coulomb atoms
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\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\newcommand{\ecpre}{the excess-charge preprint}
\newcommand{\FSectors}{Lemmas~2.1 and~2.4--2.6 of \ecpre}
\newcommand{\FGlobalDensity}{Lemma~3.3 of \ecpre}
\newcommand{\FNeutralOffset}{Proposition~12.3 of \ecpre}
\newcommand{\FLocalization}{Lemma~3.4 of \ecpre}
\newcommand{\FLocalField}{Proposition~4.5 of \ecpre}
\newcommand{\FAnnularCount}{Corollary~4.9 of \ecpre}
\newcommand{\FInnerField}{Lemma~12.1 of \ecpre}
\newcommand{\FObservations}{Lemma~5.1 of \ecpre}
\newcommand{\FEventTilt}{Lemma~5.2 of \ecpre}
\newcommand{\FAveragedTransfer}{Lemma~5.3 of \ecpre}
\newcommand{\FFixedOffset}{Proposition~10.1 of \ecpre}
\newcommand{\FIntermediate}{Proposition~11.2 of \ecpre}
\newcommand{\FNeumann}{Lemma~3.6 of \ecpre}
\newcommand{\FInsertion}{Lemma~3.5 of \ecpre}
\newcommand{\FCoulomb}{Lemma~3.1 of \ecpre}
\newcommand{\FMaximum}{Lemma~5.4 of \ecpre}
\newcommand{\FTFPair}{Lemmas~7.1 and~7.2 of \ecpre}
\newcommand{\FTFExistence}{Lemma~7.1 of \ecpre}

\begin{document}

An atom with $Z$ protons binds on the order of $Z$ electrons. The cost of
removing $m$ of them, while the nucleus is held fixed, is the $m$-th
\emph{ionization energy} $I_m(Z)$. For large $Z$, but with $m$ small compared
with $Z$, the sharp asymptotic is $I_m(Z)\sim a_{\mathrm{TF}}m^{7/3}$, where
$a_{\mathrm{TF}}$ is a universal Thomas--Fermi coefficient. Theorem~\ref{thm:main}
below establishes this in the joint limit $m\to\infty$, $Z/m\to\infty$.

The ionization energy of an atom measures the cost of removing electrons while
leaving the nucleus fixed. The first electrons removed come from the outer,
dilute part of the cloud; the leading total energy is dominated by the much
denser inner region. Thomas--Fermi theory, introduced by L.~H.~Thomas (Proc.\
Cambridge Philos.\ Soc.\ \textbf{23} (1927), 542--548,
doi:10.1017/S0305004100011683) and E.~Fermi (Rend.\ Accad.\ Naz.\ Lincei
\textbf{6} (1927), 602--607), describes the leading large-charge total energy.
Lieb and Simon (Phys.\ Rev.\ Lett.\ \textbf{31} (1973), 681--683,
doi:10.1103/PhysRevLett.31.681; Adv.\ Math.\ \textbf{23} (1977), 22--116,
doi:10.1016/0001-8708(77)90108-6, Theorem~III.1) connected it to the
many-body Schr\"odinger theory. Their energy theorem concerns terms of order
$Z^{7/3}$. Subtracting two such total-energy asymptotics does not control a
much smaller ionization energy: the remainder may exceed the difference being
sought.

The generalized ionization conjecture asserts that Thomas--Fermi theory
nevertheless fixes the leading cost of removing many outer electrons, as long
as their number is small compared with the nuclear charge. J.~P.~Solovej
(Mol.\ Phys.\ \textbf{114} (2016), 1036--1040,
doi:10.1080/00268976.2015.1130273, Section~3 Equation~(1)) formulated the
statement as an upper and a lower large-$Z$ limit followed by a large-$m$
limit. The energy part of the conjecture is sharpened below to the joint
regime in which both the number removed and the ratio $Z/m$ tend to infinity.

Earlier full-model work covers other ionization regimes. Seco, Sigal and
Solovej (Comm.\ Math.\ Phys.\ \textbf{131} (1990), 307--315,
doi:10.1007/BF02161416, Theorem~1 and p.~315) bounded the first ionization
energy of a neutral atom by $CZ^{20/21}$. Ivrii (arXiv:1210.1132v2,
Theorem~6.2.1) obtained successive one-electron ionization energies of positive
atoms that match the Thomas--Fermi chemical potential when $Z^{5/7}=o(Z-N)$
and $Z-N=o(Z)$, where $N$ is the electron count. Both leave open the joint
neutral-to-positive-ion energy asymptotic established in the present work. Uniform excess-charge
bounds were proved in Thomas--Fermi--von Weizs\"acker theory by R.~Benguria and
E.~H.~Lieb (J.\ Phys.\ B \textbf{18} (1985), 1045--1059,
doi:10.1088/0022-3700/18/6/006), and Solovej proved uniform ionization bounds
in reduced (Invent.\ Math.\ \textbf{104} (1991), 291--311,
doi:10.1007/BF01245077) and unrestricted (Ann.\ of Math.\ \textbf{158} (2003),
509--576, doi:10.4007/annals.2003.158.509, Theorems~3.8 and~1.5)
Hartree--Fock theory. Those are different variational problems; going to
arbitrary correlated fermionic states needs additional screening estimates,
obtained below through conditional many-body states.

\medskip
\noindent\textit{The theorem and its normalization.}\quad Work in atomic units.
For a positive integer $Z$ let
\begin{equation}\label{eq:hamiltonian}
 H_{Z,N}
 =\sum_{i=1}^N\Bigl(-\tfrac12\Delta_i-\tfrac{Z}{|x_i|}\Bigr)
   +\sum_{1\le i<l\le N}\frac1{|x_i-x_l|}
\end{equation}
act on $\bigwedge^N L^2(\R^3;\C^2)$ (two spin states). Write
$E_Z(N)=\inf\sigma(H_{Z,N})$ and $E_Z(0)=0$. For integers $Z>m\ge1$ define the
$m$-th ionization energy
\[
 I_m(Z)=E_Z(Z-m)-E_Z(Z).
\]
The Thomas--Fermi kinetic coefficient for this model (two spins, kinetic
$-\tfrac12\Delta$) is $T=\tfrac3{10}(3\pi^2)^{2/3}$. The corresponding
functional on a nonnegative density $\rho$ is
\[
 \mathcal T_Z(\rho)
 =T\!\int_{\R^3}\!\rho^{5/3}
 -Z\!\int_{\R^3}\!\frac{\rho(x)}{|x|}\,dx
 +\frac12\iint_{\R^3\times\R^3}\!\frac{\rho(x)\rho(y)}{|x-y|}\,dx\,dy .
\]
Let $E_Z^{\TF}(N)$ be its infimum over nonnegative densities of mass $N$ with
displayed terms finite, and set
$I_m^{\TF}(Z)=E_Z^{\TF}(Z-m)-E_Z^{\TF}(Z)$. The constant in the conjecture is
fixed by the classical Thomas--Fermi weak-ionization law of P.~B\'enilan and
H.~Brezis (J.\ Evol.\ Equ.\ \textbf{3} (2004), 673--770,
doi:10.1007/s00028-003-0117-8, Section~6 Corollary~5):
\begin{equation}\label{eq:tf-constant}
 \lim_{Z\to\infty}I_m^{\TF}(Z)=a_{\TF}m^{7/3},\qquad m>0 .
\end{equation}
The coefficient $a_{\TF}$ is recomputed below in the present normalization,
which is what fixes $a_{\TF}$ on both sides.

\begin{theorem}[Generalized ionization energy]\label{thm:main}
For the full nonrelativistic Coulomb atoms with two spin states,
\begin{equation}\label{eq:joint-limit}
 \frac{I_m(Z)}{m^{7/3}}\longrightarrow a_{\TF}>0
 \quad\text{whenever}\quad m\longrightarrow\infty,
 \qquad \frac Zm\longrightarrow\infty,
\end{equation}
with integers $Z>m\ge1$. In particular,
\begin{align}
 \lim_{m\to\infty}\frac{\limsup_{Z\to\infty} I_m(Z)}{m^{7/3}}
   &=a_{\TF},\label{eq:iterated-upper}\\
 \lim_{m\to\infty}\frac{\liminf_{Z\to\infty} I_m(Z)}{m^{7/3}}
   &=a_{\TF}.\label{eq:iterated-lower}
\end{align}
The inner limits are over integer $Z$ with $m$ fixed. No convergence of
$I_m(Z)$ at a fixed $m$ is asserted or assumed.
\end{theorem}

The iterated limits \eqref{eq:iterated-upper}--\eqref{eq:iterated-lower} are
exactly the energy part of the conjecture above; the joint statement
\eqref{eq:joint-limit} strengthens the order of limits. The energy is
the exact spectral infimum of \eqref{eq:hamiltonian}, with no restriction to
Slater determinants. For the same model, \ecpre, \emph{Uniform excess charge
for Coulomb molecules and the outer radius of neutral atoms} (2026), also
proves $0<c\le E_Z(Z-1)-E_Z(Z)\le C<\infty$ uniformly in $Z$. That
fixed first-ionization bound does not assert convergence at fixed $m$ and is
distinct from the large-$m$ limits above.

\medskip
\noindent\textit{A price per electron.}\quad The proof allows the electron
number to float. Fix $Z$, write $E_n=E_Z(n)$, and introduce the unconstrained
energy and the priced minimum
\[
 E=\inf_{n\ge0}E_n,\qquad
 P(\lambda)=\min_{n\ge0}\bigl(E_n+\lambda n\bigr)\quad(\lambda>0).
\]
The minimum is attained (Lemma~\ref{lem:priced-rough}). Locality is its point:
replacing the exterior electrons by a trial state need not preserve their
number, because the price charges the change exactly. At the length scale
$s=\lambda^{-1/4}$ the removed charge is of order $s^{-3}$. The main
intermediate theorem identifies its coefficient, for every minimizing sector,
including prices at which several sectors tie.

\begin{theorem}[Deficit at a prescribed price]\label{thm:priced-deficit}
There is a universal number $q>0$ such that for every sequence with
\[
 \lambda=s^{-4},\qquad s\longrightarrow0,\qquad Zs^3\longrightarrow\infty,
\]
and every choice of $n$ minimizing $E_n+\lambda n$, the scaled deficit
$s^3(Z-n)$ tends to $q$. Moreover $q$ is the exterior Coulomb charge of the
unique classical profile $F$ solving
\[
 \Delta F=4\pi k\bigl((F-1)_+\bigr)^{3/2}\ \ \text{on }\R^3\setminus\{0\},
 \qquad k=(5T/3)^{-3/2},
\]
tending uniformly to zero at infinity and bounded above and below by positive
multiples of $|x|^{-4}$ near $0$. The density $k((F-1)_+)^{3/2}$ has bounded
support and $F(x)=q/|x|$ outside it. Finally $a_{\TF}=\tfrac37q^{-4/3}$.
\end{theorem}

\medskip
\noindent\textit{Analytic precedents and proof structure.}\quad The local
kinetic comparison merges two classical constructions: coherent wave packets
give the upper semiclassical energy (E.~H.~Lieb, Rev.\ Modern Phys.\
\textbf{53} (1981), 603--641, doi:10.1103/RevModPhys.53.603, Section V.A.1;
also Solovej 2003, Lemma~8.2), and Neumann cube eigenvalues with their
lattice count give the matching lower comparison at the same leading
coefficient (Lieb--Simon 1977, Theorems III.10--III.13). Their remainders are
controlled by local particle counts. This is finer than the coarse
Lieb--Thirring inequality (E.~H.~Lieb and W.~E.~Thirring, Phys.\ Rev.\ Lett.\
\textbf{35} (1975), 687--689, doi:10.1103/PhysRevLett.35.687), whose universal
positive constant suffices only for preliminary density bounds. The spectral
split follows M.~Rumin (Duke Math.\ J.\ \textbf{160} (2011), 567--597,
doi:10.1215/00127094-1444305, Theorem~1.4 and Section~3.1).

Screened exterior potentials and outward Thomas--Fermi comparisons are central
to Solovej's Hartree--Fock argument (Ann.\ Math.\ \textbf{158} (2003),
Sections~6, 10, 12). The conditional states below require a separate transfer
from local quantum energies to conditional fields, while tracking changes in
particle number. The eventual singular profile belongs to the classical
Sommerfeld comparison theory (Lieb--Simon 1977, Section~IV), including the
nonspherical and chemical-potential comparisons of Solovej 2003 (Lemma~4.4,
Remark~4.5, Theorem~4.6, Corollary~4.7). The issue is to extract that profile
from the correlated quantum state and preserve its positive singularity
through conditioning.

The localization, screening, and Thomas--Fermi existence results used below
are drawn from \ecpre, \emph{Uniform excess charge for Coulomb molecules and
the outer radius of neutral atoms} (2026). Two of its consequences matter at
once: the uniform neutral-sector bound $E_Z(Z)-E\le C$ fixes the zero-price
integration constant, and the local-field estimates keep explicit control of
an arbitrary energy offset (\FNeutralOffset, \FLocalField). The positive-price
conditional comparisons and barrier propagation are proved below. The offset
dependence is essential: a minimizing priced state has energy at most
$E+Cs^{-7}$, so the allowed offset grows as $s\to0$, and a fixed-offset
conclusion from the neutral-atom theory cannot by itself yield
Theorem~\ref{thm:main}.

The argument proceeds in four stages. The price per electron and the rough
size of the deficit are analysed, and the rescaled exterior electron mass is
shown to vanish beyond a fixed radius. Noisy observations then permit a local
conditional comparison at controlled energy cost: a Thomas--Fermi trial state
in a small ball is compared with the priced many-body state, a fresh cut
records the local particles and leaves a core whose conditional screened
potential is harmonic on the ball, and matching this with the observation law
yields simultaneous upper and lower implications between local density and
local potential; two posterior laws occur and must be kept distinct. A
positive subsolution is then built at the innermost scale and propagated
through the observation scales, so the singular inner charge survives the
successive averaging; the error paid for exceptional conditional data has
vanishing expected mass, and the weak inequality is proved on open
neighborhoods of the retained branches, not across an activity interface.
Finally, conditional data are selected, rescaled by $s$, and passed to the
limit, producing a decaying solution of the unit-price Thomas--Fermi equation;
the barrier prevents loss of the singularity, a maximum-principle argument
fixes the Sommerfeld coefficient, and uniqueness forces the same deficit for
every minimizing sector. Only monotonicity of $E_Z(N)$ is then needed to
bracket the fixed particle number between two minimizing sectors (convexity
in $N$ is not used); the matching Thomas--Fermi calculation identifies
$a_{\mathrm{TF}}$, and a sequential argument gives the stated order of limits.

\medskip
The price per electron permits insertion and deletion of electrons without constraining
the exterior particle number. First, collect the screening estimates that keep
working when the energy above the unpriced minimum grows. Throughout this
section write
\[
 K(x)=\frac1{|x|},\qquad V=ZK,\qquad \nu=Z\delta_0,\qquad
 D(f)=\frac12\iint_{\R^3\times\R^3}\frac{f(x)f(y)}{|x-y|}\,dx\,dy,
\]
and use the Thomas--Fermi constants for two spins and kinetic $-\tfrac12\Delta$,
\[
 T=\frac3{10}(3\pi^2)^{2/3},\qquad k=\Bigl(\frac{5T}{3}\Bigr)^{-3/2}.
\]
For a fixed integer $Z$ abbreviate $E_n=E_Z(n)$, put $E_0=0$, and define
\[
 E=\inf_{n\ge0}E_n,\qquad
 P(\lambda)=\min_{n\ge0}(E_n+\lambda n)\quad(\lambda>0).
\]
Attainment of this minimum is proved below. The asymptotic regime used until
the fixed-particle-number analysis is
\begin{equation}\label{eq:scaling-regime}
 \lambda=s^{-4},\qquad s\longrightarrow0,\qquad Zs^3\longrightarrow\infty .
\end{equation}
Every statement about a minimizing sector holds for every choice of minimizing
index at every system in this regime.

Call a normalized finite-sector form-domain vector $\psi$ an \emph{unshifted
$\delta$-state} when its form energy is at most $E+\delta$ ($\delta\ge0$). Its
position law is $|\psi|^2$, with spins summed unless recorded; its one-particle
density is $\rho_\psi$. For real $v$ write $v_+=\max(v,0)$ and $v_-=\max(-v,0)$.
Constants $C,c>0$ may change from line to line and, unless a dependence is
shown, depend only on the fixed model and the fixed cutoff and kernel
conventions.

\medskip
\noindent\textit{Imported estimates and their hypotheses.}\par\medskip
\begin{lemma}[Sector and energy estimates]\label{lem:sector-inputs}
The sequence $E_n$ is nonincreasing. If $E_n<E_{n-1}$ there is a normalized
ground state in sector $n$, and $n\le2Z+1\le3Z$. There is a largest such index
$N_*$ with $Z\le N_*\le3Z$ and
\[
 E=E_{N_*}=E_n\quad(n\ge N_*).
\]
There are universal $C,Z_0$ with
\begin{equation}\label{eq:neutral-offset}
 E_Z(Z)\le E+C\qquad(Z\ge Z_0).
\end{equation}
If $\Psi$ is a normalized sector ground state and $F$ is a bounded real symmetric
Lipschitz function of the spatial configuration, then
\begin{equation}\label{eq:ground-multiplier}
 \langle F\Psi,H_{Z,n}F\Psi\rangle-E_n\|F\Psi\|_2^2
       =\tfrac12\,\E_\Psi|\nabla F|^2,
\end{equation}
as a quadratic form, the gradient including all spatial coordinates. Finally,
every unshifted $\delta$-state in a sector $n\le3Z$ satisfies
\begin{equation}\label{eq:global-density-input}
 \int_{\R^3}\rho_\psi^{5/3}\le C(Z^{7/3}+\delta).
\end{equation}
\end{lemma}

\begin{proof}
The sector assertions and \eqref{eq:ground-multiplier} are \FSectors; the
density bound \eqref{eq:global-density-input} is \FGlobalDensity; the stronger
$E_Z(Z-1)\le E+C$ is \FNeutralOffset. Monotonicity gives
\eqref{eq:neutral-offset}. All use exactly the Hamiltonian and two-spin
convention fixed above.
\end{proof}

A \emph{one-particle square cut} is a pair of nonnegative bounded-gradient
Lipschitz functions $o,c$ with $o^2+c^2=1$. Given the positions, assign particles
independently to the out and core groups with probabilities $o^2$ and $c^2$; the
\emph{cut data} record the out labels, positions, and spins (and, for a finite
sequence of cuts, the stage labels). Core states are the corresponding products
of cut factors, so antisymmetry holds within each group. This is the
localization convention of \FLocalization.

The local estimate needed later has a quantitative condition on 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
\[
 \begin{gathered}
 m_y=\max(a_y^{-3},1)+\sqrt{\delta a_y},\qquad
 J_y=V(y)-\sum_{|x_i-y|\ge Aa_y}\frac1{|y-x_i|},\\
 P_*=\max\Bigl(1,\ \sup_{y\ne0}\frac{\E\bigl[\#\{i:x_i\in B_y\}^2\bigr]}{m_y^2}\Bigr).
 \end{gathered}
\]
$P_*$ is finite for each finite system, since the count is at most the sector
particle number and $m_y\ge1$.

\begin{lemma}[Local fields for arbitrary offsets]\label{lem:local-field-input}
For every normalized unshifted $\delta$-state of finite particle number,
$P_*\le C$. Fix $y\ne0$ and $C_0\ge1$. The data $X$ may be trivial; otherwise
suppose the initial cut, or finite sequence of cuts $(o_h,c_h)$, has total out
support covered by at most $C_0$ cells $B_z$ with
\[
 C_0^{-1}\le a_z/a_y\le C_0,\quad
 \sum_h\bigl(|\nabla o_h|^2+|\nabla c_h|^2\bigr)
       \le\sum_z D_z^2a_z^{-2}\ind_{B_z},\quad
 \frac{D_z^2}{a_zm_z}\le C_0\sqrt{P_*}.
\]
Then for the complete cut data
\begin{equation}\label{eq:local-field-input}
 \bigl\|(\E[J_y\mid X])_+\bigr\|_2\le C(C_0)\,\frac{m_y}{a_y}.
\end{equation}
The constant is independent of $Z$, the sector, the state, $y$, and $\delta$,
apart from the displayed $m_y$-dependence.
\end{lemma}

\begin{proof}
This is \FLocalField, whose offset is measured from the minimum $E$ over all
sectors, with no bound required on the offset along a sequence. All later applications
always verify the stronger cut condition $D_z^2/(a_zm_z)\le C_0$.
\end{proof}

\begin{lemma}[Annular screening estimates]\label{lem:screening-inputs}
Let $\psi$ be a normalized unshifted $\delta$-state of finite particle number.
For fixed $0<\alpha<\beta<\infty$ and every $u>0$,
\begin{equation}\label{eq:annular-count}
 \bigl\|\#\{i:\alpha u\le|x_i|\le\beta u\}\bigr\|_2
   \le C_{\alpha,\beta}\bigl(\max(u^{-3},1)+\sqrt{\delta u}\bigr).
\end{equation}
Suppose also $n\le3Z$ and set
$B_{u/8}(x)=V(x)-\sum_{|x_i|<u/8}|x-x_i|^{-1}$. Let $Y$ be trivial, or the data
of a square cut whose out support lies in $\{u/4\le|x|\le8u\}$ and whose squared
gradient sum is at most $C/u^2$. There are conditional versions harmonic on
$u/4<|x|<6u$ with
\begin{equation}\label{eq:conditional-inner-field}
 \Bigl\|\sup_{u/2\le|x|\le4u}\bigl(\E[B_{u/8}(x)\mid Y]\bigr)_+\Bigr\|_2
 \le C\Bigl(\frac{\max(u^{-3},1)}u+\sqrt{\frac\delta u}\Bigr).
\end{equation}
The constant may depend on the fixed gradient bound. Both allow every finite
$\delta$.
\end{lemma}

\begin{proof}
\eqref{eq:annular-count} is \FAnnularCount. Boundary spheres have zero probability
(the density is absolutely continuous), so open or closed annuli agree. For
\eqref{eq:conditional-inner-field} use \FInnerField with inner cutoff $h=u/8$,
broad annulus $u/4<|x|<6u$, smaller annulus $u/2\le|x|\le4u$; its series is
\[
 \frac{\max(u^{-3},1)+\sqrt{\delta u}}u
 +\sum_{j\ge0}\frac{\max((2^jh)^{-3},1)+\sqrt{\delta\,2^jh}}{\max(u,2^jh)},
\]
which with $h=u/8$ geometrically sums to the right side of
\eqref{eq:conditional-inner-field}, also for $u<1$. The cut support and gradient
bound meet the fixed-annulus hypotheses. The conditional harmonic versions
follow by integration against the conditional law: for each fixed finite system
the inner sources stay uniformly separated from the annulus, so differentiation
under the integral is valid. No minimizing property is used.
\end{proof}

\medskip
\noindent\textit{The rough charge deficit.}\par\medskip
\begin{lemma}\label{lem:priced-rough}
$P(\lambda)$ is attained. Along \eqref{eq:scaling-regime} every minimizing index
$n$ is eventually nonzero and strictly bounded. For a normalized ground state
$\Psi$ in that sector put $D_0=E_n-E$. Uniformly over all choices,
\begin{equation}\label{eq:rough-deficit}
 |Z-n|\le Cs^{-3},\qquad 0\le D_0\le Cs^{-7}.
\end{equation}
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:sector-inputs}, $E_n$ is constant for $n\ge N_*$, and since
$\lambda>0$ the objective is strictly increasing there, so the minimum is
attained. A positive minimizing index compares with $n-1$ to give
$E_{n-1}-E_n\ge\lambda>0$, so it is strictly bounded, has a ground state, and
satisfies $n\le3Z$. A normalized exponential one-electron trial of radius
$Z^{-1}$ has energy $-Z^2/2$, and $\lambda/Z^2=s^2/(Zs^3)^2\to0$, so
$P(\lambda)<0$ eventually and the vacuum is excluded.

The neutral sector competes. By \eqref{eq:neutral-offset},
\begin{equation}\label{eq:priced-offset}
 0\le D_0\le C+s^{-4}(Z-n).
\end{equation}
For fixed inner configuration the spherical average of $B_{s/8}$ on $|x|=s$ is
$(Z-\#\{|x_i|<s/8\})/s$, with expectation at least $(Z-n)/s$. Apply
\eqref{eq:conditional-inner-field} with trivial data and offset $D_0$: for small
$s$,
\[
 Z-n\le C\bigl(s^{-3}+\sqrt{D_0s}\bigr).
\]
If $d=Z-n\ge0$ and $z=s^3d$, then \eqref{eq:priced-offset} gives
$D_0s^7\le Cs^7+z$, hence $z\le C(1+\sqrt{z+Cs^7})$, bounding $z$. If $d<0$,
\eqref{eq:priced-offset} gives $d\ge-Cs^4$. These two cases prove the deficit
estimate, and substitution in \eqref{eq:priced-offset} the bound for $D_0$.
\end{proof}

\medskip
\noindent\textit{A cutoff at one fixed exterior scale.}\par\medskip
\begin{lemma}\label{lem:exterior-cutoff}
There is a sufficiently large fixed $R_0$ such that, for every sequence of
minimizing sectors and ground states in \eqref{eq:scaling-regime},
\begin{equation}\label{eq:exterior-cutoff}
 s^3\,\E_\Psi\#\{i:|x_i|>R_0s\}\longrightarrow0.
\end{equation}
\end{lemma}

\begin{proof}
Take smooth $0\le\chi\le1$ supported in $[1/2,4]$ and equal to one on $[1,2]$. At
shell scale $u>0$ put $w_i=\chi(|x_i|/u)^2$, $S_u=\sum_iw_i$, $m_u=\E S_u$, and
$N_u=\E\#\{i:u/2\le|x_i|\le4u\}$. Test the weak eigenvalue equation with
$S_u\Psi$. In the $i$-th summand the other coordinates have energy at least
$E_{n-1}$, and integration by parts gives the kinetic piece
\[
 \tfrac12\operatorname{Re}\!\int\nabla_i\overline\Psi\cdot\nabla_i(w_i\Psi)
 =\tfrac12\int w_i|\nabla_i\Psi|^2-\tfrac14\E\Delta_iw_i
 \ge-\tfrac14\E\Delta_iw_i.
\]
Keep only repulsions from $|x_l|<u/8$ (the selected particle is outside that ball
whenever $w_i>0$). Since $E_{n-1}-E_n\ge s^{-4}$,
\begin{equation}\label{eq:shell-eigen-equation}
 s^{-4}m_u\le Cu^{-2}N_u+\E\sum_iw_iB_{u/8}(x_i).
\end{equation}
All tests are bounded symmetric form-domain multipliers for each fixed system.

Suppose $m_u>0$ and bias the position law by $S_u/m_u$, i.e. use
$\Psi_u=\sqrt{S_u/m_u}\,\Psi$. The function $\sqrt{S_u}$ is the Euclidean norm of
the cutoff vector, Lipschitz including at its zeros, with
$|\nabla\sqrt{S_u}|^2\le Cu^{-2}\#\{i:u/2\le|x_i|\le4u\}$. By
\eqref{eq:ground-multiplier} $\Psi_u$ is unshifted with offset at most
\begin{equation}\label{eq:shell-bias-offset}
 \delta_u=D_0+Cu^{-2}N_u/m_u.
\end{equation}
In both laws use the same auxiliary square cut, full out on $u/2\le|x|\le4u$,
supported in $u/4\le|x|\le8u$, gradient $\le C/u$. Its data $Y$ determine
$S_u$ and every positive-weight position, and biasing by the $Y$-measurable
$S_u/m_u$ leaves the conditional raw law on $S_u>0$ unchanged, so the conditional
harmonic fields there coincide. Let
$M(Y)=\sup_{u/2\le|x|\le4u}(\E[B_{u/8}(x)\mid Y])_+$. Conditioning on $Y$,
\[
 \E\sum_iw_iB_{u/8}(x_i)\le\E[S_uM(Y)]=m_u\,\E_{\Psi_u}M(Y).
\]
Applying \eqref{eq:conditional-inner-field} to $\Psi_u$ with
\eqref{eq:shell-bias-offset},
\begin{align}
 \E\sum_iw_iB_{u/8}(x_i)
 &\le Cm_u\Bigl(\frac{\max(u^{-3},1)}u+\sqrt{D_0/u}\Bigr)
       +Cu^{-3/2}\sqrt{m_uN_u}. \label{eq:shell-field-bound}
\end{align}
If $m_u=0$ the shell estimate is immediate.

By \eqref{eq:rough-deficit}, for $u\ge R_0s$,
$s^4\max(u^{-3},1)/u\le R_0^{-4}+s^4$ and
$s^4\sqrt{D_0/u}\le C\sqrt{s/u}\le CR_0^{-1/2}$. Choose $R_0$ large, then $s$
small, and absorb the $m_u$-terms of \eqref{eq:shell-field-bound} into
\eqref{eq:shell-eigen-equation}. Since $m_u\le N_u$ the mixed term is at most
$Cu^{-3/2}N_u$, so
\[
 m_u\le Cs^4(u^{-2}+u^{-3/2})N_u,
 \qquad
 N_u\le C\bigl(u^{-3}+1+s^{-7/2}\sqrt u\bigr).
\]
The full-weight shells $u=2^jR_0s$, $j\ge0$, cover the exterior; Tonelli and the
two bounds give
\begin{align*}
 s^3\,\E\#\{|x_i|>R_0s\}
 &\le C s^7\sum_{j\ge0}
   (u^{-2}+u^{-3/2})(u^{-3}+1+s^{-7/2}u^{1/2})\Big|_{u=2^jR_0s}\\
 &\le C_{R_0}(s^2+s^{5/2}+s^5+s^{11/2})\longrightarrow0.
\end{align*}
Every $u$-power in the sum is negative, so the estimate covers arbitrarily distant
shells with the same fixed $R_0$.
\end{proof}

\medskip
\label{sec:comparison}

A local Thomas--Fermi relation is derived for a conditional density. Conditioning
keeps information about the electron configuration while allowing a controlled
change of state on any event of appreciable probability. Two different
conditional laws enter: observations define the density to be estimated, while a
fresh spatial cut defines the core used in the energy comparison. The two are kept
separate and match their potentials only after taking expectation.

Throughout, $\Psi$ is a normalized ground state in any sector $n$ minimizing
$P(\lambda)$, with $\lambda=s^{-4}$ and \eqref{eq:scaling-regime} in force; then
$n\le3Z$ and $D_0=E_n-E\le Cs^{-7}$ by Lemma~\ref{lem:priced-rough}. Statements
about sufficiently small scales are uniform in the choice of minimizing sector.

\medskip
\noindent\textit{Nested observations and their energy cost.}\par\medskip
Fix $0<\eps<1$, to be chosen later, and set
\begin{equation}
 r_0=\eps Z^{-1/3},\quad r_j=2^jr_0,\quad
 J_*=\max\{j\ge0:r_j\le s\},\quad
 \ell_j=r_j^{1.01}\ (0\le j<J_*).
 \label{eq:observation-scales}
\end{equation}
Eventually $2r_0\le s$. At each scale $j<J_*$ observe the unordered array of
points $x_i+\ell_jU_{ji}$, all coordinates of all $U_{ji}$ independent and
independent of the raw configuration, with density
$\zeta(u)=c_\zeta\exp(-1/(1-u^2))\ind_{\{|u|<1\}}$. Labels are forgotten in each
array. Let $\mathcal F_j$ be the information in arrays $j,\dots,J_*-1$ and
$\mathcal F_{J_*}$ be trivial; these sigma-fields decrease with $j$.

Use $L=10^5$, $A=40$, $a_y=|y|/L$ from Lemma~\ref{lem:local-field-input}. Fix a
radial $g\in C_c^\infty(B(0,1))$ with $\int g^2=1$, set
$g_b(x)=b^{-3/2}g(x/b)$, and choose $w=10^{-5}$, $0<c_1<(10L)^{-1}$. Writing
$d_x=|x|$, define master packets and conditional fields by
\begin{equation}
 \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)&=\E\Bigl[\sum_i\mathcal K_{x_i}(y)\,\Big|\,\mathcal F_j\Bigr],\\
 H'_j(y)&=V(y)-\lambda-(K*\mu_j)(y).
 \end{aligned}
 \label{eq:master-kernel}
\end{equation}
The superscript $\mathrm b$ denotes the lower clamp. The width is globally
Lipschitz with $\operatorname{Lip}(t)\le c_1(1+w)s^w$ and
$t_x\ge c_1r_0^{1+w}$. Hence 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 continuous and
deterministically bounded. These fixed-system bounds justify conditional
integration and are not uniform in $Z$.

The versions used are obtained by integrating against the conditional law of the raw
configuration (specified, e.g., by the finite-array likelihoods of
\FObservations). They satisfy, simultaneously in space,
\begin{equation}
 \E[\mu_j\mid\mathcal F_{j+1}]=\mu_{j+1},\quad
 \E[H'_j\mid\mathcal F_{j+1}]=H'_{j+1},\quad
 \Delta H'_j=-4\pi\nu+4\pi\mu_j.
 \label{eq:observation-tower}
\end{equation}
Continuity extends identities first proved on a countable dense set; in
particular $H'_j$ is subharmonic off the nucleus.

\begin{lemma}[Cost of an observation event]\label{lem:event-tilt}
Let $j<J_*$ and let $A'\in\mathcal F_j$ have probability $p>0$. There is a
normalized antisymmetric state $\Psi_{A'}$ in the same sector whose raw law is
the raw marginal conditioned on $A'$ and whose form energy satisfies
\begin{equation}
 \langle\Psi_{A'},H_{Z,n}\Psi_{A'}\rangle
 \le E_n+C r_j^{-2.02}\log^5(e/p).
 \label{eq:event-tilt-cost}
\end{equation}
The constant is independent of the particle number, the number of observed
arrays, and $D_0$.
\end{lemma}

\begin{proof}
If $p_{A'}(x)$ is the probability of the event given raw positions, take
$\Psi_{A'}=\sqrt{p_{A'}(x)/p}\,\Psi$. The multiplier is symmetric and preserves
the form domain, and the first inequality of \FEventTilt with $\ell_j=r_j^{1.01}$
is exactly \eqref{eq:event-tilt-cost}. Its proof uses the ground-state multiplier
identity in sector $n$; it measures the cost above $E_n$, independently of the
offset of $E_n$ from $E$.
\end{proof}

So the unshifted offset of the tilted state is at most
$D_0+Cr_j^{-2.02}\log^5(e/p)$, while its shifted energy is at most
\begin{equation}
 P(\lambda)+Cr_j^{-2.02}\log^5(e/p).
 \label{eq:shifted-tilt-cost}
\end{equation}
This distinction permits the local comparison below for $D_0$ up to order $s^{-7}$.

\medskip
\noindent\textit{Statement and preliminary bounds.}\par\medskip
\begin{proposition}[Simultaneous inverse comparison]
\label{prop:inverse-comparison}
There is a universal constant $C_{\mathrm{cap}}$ with the following property.
Fix $M\ge2$, $0<h_l<h_h<\infty$, and $0<\xi<h_l/16$. For all sufficiently far
systems along \eqref{eq:scaling-regime}, at every $j<J_*$ there is an event
$\mathcal G_j\in\mathcal F_j$ such that, with $r=r_j$,
$\Prob(\mathcal G_j^c)\le Cr^{25}$. On $\mathcal G_j$, simultaneously for
$r\le d_y\le4Mr$ and $h_l\le h\le h_h$,
\begin{equation}
 \begin{split}
 d_y^4H'_j(y)&\le C_{\mathrm{cap}},\\
 d_y^6\mu_j(y)>kh^{3/2}
       &\Longrightarrow d_y^4H'_j(y)\ge h-\xi,\\
 d_y^6\mu_j(y)<kh^{3/2}
       &\Longrightarrow d_y^4H'_j(y)\le h+\xi.
 \end{split}
 \label{eq:inverse-comparison}
\end{equation}
The probability constant and the far-system threshold may depend on the displayed
fixed parameters and the packet conventions; $C_{\mathrm{cap}}$ is independent of
the band and accuracy parameters.
\end{proposition}

The two implications compare the field with each positive density height from
opposite sides; they yield an approximate Thomas--Fermi relation in
the limit analysis while allowing the field to be arbitrarily negative where the
density is small.

The proposition is proved in the rest of the section, using the arbitrary-offset
screening estimates of the price-locality discussion and the cost \eqref{eq:event-tilt-cost};
the fixed-offset comparison \FFixedOffset with a growing $D_0$ is not applied.

Fix $j<J_*$, $r=r_j$. On the enlarged band $r/2\le d_y\le8Mr$ write
$a=a_y$, $t=t_y$, $\tau=t/a$. Uniformly there, for fixed $M$,
\begin{equation}
 a\asymp_M r,\qquad c_Ma^w\le\tau\le Cs^w.
 \label{eq:packet-scale-ratios}
\end{equation}
If $\mathcal K_x(y)\ne0$, the width Lipschitz bound gives $|x-y|\le2t$ and
$t_x/t=1+O(s^w)$; hence by change of variables
\begin{equation}
 \int\mathcal K_z(y)\,dz=1+O(s^w).
 \label{eq:packet-approximate-mass}
\end{equation}
On the local support the kernel is bounded by $Ct^{-3}$ and Lipschitz in either
argument with constant $Ct^{-4}$.

\paragraph{A count event.}
Except on an event of probability at most $r^{40}$, the $j$-th array has at most
$C_Mr^{-3}$ entries in $[r/4,12Mr]$. To see this, condition on exceeding a large
threshold; if its probability exceeds $r^{40}$, compact noise support forces every
compatible raw configuration to have that many points in $[r/8,13Mr]$, whose
conditional expected count by Lemma~\ref{lem:event-tilt} and
\eqref{eq:annular-count} is at most $C_Mr^{-3}$; indeed
$\sqrt{D_0r}\le Cr^{-3}(r/s)^{7/2}\le Cr^{-3}$ and
$r^{-0.51}\log^{5/2}(e/r^{40})=o(r^{-3})$, a contradiction for a large fixed
threshold. Let $\mathcal C_j$ be the retained event. On it every compatible raw
configuration has at most $C_Mr^{-3}$ relevant centers, so integration against the
observation posterior gives
\begin{equation}
 0\le\mu_j(y)\le C_Mr^{-6-3w},\qquad
 |\nabla\mu_j(y)|\le C_Mr^{-7-4w}.
 \label{eq:conditional-density-bounds}
\end{equation}

\paragraph{A universal field cap.}
At a fixed $y$ of the enlarged band condition on $\{d_y^4H'_j(y)>b_0\}$, of
probability $p>0$. Packets centered outside the exclusion ball in $J_y$ do not
contain $y$, and their potentials at $y$ equal the point potentials by radial
averaging. Dropping the other nonnegative packet potentials and using the tower
bounds the event-mean of $H'_j(y)$ above by the tilted mean of $J_y-\lambda$; the
trivial-data case of Lemma~\ref{lem:local-field-input} gives
\begin{equation}
 b_0\le C+C(d_y/s)^{7/2}-(d_y/s)^4+C_Mr^{2.49}\log^{5/2}(e/p).
 \label{eq:field-cap-tail}
\end{equation}
In this setting $a<1$ eventually, and the polynomial-power difference has a universal upper
bound. Choose a universal $C'$ exceeding the whole deterministic bound by one;
solving \eqref{eq:field-cap-tail} for $p$ and integrating its stretched-exponential
tail gives, for every fixed $P>0$,
\begin{equation}
 \E\bigl(d_y^4H'_j(y)-C'\bigr)_+\le C_{P,M}r^P.
 \label{eq:cap-excess}
\end{equation}
Subharmonicity makes this simultaneous. For $3r/4\le d_y\le6Mr$ take a ball of
radius $d_y/10$ in the enlarged band; its mean bounds $H'_j(y)$ above. The
$C'd_z^{-4}$ contribution after multiplying by $d_y^4$ is bounded universally, and
the rest is at most
$Cr^{-3}\int_{\{r/2\le d_z\le8Mr\}}(d_z^4H'_j(z)-C')_+dz$, which by
\eqref{eq:cap-excess} and Markov is at most one off an event of probability at most
$C_Mr^{40}$. This is the simultaneous cap
\begin{equation}
 H'_j(y)\le C_{\mathrm{cap}}d_y^{-4}\quad(3r/4\le d_y\le6Mr)
 \label{eq:enlarged-field-cap}
\end{equation}
with universal $C_{\mathrm{cap}}$.

\medskip
\noindent\textit{A fine patch comparison with the price included.}\par\medskip
Fix $y$ in the original band and $h_l/2\le h\le2h_h$. Suppose one of the events,
intersected with $\mathcal C_j$, has probability $p\ge r^{42}$:
\begin{equation}
 \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}
 \label{eq:fixed-point-failures}
\end{equation}
Let $Q$ be the original joint law conditioned on the event. When cuts are
introduced, enlarge this law by fresh labels with ordinary conditional
probabilities given the raw positions; its raw marginal is the state of
Lemma~\ref{lem:event-tilt}. Set
$\delta'=Cr^{-2.02}\log^5(e/r^{42})$, $\delta=D_0+\delta'$. Then the raw state has
unshifted offset at most $\delta$, shifted offset at most $\delta'$, and
\begin{equation}
 m_y\le C_Ma^{-3},\qquad \delta'=o(a^{-6.98}).
 \label{eq:patch-offsets}
\end{equation}
Constants below may depend on the fixed band parameters.

Put $\beta=a^{0.2}$, $b=\beta a$, with $s$ small enough that $\beta<1/100$. For
fixed $M$ the scales satisfy $b\ll\ell_j\ll t\ll a$, each consecutive ratio tending
to zero uniformly in $j$, also when $r/s\to0$, by \eqref{eq:packet-scale-ratios}.
The fine width $b$ makes semiclassical errors small; the broader master width $t$
absorbs both this smearing and the displacement $\ell_j$, while $t/a\to0$ permits
the local field comparison.

Choose a square cut, using the sine and cosine of a smooth angle, full out on
$B(y,t_0)$, supported out in $B(y,t_0+b)$, with gradients bounded by $C/b$, where
$5a\le t_0\le6a$ is selected below. Retain the out particles with
$|x_i-y|<t_0-7b$; call the others deleted. Let $n_o$ be the total out count and
$n_{\mathrm{del}}$ the deleted count. Then $t_0$ may be chosen so that
\begin{equation}
 \E_Q n_o^2\le C_Ma^{-6},\qquad \E_Q n_{\mathrm{del}}^2\le C_M\beta a^{-6}.
 \label{eq:patch-counts}
\end{equation}
For every candidate $t_0$ the total out count is bounded by the raw count in
$B(y,7a)$, in a fixed annulus of radius $\sim a$ about the nucleus; the annular
count \eqref{eq:annular-count} with \eqref{eq:patch-offsets} bounds its expected
square by $C_Ma^{-6}$ uniformly in $t_0$. The deleted particles lie in the strip
$t_0-7b\le|x_i-y|\le t_0+b$; integrating its squared count over $t_0\in[5a,6a]$,
each ordered pair contributes for lengths at most $8b$, so averaging selects $t_0$
for the second bound of \eqref{eq:patch-counts} while preserving the first.

\paragraph{The fresh conditional core.}
In the cut marginal of the tilted state let $X$ record the out labels, positions,
and spins; slice only on these fresh cut data. Denote the normalized core slice's
density by $\rho_X^c$ and its energy, with price per core electron, by
$e_X^\lambda$. Then $e_X^\lambda\ge P(\lambda)$ and
$\Phi=V-\lambda-K*\rho_X^c$. Localization products preserve antisymmetry within
each group, and the slices lie in their sector form domains a.e. by
\FLocalization.

There is a nonnegative $F=F(X)$ with
\begin{equation}
 \Phi\le F\text{ on }B(y,7a),\qquad \|F\|_{L^2(Q)}\le C_Ma^{-4}.
 \label{eq:patch-field-envelope}
\end{equation}
Condition the field that subtracts only electrons outside $B(y,20a)$ (all in the
core); it is harmonic on $B(y,10a)$ and bounds $\Phi$ above there. At
$z\in B(y,10a)$ the inclusion $B(y,20a)\subset B(z,Aa_z)$ bounds it above by
$\E_Q[J_z\mid X]$. The cut satisfies Lemma~\ref{lem:local-field-input} uniformly:
a bounded number of comparable cells covers its out support, $D_z^2\le C\beta^{-2}$,
and $a_zm_z\ge ca^{-2}$, so $D_z^2/(a_zm_z)\le Ca^{1.6}\le C$. That lemma bounds
the positive part in $L^2(Q)$ at each $z$; submeans on balls of radius $a$ and
Minkowski's integral inequality give \eqref{eq:patch-field-envelope}.

\paragraph{The patch minimizer.}
On $\Omega=B(y,t_0-4b)$ minimize
$\mathcal E_\Phi(f)=T\int f^{5/3}-\int\Phi f+D(f)$, $f\ge0$,
$\supp f\subset\overline\Omega$. The field $\Phi$ is bounded and harmonic on a
neighborhood of $\overline\Omega$ (core excluded from $B(y,t_0)$, nucleus far
outside). Coercivity in $L^{5/3}(\Omega)$, weak lower semicontinuity, and strict
convexity give a unique minimizer $\rho=\rho_X$ with Euler equation
\begin{equation}
 W=\Phi-K*\rho,\qquad \rho=kW_+^{3/2}\text{ on }\Omega.
 \label{eq:patch-euler}
\end{equation}
Nonnegative variations give the one-sided condition where $\rho=0$; two-sided
variations on positive density give equality. Comparison with zero and
$D(\rho)\ge(\int\rho)^2/(24a)$ give
\begin{equation}
 \int\rho\le CaF.
 \label{eq:patch-mass-bound}
\end{equation}
These minimizers are measurable in $X$: they depend continuously in $L^{5/3}$ on a
field in $L^{5/2}(\Omega)$; coercivity bounds a converging sequence, weak limits
minimize by lower semicontinuity and comparison with the minimizer, uniqueness
identifies the limit, convergence of minimum values forces $L^{5/3}$-convergence,
and uniform convexity gives strong convergence. The Coulomb form is continuous
positive quadratic on $L^{5/3}(\Omega)$.

Let $\vartheta_b$ be the radial probability kernel of \FNeumann (a cube kernel
averaged over translations and rotations), bounded by $b^{-3}$, supported in
$B(0,\sqrt3b)$, and set
$\sigma=\sum_{i\ \mathrm{retained}}\vartheta_b(\cdot-x_i)$; each smear lies in
$\Omega$ by the $7b$ margin. The key estimate reads
\begin{equation}
 \E_Q\Bigl[D(\sigma-\rho)+\int W_-\sigma\Bigr]\le C_Ma^{-7+.02},\quad
 \int_{B(y,4a)}\rho_{\mathrm{tilt}}^{5/3}\le C_Ma^{-7},
 \label{eq:patch-comparison}
\end{equation}
where $\rho_{\mathrm{tilt}}$ is the ordinary density of the tilted state (not the
conditional patch density $\rho$).

\paragraph{Lower and upper energy comparisons.}
For the fresh square cut $(o,c)$ and each out-label set
$I\subset\{1,\dots,n\}$ set
$\Psi_I=\prod_{i\in I}o(x_i)\prod_{i\notin I}c(x_i)\Psi_{A'}$ and
$T_o=\tfrac12\sum_{I}\sum_{i\in I}\|\nabla_i\Psi_I\|_2^2$ (total out kinetic
energy). Adding price to the exact localization identity of \FLocalization,
\begin{equation}
 \langle\Psi_{A'},H_{Z,n}\Psi_{A'}\rangle+\lambda n+\operatorname{err}_{\mathrm{loc}}
 =\E_Qe_X^\lambda+T_o-\E_Q\sum_{i\ \mathrm{out}}\Phi(x_i)
       +\E_Q\sum_{i<l\ \mathrm{out}}K(x_i-x_l),
 \label{eq:priced-localization-identity}
\end{equation}
with $0\le\operatorname{err}_{\mathrm{loc}}\le C_Mb^{-2}a^{-3}$. The price sits in
the core energy and in $\Phi$, so it contributes no localization error.
\FNeumann (for any measurably selected retained subset) gives
\begin{equation}
 \begin{split}
 T_o&\ge\E_Q\Bigl[T\int\sigma^{5/3}-Cb^{-2}(n_o^{4/3}+n_o)\Bigr],\\
 \sum_{i<l\ \mathrm{out}}K(x_i-x_l)&\ge D(\sigma)-Cn_o/b,
 \end{split}
 \label{eq:patch-lower-comparison}
\end{equation}
with leading kinetic coefficient exactly $T$ for two spins and $-\tfrac12\Delta$.

For a.e. fixed slice the priced upper comparison is
\begin{equation}
 P(\lambda)\le e_X^\lambda+T\int\rho^{5/3}+Cb^{-2}\int\rho-\int\Phi\rho+D(\rho).
 \label{eq:patch-upper-comparison}
\end{equation}
The packet construction is standard (Solovej 2003, Lemma~8.2); the
insertion proof of \FInsertion gives this priced version. Integrate the rank-one
matrices of $g_b(x-z)e^{ip\cdot x}$, both spins, over
$|p|\le(3\pi^2\rho(z))^{1/3}$ with measure $dz\,dp/(2\pi)^3$; the resulting
$0\le\gamma\le1$ has $\Tr\gamma=\int\rho$, $\rho_\gamma=g_b^2*\rho$, and
$\Tr(-\tfrac12\Delta)\gamma=T\int\rho^{5/3}+Cb^{-2}\int\rho$ by Plancherel and
momentum-ball integration. Truncate the spectral decomposition to finitely many
eigenvectors and occupy them independently with the eigenvalues: each realization
is a finite Slater determinant with integer electron count and mean repulsion at
most the direct energy of the truncated density (direct-minus-exchange identity).
Its support lies strictly inside $B(y,t_0)$, separated from core and nucleus, so it
may be wedged with the core; every state has shifted energy at least $P(\lambda)$,
and the mean price paid is $\lambda\Tr\gamma$. Positive truncations increase
densities and converge traces; potentials are bounded on the packet region for
each fixed slice, so one-body terms converge and direct energies by monotone
convergence. Radial smearing does not increase $D$, and harmonic means replace
$\int\Phi(g_b^2*\rho)$ by $\int\Phi\rho$ (including the constant price). Averaging
the integer-sector trials and passing to the limit gives
\eqref{eq:patch-upper-comparison}, needing neither integer $\int\rho$ nor a
measurable choice of determinant as the slice varies.

For retained particles the harmonic means also give
$\sum_{i\ \mathrm{retained}}\Phi(x_i)=\int\Phi\sigma$; the deleted attractions
cost at most $Fn_{\mathrm{del}}$. Subtract \eqref{eq:patch-upper-comparison} in
\eqref{eq:priced-localization-identity}, use
\eqref{eq:shifted-tilt-cost}, \eqref{eq:patch-lower-comparison}, and expand about
the minimizer. Convexity of the kinetic term leaves $D(\sigma-\rho)$ and
\[
 \int\Bigl(\tfrac{5T}{3}\rho^{2/3}-W\Bigr)(\sigma-\rho)=\int W_-\sigma,
\]
the equality following from \eqref{eq:patch-euler}: the coefficient vanishes where
$\rho>0$ and equals $W_-$ where $\rho=0$. By \eqref{eq:patch-counts},
\eqref{eq:patch-field-envelope}, \eqref{eq:patch-mass-bound},
\begin{equation}
 \E_Q\Bigl[D(\sigma-\rho)+\int W_-\sigma\Bigr]
 \le\delta'+C_M\bigl(\sqrt\beta\,a^{-7}+b^{-1}a^{-3}+b^{-2}(a^{-4}+a^{-3})\bigr).
 \label{eq:patch-error-budget}
\end{equation}
All terms are integrable: $\int\rho\le CaF$ and
$\mathcal E_\Phi(\rho)\le0$ bound kinetic and direct terms by $CaF^2$ in
expectation; the exact cut identity has integrable absolute Coulomb terms by
form-domain admissibility. Since $b=a^{1.2}$, the errors of
\eqref{eq:patch-error-budget}, divided by $a^{-7}$, are at most
$C_M(a^{4.98}\log^5(e/a)+a^{.1}+a^{2.8}+a^{.6}+a^{1.6})$, proving the first part
of \eqref{eq:patch-comparison}. For the second, use $e_X^\lambda\ge P(\lambda)$
directly in \eqref{eq:priced-localization-identity}: dropping the nonnegative out
repulsion and $\E_QFn_o\le C_Ma^{-7}$ give $T_o\le C_Ma^{-7}$, and the coarse
kinetic-density inequality of \FLocalization on the ball containing $B(y,4a)$
gives the claim.

\medskip
\noindent\textit{Recovering the conditional density from the patch.}\par\medskip
By \eqref{eq:patch-euler} the patch field satisfies
$\Delta W=4\pi kW_+^{3/2}$ on $B(y,4a)$. The following bounds are used:
\begin{equation}
 W\le Ca^{-4}\text{ on }B(y,3a),\quad
 \sup_{B(y,2t)}|W-W(y)|\le C\tau\bigl(a^{-4}+W(y)_-\bigr).
 \label{eq:patch-field-oscillation}
\end{equation}
For the first compare on the ball of radius $R=4a$ with
$U(x)=CR^4/(R^2-|x-y|^2)^4$: for large universal $C$,
$\Delta U\le4\pi kU^{3/2}$ and $U$ diverges at the boundary, so testing with the
positive part of $W-U$ proves it. For the second, the Coulomb potential of
$\rho\ind_{B(y,2a)}$ is $O(a^{-4})$ with gradient $O(a^{-5})$ there; adding it to
$W$ leaves a harmonic function bounded above by $Ca^{-4}$, and the interior
gradient estimate on its nonnegative harmonic distance from this bound gives
center value at most $Ca^{-4}+W(y)_-$. Compact $L^{5/3}$ densities have continuous
potentials, so $W$ is continuous.

Define $\mathcal Rf(y)=\int\mathcal K_z(y)f(z)\,dz$. On every compatible path under
$Q$ (including fresh cut labels),
\begin{equation}
 |\mathcal R\sigma(y)-\mu_j(y)|\le C_Mr^{-3}(\ell_j+b)t^{-4}=o(a^{-6}).
 \label{eq:noisy-array-sandwich}
\end{equation}
Match any compatible raw configuration to the $j$-th observed multiset: each
displacement is at most $\sqrt3\ell_j$, so the raw kernel sum differs by at most
$C_Mr^{-3}\ell_jt^{-4}$ from the sum at that multiset, and the same holds for its
posterior average $\mu_j$. Only $C_Mr^{-3}$ entries contribute (event
$\mathcal C_j$), all relevant centers lie in the fully retained region
($t/a\to0$, $b/a\to0$, $\ell_j/a\to0$), and replacing each by its fine smear
changes the sum by at most $C_Mr^{-3}bt^{-4}$. This proves
\eqref{eq:noisy-array-sandwich} without identifying the two posteriors; the ratios
to $a^{-6}$ are $C_M(a^{.01-4w}+a^{.2-4w})=o(1)$.

Let $G=\{D(\sigma-\rho)\le a^{-7+.01}\}$, so $Q(G^c)\le C_Ma^{.01}$. The function
$z\mapsto\mathcal K_z(y)$ is a compactly supported Lipschitz test with gradient
norm at most $Ct^{-5/2}$; Coulomb duality \FCoulomb gives, on $G$,
\begin{equation}
 |\mathcal R\rho(y)-\mathcal R\sigma(y)|\le Ca^{-7/2+.005}t^{-5/2}=o(a^{-6}),
 \label{eq:local-coulomb-duality}
\end{equation}
the relative gain at least $a^{.005-(5/2)w}$. All small errors are uniform for
fixed parameters along \eqref{eq:scaling-regime}, including arbitrarily small
$r/s$.

On $G$ the high-density antecedent in \eqref{eq:fixed-point-failures} forces
\begin{equation}
 W(y)\ge(h-\xi/16)d_y^{-4}.
 \label{eq:high-patch-field}
\end{equation}
Otherwise \eqref{eq:patch-field-oscillation} gives
$W(z)\le(h-\xi/32)d_y^{-4}$ on the averaging kernel's support (still true for
arbitrarily negative $W(y)$, since the upper estimate contains the nonpositive
$-(1-C\tau)W(y)_-$), and \eqref{eq:patch-euler},
\eqref{eq:packet-approximate-mass}, \eqref{eq:noisy-array-sandwich},
\eqref{eq:local-coulomb-duality} contradict the antecedent. Likewise the
low-density antecedent forces, on $G$,
\begin{equation}
 W(y)\le(h+\xi/16)d_y^{-4}.
 \label{eq:low-patch-field}
\end{equation}
The upper bound in \eqref{eq:patch-field-oscillation} passes
\eqref{eq:low-patch-field} to expectation with $o(a^{-4})$ error.

The high case needs a control of the negative field on $G^c$. The high antecedent
and \eqref{eq:noisy-array-sandwich} hold on all paths under $Q$ and give
$\int_{B(y,2t)}\sigma\ge ca^{-6}t^3$; using \eqref{eq:patch-field-oscillation} on
this ball and absorbing $C\tau W(y)_-$,
\begin{equation}
 W(y)_-\le C(a^{-6}t^3)^{-1}\int W_-\sigma+C\tau a^{-4}.
 \label{eq:negative-patch-control}
\end{equation}
Hence \eqref{eq:patch-comparison} gives $\E_Q[\ind_{G^c}W(y)_-]=o(a^{-4})$, since
its ratio to $a^{-4}$ is at most
$C_Ma^{.02}\tau^{-3}+C\tau\le C_Ma^{.02-3w}+Cs^w=o(1)$. So
\eqref{eq:high-patch-field} also passes to expectation with $o(a^{-4})$ error,
using no lower cap on the conditional field.

\medskip
\noindent\textit{Matching the potentials of the two conditional laws.}\par\medskip
Adapting the averaged field transfer \FAveragedTransfer to the priced fields (the
patch estimates give its small errors) yields
\begin{equation}
 \bigl|\E_Q(H'_j-W)(y)\bigr|=o(a^{-4}).
 \label{eq:potential-matching}
\end{equation}
Write $P^{\mathrm{raw}},P^{\mathrm{ms}}$ 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. Since the event defining $Q$ lies in
$\mathcal F_j$, the observation tower gives
$\E_Q(K*\mu_j)(y)=\E_QP^{\mathrm{ms}}$; the core in $W$ uses the fresh-cut tower
in the tilted raw law. Combining and cancelling nucleus and price, $\E_Q(H'_j-W)(y)$
is the expectation of
\begin{equation}
 (P^{\mathrm{raw}}-P^{\mathrm{ms}})-P^{\mathrm{del}}-(P^{\mathrm{ret}}-K*\sigma)(y)
    +K*(\rho-\sigma)(y).
 \label{eq:two-posterior-potential-identity}
\end{equation}
By H\"older and the density bound in \eqref{eq:patch-comparison}, for $0<u\le4a$,
\begin{equation}
 \int_{B(y,u)}\frac{\rho_{\mathrm{tilt}}(z)}{|z-y|}\,dz\le C_Ma^{-4}(u/a)^{1/5}.
 \label{eq:raw-short-range-potential}
\end{equation}
Radial smearing never increases a point potential and leaves it unchanged outside
the smear radius, so the master-smearing difference comes only from centers within
$2t$ of $y$ and the fine-smearing difference only from centers within $\sqrt3b$;
\eqref{eq:raw-short-range-potential} bounds both by $o(a^{-4})$. Deleted centers
are at distance $\sim a$, so \eqref{eq:patch-counts} gives
$\E_QP^{\mathrm{del}}\le C_Ma^{-4}\sqrt\beta$.

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 $O(t^{-1/2})$,
so Coulomb duality with \eqref{eq:patch-comparison} gives expected absolute
truncated error at most $C_Mt^{-1/2}a^{-7/2+.01}=o(a^{-4})$ (relative gain
$a^{.01}\tau^{-1/2}\le C_Ma^{.01-w/2}$). The truncation loss for $\rho$ is at most
$Ca^{-6}t^2$ by \eqref{eq:patch-field-oscillation}; for $\sigma$ it is at most
$C_Ma^{-4}((t+\sqrt3b)/a)^{1/5}$, also $o(a^{-4})$ by
\eqref{eq:packet-scale-ratios}. This proves \eqref{eq:potential-matching}.

In the high case \eqref{eq:high-patch-field}, \eqref{eq:negative-patch-control} give
$\E_QW(y)\ge(h-\xi/16)d_y^{-4}-o(a^{-4})$, while the failed comparison gives
$\E_QH'_j(y)\le(h-\xi/4)d_y^{-4}$; \eqref{eq:potential-matching} contradicts these
eventually. The low case is the same from \eqref{eq:low-patch-field} and the upper
bound on $W$. So each fixed-point failure in \eqref{eq:fixed-point-failures}
intersected with $\mathcal C_j$ has probability less than $r^{42}$.

\medskip
\noindent\textit{Spatial and height nets.}\par\medskip
Take a spatial net of mesh at most $r^2$ in the original band ($O_M(r^{-3})$
points) and a fixed 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: total probability $O_M(r^{39})+O_M(r^{40})\le Cr^{25}$,
all $\mathcal F_j$-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
\eqref{eq:conditional-density-bounds}. For the field, on balls of radius a small
fixed multiple of $r$ within the enlarged cap band, adding the Coulomb potential of
$\mu_j$ makes $H'_j$ harmonic and bounds that harmonic function above by
$C_Mr^{-4-3w}$ by \eqref{eq:conditional-density-bounds}, \eqref{eq:enlarged-field-cap};
the positive-harmonic gradient argument of \eqref{eq:patch-field-oscillation} shows
$v(y)=d_y^4H'_j(y)$ varies over $O(r^2)$ by at most
\begin{equation}
 C_Mr^{1-3w}+C_Mr\,v(y)_-.
 \label{eq:net-field-variation}
\end{equation}
Suppose a high-density failure with tolerance $\xi$ occurs at $(y,h)$. If
$v(y)\ge0$, \eqref{eq:net-field-variation} puts the nearby net value below
$h-3\xi/4$ for small $s$; if $v(y)<0$ the upper estimate is at most
$(1-C_Mr)v(y)+C_Mr^{1-3w}$, same conclusion since $h\ge h_l>16\xi$. A height-net
value between $h-\xi/4$ and $h-\xi/8$ preserves the strict high antecedent (small
density variation) while the field violates its comparison with tolerance
$\xi/4$: an excluded net failure. A low-density failure has $v(y)>h+\xi>0$, its
nearest net value above $h+3\xi/4$ by \eqref{eq:net-field-variation}, and a
height-net value between $h+\xi/8$ and $h+\xi/4$ preserves the low antecedent:
again excluded. So both comparisons hold everywhere on the band for every
$h_l\le h\le h_h$, which with \eqref{eq:enlarged-field-cap} proves
Proposition~\ref{prop:inverse-comparison}.

\medskip
The local density--field comparison of Proposition~\ref{prop:inverse-comparison}
alone does not exclude a limiting field that is too small near the nucleus. A
positive barrier is carried from the nuclear scale to an arbitrary intermediate
scale, keeping the actual conditional density inside the current radius and
using the Thomas--Fermi reaction outside it. Rare failures of the local
comparison are retained as an explicit small source error. This adapts the
intermediate-subsolution construction \FIntermediate and uses the positive-price
comparison proved above, so the growing offset $D_0=O(s^{-7})$ is allowed.

Write $d=|y|$ and put
\begin{equation}\label{eq:barrier-function}
 f(t)=4\pi k(t_+)^{3/2},\qquad
 b_r(y)=Bd^{-4}\Bigl(1-\frac r{8d}\Bigr)\quad(d\ge r),
\end{equation}
with $B>0$ fixed below. Direct differentiation gives
$\Delta b_r=Bd^{-6}(12-\tfrac{20r}{8d})\ge\tfrac{19}{2}Bd^{-6}$, so
$\Delta b_r\ge f(b_r)$ whenever $4\pi k\sqrt B\le19/2$. For $R=2r$,
\begin{equation}\label{eq:barrier-gap}
 b_r-b_R\ge\gamma R^{-4}\quad(R\le d\le1.1R),\qquad
 \gamma=\frac{B}{16(1.1)^5}>0.
\end{equation}

\begin{proposition}[Conditional barrier invariant]\label{prop:barrier-invariant}
There are fixed positive constants $\eps,B,C_{\mathrm{inv}}$ and $M>2$ with the
following property. For every sufficiently far system in
\eqref{eq:scaling-regime} use the observations and kernels of
\eqref{eq:observation-scales}, \eqref{eq:master-kernel} with this $\eps$. For
every $0\le j\le J_*$ there exist $\mathcal F_j$-measurable $u_j,p_j$ such that,
with $r=r_j$,
\begin{equation}\label{eq:barrier-invariant}
 \begin{aligned}
 \Delta u_j&\ge-4\pi\nu+4\pi\ind_{\{d<r\}}\mu_j-4\pi p_j
       +\ind_{\{d\ge r\}}f(u_j),\\
 b_r\le u_j&\le C_{\mathrm{inv}}d^{-4}\ (d\ge r),\qquad
 u_j=b_r\ (d>Mr).
 \end{aligned}
\end{equation}
The first line holds in distributions on $\R^3$. For each fixed system $u_j-V$ is
continuous and locally Lipschitz with deterministic compact-set bounds, and $p_j$
is a nonnegative bounded density supported in $\{d\le r_j\}$ with a deterministic
bound. Its expected total mass satisfies
\begin{equation}\label{eq:barrier-error-mass}
 \sup_{0\le j\le J_*}\E\!\int p_j\le C\bigl(r_0^{32}+s^{19/2-3w}\bigr)=o(1).
\end{equation}
The fixed-system regularity bounds need not be uniform in $Z$; the constants in
\eqref{eq:barrier-error-mass} are uniform.
\end{proposition}

\begin{proof}
The weak maximum rule \FMaximum is used. In nonsingular form, if finitely many
continuous locally $H^1$ functions satisfy
$\Delta v_i\ge g(y)+F(y,v_i)$ with $g$ locally bounded and $F$ measurable in $y$,
continuous in its second variable, locally bounded on bounded ranges, then their
maximum satisfies the same inequality with its own value in the reaction. The
lemma also allows a common singular $-4\pi\nu$ when all branches have continuous
locally $H^1$ offsets from $V$ and the reaction vanishes near the origin. These
are the forms used below.

\paragraph{Initialization.}
For small fixed $\eps$ the initial conditional density satisfies, off an event of
probability at most $Cr_0^{35}$,
\begin{equation}\label{eq:initial-potential}
 P_0(y):=K*(\ind_{\{d<r_0\}}\mu_0)(y)\le0.1\,Z/r_0
 \quad(r_0\le|y|\le1.1r_0).
\end{equation}
Fix $y$ in that annulus and the $\mathcal F_0$-event
$A_y=\{P_0(y)>0.05Z/r_0\}$. If $\Prob(A_y)>r_0^{40}$,
Lemma~\ref{lem:event-tilt} realizes the raw conditional marginal as a normalized
state in sector $n$ with unshifted offset at most
$D_0+Cr_0^{-2.02}\log^5(e/r_0^{40})$. By \eqref{eq:rough-deficit}
$D_0\le Cs^{-7}$; since $r_0/s\to0$ this offset is at most $Z^{7/3}$ eventually,
so the global density estimate \FGlobalDensity applies (normalized antisymmetric
state, $n\le3Z$): $\int\rho_{A_y}^{5/3}\le CZ^{7/3}$. Only centers $|x|<2r_0$
contribute to $P_0$; a truncated packet's potential is at most the full packet's,
which radial smearing bounds by $K(y-x)$. The tower and H\"older give
\[
 \E[P_0(y)\mid A_y]\le\int_{|x|<2r_0}\frac{\rho_{A_y}(x)}{|y-x|}\,dx
 \le CZ^{7/5}r_0^{1/5}=C\eps^{6/5}Z/r_0.
\]
Choose $\eps$ so that the last coefficient is below $0.05$; this contradicts the
event definition, so $\Prob(A_y)\le r_0^{40}$.

The minimum packet width is $c_1r_0^{1+w}$; integrating $|y-z|^{-2}$ against each
bounded packet gives $\|\nabla P_0\|_\infty\le CZr_0^{-2-2w}$. A spatial net of
spacing $r_0^2$ in the annulus has $O(r_0^{-3})$ points, with union failure
probability $O(r_0^{37})$ and interpolation error over $Z/r_0$ of order
$O(r_0^{1-2w})$. This proves \eqref{eq:initial-potential} with the stated (weaker)
exponent; the good event is measurable since $P_0$ is continuous in space.

Let $G_0$ be this good event and set
$\mu^{\mathrm{in}}=\ind_{G_0}\ind_{\{d<r_0\}}\mu_0$,
$p_0=\ind_{G_0^c}\ind_{\{d<r_0\}}\mu_0$. On $d<2r_0$ consider
\[
 v=V-K*\mu^{\mathrm{in}}-0.7Z/r_0+C_3(Z/r_0)^{3/2}d^2.
\]
Fix $C_3$ with $6C_3\ge4\pi k\,2^{3/2}$ and decrease $\eps$ so that
$4C_3\eps^{3/2}\le0.05$ (using $Zr_0^3=\eps^3$, the quadratic term is at most
$0.05Z/r_0$ on this ball). On $r_0\le d<2r_0$, the bound $v\le2Z/r_0$ holds and
$\Delta v=-4\pi\nu+4\pi\mu^{\mathrm{in}}+6C_3(Z/r_0)^{3/2}$ with
$6C_3(Z/r_0)^{3/2}\ge f(v)$. On the lower overlap $r_0\le d<1.06r_0$,
$v\ge(1/1.06-0.1-0.7)Z/r_0>0.14Z/r_0$ (on $G_0^c$ drop the $0.1$); on
$1.9r_0<d<2r_0$, $v\le(1/1.9-0.7+0.05)Z/r_0<0$. Fix $B$ from the subsolution
condition and $0<B\le0.1\eps^3$. Define $u_0$ on an open cover by
\[
 u_0=\begin{cases}
 v,&d<1.06r_0,\\
 \max(v,b_{r_0}),&r_0<d<2r_0,\\
 b_{r_0},&d>1.9r_0.
 \end{cases}
\]
The pieces agree on overlaps. Near $d=r_0$ the branch $v$ alone is active and
supplies both source inequalities; elsewhere the weak maximum rule proves
\eqref{eq:barrier-invariant}. The exterior cap holds for
$C_{\mathrm{inv}}\ge\max(B,32\eps^3)$, and $\E\int p_0\le n\Prob(G_0^c)\le
Cr_0^{32}$.

\paragraph{Choice of remaining constants.}
Increase $C_{\mathrm{inv}}$ past the universal cap $C_{\mathrm{cap}}$ of
Proposition~\ref{prop:inverse-comparison}. Choose
$0<\lambda_2<\lambda_1<\gamma/2$, $M>2$, $C_{\mathrm{inv}}M^{-4}<\lambda_2$ (the
$\lambda_1,\lambda_2$ are barrier shifts, distinct from the price $s^{-4}$). Fix
$0<h_l<B/4$ and $h_h>C_{\mathrm{inv}}$, then $e_0,\xi>0$ so small that
\begin{equation}\label{eq:barrier-parameters}
 2e_0(2M)^4<B/4,\qquad 16\xi+2e_0<\lambda_1-\lambda_2,\qquad
 \xi<\min(\lambda_2,h_l/16).
\end{equation}
All constants are now fixed, and Proposition~\ref{prop:inverse-comparison}
applies with exactly these parameters along the regime.

\paragraph{One outward step.}
Suppose the invariant holds at $r=r_j$, $j<J_*$, and set $R=2r$, $u=u_j$. On the
good event of Proposition~\ref{prop:inverse-comparison} for the band $r\le
d\le4Mr$ use $v_1=u-\lambda_1R^{-4}$ and $v_2=H'_j-\lambda_2R^{-4}$. The two
branches are complementary: on the gained annulus $r\le d<R$ the old reaction
must supply the actual density, so a retained $v_1$ must satisfy
$f(u)\ge4\pi\mu_j$; outside $R$ the field branch already has Laplacian
$4\pi\mu_j$, which must instead dominate $f(v_2)$. The opposite inverse
implications, with unequal shifts, give these on the neighborhoods below. On bad
data omit $v_2$. Define
\begin{equation}\label{eq:barrier-open-cover}
 \begin{array}{c|c}
 \text{region}&\widetilde u\\ \hline
 d<1.08R&\max(v_1,v_2)\\
 1.04R<d<2MR&\max(v_1,v_2,b_R)\\
 d>MR&b_R
 \end{array}
\end{equation}
with $v_2$ omitted on bad data, and
\begin{equation}\label{eq:barrier-added-error}
 \widetilde p=p_j+\ind_{\{\mathrm{bad}\}}\ind_{\{r\le d<R\}}\mu_j.
\end{equation}
This is the schematic of the continuation step from $r$ to $R=2r$: the shifted
branches $v_1=u_j-\lambda_1R^{-4}$ and $v_2=H'_j-\lambda_2R^{-4}$ (second omitted
on bad data) cover $d<1.08R$; the larger region $1.04R<d<2MR$ adds $b_R$; beyond
$MR$ only $b_R$ survives; the reaction source changes on the shaded annulus
$r\le d<R$. On the lower overlap \eqref{eq:barrier-gap} gives
$v_1\ge b_r-\lambda_1R^{-4}>b_R$; on the upper overlap the caps and
$C_{\mathrm{inv}}M^{-4}<\lambda_2$ put both shifted branches below $b_R$. So the
pieces agree, giving $b_R\le\widetilde u\le C_{\mathrm{inv}}d^{-4}$ on $d\ge R$
and $\widetilde u=b_R$ on $d>MR$ (for $R\le d<1.08R$ the lower bound already
follows from $v_1$).

It remains to prove the weak inequality before averaging. Do so on open
neighborhoods, so no derivative is taken across the maximizing sets. At a point
of an open region of \eqref{eq:barrier-open-cover}, retain the candidates within
$e_0R^{-4}$ of the maximum; by continuity and finiteness there is a neighborhood
where every discarded candidate stays strictly below a fixed retained maximizer
and every retained $v$ satisfies
\begin{equation}\label{eq:retained-branches}
 \widetilde u-v<2e_0R^{-4},
\end{equation}
with the maximum still $\widetilde u$. At the origin use the continuous offsets
from $V$. Every retained branch satisfies, on that whole neighborhood,
\begin{equation}\label{eq:common-branch-inequality}
 \Delta v\ge-4\pi\nu+4\pi\ind_{\{d<R\}}\mu_j-4\pi\widetilde p
       +\ind_{\{d\ge R\}}f(v).
\end{equation}
For $v_1$ the old inequality suffices on $d<r$, and on $d\ge R$ use
$f(u)\ge f(v_1)$; the only new source is on $r\le d<R$, where bad data are paid
by \eqref{eq:barrier-added-error} and on good data $v_2$ is a candidate, so
\eqref{eq:retained-branches} gives
$H'_j\le u-(\lambda_1-\lambda_2-2e_0)R^{-4}$. Also
$h_l<7B/8\le d^4u\le C_{\mathrm{inv}}<h_h$. If $4\pi\mu_j>f(u)$, the high
implication of \eqref{eq:inverse-comparison} at height $h=d^4u$ gives
$H'_j\ge u-\xi d^{-4}\ge u-16\xi R^{-4}$, contradicting
\eqref{eq:barrier-parameters}. Hence $4\pi\mu_j\le f(u)$, which supplies the new
density. For $v_2$ its exact identity $\Delta v_2=-4\pi\nu+4\pi\mu_j$ suffices on
$d<R$. Where it is retained, the splice or the included barrier gives
$\widetilde u\ge b_R$ and $d\le2MR$, so
$d^4v_2\ge7B/8-2e_0(2M)^4>5B/8>h_l$ and $d^4v_2\le C_{\mathrm{cap}}<h_h$. If
$4\pi\mu_j<f(v_2)$, the low implication gives
$H'_j\le v_2+\xi d^{-4}\le v_2+\xi R^{-4}<v_2+\lambda_2R^{-4}=H'_j$, a
contradiction; hence $4\pi\mu_j\ge f(v_2)$ there. The branch $b_R$ occurs only
outside $R$ and satisfies \eqref{eq:common-branch-inequality} by the subsolution
property. These comparisons concern zero-order densities a.e. on the same open
neighborhood, including across $d=R$; the common reaction
$F(y,t)=\ind_{\{d\ge R\}}f(t)$ meets the rule's hypotheses (near zero the common
$V$ singularity cancels and the reaction vanishes). The weak maximum rule proves
\eqref{eq:common-branch-inequality} for $\widetilde u$ locally; a partition of
unity proves it globally.

\paragraph{Forgetting one observation.}
Define $u_{j+1}=V+\E[\widetilde u-V\mid\mathcal F_{j+1}]$ and
$p_{j+1}=\E[\widetilde p\mid\mathcal F_{j+1}]$. For a fixed system the positive
minimum packet width gives deterministic bounds for $\mu_j$, $K*\mu_j$, and its
first derivatives; the initial truncated potential and the offsets have the same
bounds; subtracting constants, taking finite maxima, and gluing on the fixed open
cover preserve them; where $b_R$ is used its offset from $V$ is smooth and
separated from the origin; the errors stay bounded densities supported in
deterministic balls. Conditional integration preserves these bounds. All fields
and flags are jointly measurable (the initial supremum tested on a countable
dense set), so Fubini permits integration against each smooth compactly supported
test, and a countable dense family of nonnegative tests with local approximation
supplies simultaneous a.s. weak inequalities. The tower gives $\mu_{j+1}$ as the
new inner density, and convexity gives
$\E[f(\widetilde u)\mid\mathcal F_{j+1}]\ge f(V+\E[\widetilde u-V\mid\mathcal
F_{j+1}])$, proving the first line at $R$; the bounds, the equality beyond $MR$,
and the error support also pass to the average.

\paragraph{Total error.}
For $r\le|y|<2r$ every master packet behind $\mu_j(y)$ has center in a fixed
annulus comparable to $r$ and width comparable to $r^{1+w}$. Conditional Jensen,
the packet bound, and \eqref{eq:annular-count}, \eqref{eq:rough-deficit} give
\[
 \|\mu_j(y)\|_{L^2(\Prob)}\le Cr^{-3-3w}\bigl(r^{-3}+\sqrt{D_0r}\bigr)
 \le Cr^{-6-3w},
\]
since $r\le s$ and $D_0\le Cs^{-7}$. The bad-band event has probability at most
$Cr^{25}$; Cauchy--Schwarz and integration over the gain annulus give
\[
 \E\!\int\ind_{\{\mathrm{bad}\}}\ind_{\{r\le d<R\}}\mu_j\le Cr^{25/2+3-6-3w}
 =Cr^{19/2-3w}.
\]
The sum over the geometric scales is at most $Cs^{19/2-3w}$; conditional
averaging preserves expected total mass. With the initial estimate this proves
\eqref{eq:barrier-error-mass} and completes the induction.
\end{proof}

\medskip
The charge and profile assertions of
Theorem~\ref{thm:priced-deficit} remain to be proved; Proposition~\ref{prop:tf-identification}
identifies the energy coefficient. The barrier is available at every
intermediate observation scale, before all observations are forgotten, which
permits selecting conditional fields satisfying both the local Thomas--Fermi
equation and the small-error barrier.

\medskip
\noindent\textit{Selecting conditional data.}\par\medskip
Take an arbitrary sequence in \eqref{eq:scaling-regime}, with an arbitrary
minimizing index $n$. By \eqref{eq:rough-deficit}, after a subsequence
$q_s:=s^3(Z-n)\to q_*$. Choose integers $l\to\infty$ slowly and $j<J_*$ with
\begin{equation}\label{eq:intermediate-scale}
 \frac1{2l}\le a_*:=\frac{r_j}{s}\le\frac1l;
\end{equation}
possible since $r_0/s\to0$ and the radii are dyadic. With the fixed barrier
parameters, apply Proposition~\ref{prop:inverse-comparison} at this scale with
$M=l^2$, a fixed upper height above $C_{\mathrm{cap}}+1$, and lower height and
tolerance small in terms of $l$. On its good event,
\begin{equation}\label{eq:local-tf-approximation}
 \bigl|\mu_j(y)-k\bigl(H'_j(y)_+\bigr)^{3/2}\bigr|\le l^{-8}|y|^{-6}
 \qquad(r_j\le|y|\le ls).
\end{equation}
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 (the high implication near that
endpoint would contradict the cap). Above the lower endpoint the two
implications near height $v$ squeeze $u$ within the tolerance of $v$; at or below
it they bound $u$ above by that endpoint plus tolerance. So $|u_+-v|\le h_l+\xi$
with both in $[0,h_h+1]$, using no lower bound on $u$; uniform continuity of
$t\mapsto t^{3/2}$ there gives \eqref{eq:local-tf-approximation} for small
$h_l,\xi$. The spatial range is covered since $4Mr_j\ge2ls$.

For each fixed $l$ these are fixed parameters of
Proposition~\ref{prop:inverse-comparison}: thresholds hold eventually and the
exceptional probability tends to zero, so a slow diagonal choice makes
\eqref{eq:local-tf-approximation} hold off probability tending to zero while
ensuring \eqref{eq:intermediate-scale}.

Data are also selected on which
\begin{equation}\label{eq:selected-data}
 \int p_j\le1,\qquad
 s^3\!\int_{|y|>2R_0s}\mu_j(y)\,dy\le\eta_s,\qquad\eta_s\longrightarrow0.
\end{equation}
The first excluded probability tends to zero by \eqref{eq:barrier-error-mass}.
For the second, a master packet centered in $|x|\le R_0s$ lies in $|y|\le2R_0s$
for far systems; the tower and \eqref{eq:exterior-cutoff} show its left side has
expectation tending to zero uniformly in $j$, and Markov supplies a threshold
$\eta_s\to0$. These events and the good-comparison event have positive joint
probability; pick a point in their intersection where all identities and barrier
inequalities hold. Every selected density still has total mass exactly $n$.

\medskip
\noindent\textit{Compactness and the exterior boundary condition.}\par\medskip
On 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; the screened
field is controlled by its spherical mean. Newton's identity and $\int\mu_j=n$ give
\begin{equation}\label{eq:spherical-mean}
 \overline F_s(d):=\frac1{4\pi}\int_{\mathbb S^2}F_s(d\omega)\,d\omega
 =\frac{q_s}{d}+\int_{|x|>d}\Bigl(\frac1d-\frac1{|x|}\Bigr)\rho_s(x)\,dx
 \ge\frac{q_s}{d}.
\end{equation}
On every compact annulus, Proposition~\ref{prop:inverse-comparison} gives the
uniform pointwise bound $F_s\le1+C_{\mathrm{cap}}|x|^{-4}$ for far systems;
with \eqref{eq:spherical-mean} and boundedness of $q_s$ this bounds $F_s$ in
local $L^1$ away from zero (integrate $|F_s|=2(F_s)_+-F_s$ over spheres, then
radii). Equation~\eqref{eq:local-tf-approximation} also bounds $\rho_s$ on these
annuli.

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
there, and the added potential and gradient are uniformly bounded by integrals of
$|x-y|^{-1}$ and $|x-y|^{-2}$ against a bounded density. Interior harmonic
estimates with the local $L^1$ bound give uniform bounds and equicontinuity on the
smaller ball; a diagonal Arzel\`a--Ascoli argument gives a subsequence converging
locally uniformly on $\R^3\setminus\{0\}$ to $F_\infty$. The scaled error in
\eqref{eq:local-tf-approximation} vanishes locally uniformly, so $\rho_s\to
k((F_\infty-1)_+)^{3/2}$ there, hence
\begin{equation}\label{eq:limit-pde}
 \Delta F_\infty=4\pi k\bigl((F_\infty-1)_+\bigr)^{3/2},\qquad
 F_\infty(x)\le1+C_{\mathrm{cap}}|x|^{-4},
\end{equation}
classically off zero (bounded local sources give local H\"older regularity of the
field, the reaction is then locally H\"older, and interior Poisson regularity
finishes).

The selected tail bound says $\int_{|x|>2R_0}\rho_s\to0$; local uniform density
convergence makes $F_\infty$ harmonic on $|x|>2R_0$, and for $d>2R_0$ the last
term of \eqref{eq:spherical-mean} is between $0$ and $d^{-1}$ times this tail.
Hence
\begin{equation}\label{eq:limit-exterior-mean}
 \overline F_\infty(d)=\frac{q_*}{d}\qquad(d>2R_0).
\end{equation}
These facts give the uniform boundary condition
\begin{equation}\label{eq:limit-decay}
 F_\infty(x)\longrightarrow0\qquad(|x|\to\infty).
\end{equation}
No positivity is used yet. On large annuli rescaled to a fixed annulus the cap
bounds the positive part and \eqref{eq:limit-exterior-mean} bounds the integral of
the negative part, so interior harmonic estimates give a uniform absolute bound
throughout the exterior. For any $R_i\to\infty$ a subsequence of the dilates
$F_\infty(R_ix)$ converges locally on punctured space to a bounded harmonic
function (same absolute bound on every punctured compact set), whose singularity
at zero is removable; Liouville makes it constant, its spherical mean is zero by
\eqref{eq:limit-exterior-mean}, so it is zero. If \eqref{eq:limit-decay} failed, a
convergent subsequence at offending radii would contradict this.

\medskip
\noindent\textit{Transferring the positive barrier.}\par\medskip
The claim is
\begin{equation}\label{eq:limit-positive-barrier}
 F_\infty(x)\ge B|x|^{-4}\qquad(x\ne0),
\end{equation}
the field being without the subtracted price, which permits the reaction
comparison. Fix $S>0$ and write $U_s(x)=s^4u_j(sx)$. Inside $a_*$ the
distributional inequality for $U_s$ and the equation for $F_s$ share the same
point-nucleus and density terms, with the rescaled $p_j$ error. Outside $a_*$,
\eqref{eq:local-tf-approximation} gives, once $l>S$,
$\rho_s(x)\le k((F_s(x)-1)_+)^{3/2}+l^{-8}|x|^{-6}$. Define the nonnegative
correction potential
\begin{equation}\label{eq:barrier-correction}
 L_s=K*\Bigl(s^6p_j(s\,\cdot)+l^{-8}|\cdot|^{-6}\ind_{\{a_*\le|\cdot|\le S\}}\Bigr).
\end{equation}
Scaling \eqref{eq:barrier-invariant}, subtracting the equation for $F_s$, and
using $\Delta K=-4\pi\delta_0$ give on $B(0,S)$
\begin{equation}\label{eq:barrier-transfer-inequality}
 \Delta(U_s-F_s-L_s)\ge\ind_{\{|x|\ge a_*}}\bigl[f(U_s)-f(F_s-1)\bigr].
\end{equation}
Subtracting $L_s$ cancels both the accumulated source error and the approximation
error with the right sign. Choose any
$c>(C_{\mathrm{inv}}S^{-4}-\min_{|x|=S}F_\infty(x))_+$. For far systems, local
uniform convergence, the upper bound for $U_s$, and $L_s\ge0$ give
$U_s-F_s-L_s\le c$ on the sphere, hence throughout the ball: test
\eqref{eq:barrier-transfer-inequality} with $(U_s-F_s-L_s-c)_+$. On the positive
set $U_s>F_s+L_s+c>F_s-1$ so the reaction difference is nonnegative, giving
$\int_{B(0,S)}|\nabla(U_s-F_s-L_s-c)_+|^2\le0$. The test is legitimate on the full
ball: nuclear singularities cancel, remaining offsets are continuous locally $H^1$,
the correction is a bounded compactly supported potential for each fixed system,
and all right sides are locally bounded after cancellation since the reaction
vanishes on $|x|<a_*$.

For each fixed $x\ne0$, $L_s(x)\to0$: the first density has mass at most $s^3$ by
\eqref{eq:selected-data} supported in $|x|\le a_*$, and the second has mass at
most $Cl^{-8}a_*^{-3}\le Cl^{-5}$ by \eqref{eq:intermediate-scale}, its density
near a fixed nonzero point being $O_x(l^{-8})$; split the potential integral into a
small ball and its complement. Finally for fixed $0<|x|<S$ and far systems
$U_s(x)\ge B|x|^{-4}(1-a_*/(8|x|))$. Pass to the limit in $U_s\le F_s+L_s+c$,
decrease $c$ to its permitted endpoint, and let $S\to\infty$ using
\eqref{eq:limit-decay}: this proves \eqref{eq:limit-positive-barrier}.

\medskip
\noindent\textit{The singular profile and uniqueness.}\par\medskip
The dilation argument is the Sommerfeld comparison principle; homogeneous
zero-price profiles are covered by Solovej 2003, Lemma~4.4 and Remark~4.5
(including nonspherical fields). Dilation removes the unit price near the
singularity; a separate comparison proves uniqueness for the priced equation.

\begin{lemma}[Unique decaying singular profile]\label{lem:singular-profile}
There is at most one classical solution $F$ of
$\Delta F=4\pi k((F-1)_+)^{3/2}$ on $\R^3\setminus\{0\}$ tending uniformly to
zero at infinity and satisfying $c|x|^{-4}\le F(x)\le C|x|^{-4}$ near zero for
some $c,C>0$. Every such solution has the uniform singular asymptotic
\begin{equation}\label{eq:sommerfeld-coefficient}
 |x|^4F(x)\longrightarrow A_{\mathrm S}:=\Bigl(\frac{12}{4\pi k}\Bigr)^2
 \qquad(|x|\downarrow0).
\end{equation}
If a solution exists, it is radial.
\end{lemma}

\begin{proof}
Let $a\le b$ be the lower and upper limits of $|x|^4F(x)$ at zero
($0<a\le b<\infty$). Take points approaching the upper limit, radii $t$, and
dilates $u_t(x)=t^4F(tx)$ satisfying
$\Delta u_t=4\pi k((u_t-t^4)_+)^{3/2}$, bounded above and below by positive
multiples of $|x|^{-4}$ on compact annuli. The local potential and harmonic
compactness argument above gives a subsequential classical limit $u$ with
$\Delta u=4\pi ku^{3/2}$; $u(x)\le b|x|^{-4}$ for all $x\ne0$, with equality at a
unit point after a subsequence of unit directions. There
$u-b|x|^{-4}$ has a local maximum, so $4\pi kb^{3/2}\le12b$. The dilation at
points approaching the lower limit touches $a|x|^{-4}$ from above and gives
$4\pi ka^{3/2}\ge12a$. Both constants positive give
$b\le(12/(4\pi k))^2\le a$, i.e.\ \eqref{eq:sommerfeld-coefficient}; the contact
argument does not assume radiality.

For two such solutions $F_1,F_2$ and $\alpha>1$, $\delta>0$, the function
$G=\alpha F_1+\delta$ is an upper comparison:
$\Delta G=\alpha\,4\pi k((F_1-1)_+)^{3/2}\le4\pi k((G-1)_+)^{3/2}$, immediate if
$F_1\le1$ and otherwise from $G-1\ge\alpha(F_1-1)$, $\alpha^{3/2}\ge\alpha$. The
common asymptotic coefficient makes $G$ dominate $F_2$ on small spheres; uniform
decay and $\delta>0$ give domination on large spheres; testing with $(F_2-G)_+$
using the nondecreasing reaction proves $F_2\le G$ everywhere. Let
$\delta\downarrow0$, $\alpha\downarrow1$, and interchange to get equality.
Rotations preserve equation and both boundary conditions, so uniqueness gives
radiality.
\end{proof}

\begin{proof}[The charge assertion of Theorem~\ref{thm:priced-deficit}]
The field $F_\infty$ satisfies the lemma's hypotheses by
\eqref{eq:limit-pde}, \eqref{eq:limit-decay}, \eqref{eq:limit-positive-barrier}:
a solution exists, is unique, and is radial. Its exterior mean fixes $q_*$
uniquely by \eqref{eq:limit-exterior-mean}; at any fixed $d>2R_0$,
$q_*/d=\overline F_\infty(d)\ge Bd^{-4}>0$. Denote this positive number by $q$.
The reaction density vanishes beyond $2R_0$, and radiality with
\eqref{eq:limit-exterior-mean} gives $F_\infty(x)=q/|x|$ there. Every sequence and
every choice of minimizing indices has bounded scaled deficits by
\eqref{eq:rough-deficit}; from any convergent subsequence this construction
produces the same unique profile and limit $q$. Hence
$s^3(Z-n)\to q>0$ along \eqref{eq:scaling-regime}, for an arbitrary minimizing
index at every stage. The remaining coefficient identity is
Proposition~\ref{prop:tf-identification}.
\end{proof}

\medskip
\label{sec:fixed-sectors}

First, fixed-sector energy differences are recovered from the priced deficit of
the limit analysis. This uses monotonicity of the sector energies, not
convexity, and does not require every sector to minimize at some price. The
priced minimum is concave in the price, and its minimizing sectors bound its
secant slopes even when the minimizer changes. Integrating the deficit law
therefore pins the energy at those sectors; monotonicity brackets the wanted
particle number. The coefficient is then identified by the matching Thomas--Fermi
calculation.

\medskip
\noindent\textit{Integrating the price and bracketing the sector.}\par\medskip
\begin{proposition}\label{prop:fixed-sector-limit}
Let $q$ be the deficit coefficient of Theorem~\ref{thm:priced-deficit}. For every
sequence of integers $Z>m\ge1$ with $m\to\infty$ and $Z/m\to\infty$,
\begin{equation}\label{eq:fixed-sector-limit}
 \frac{E_Z(Z-m)-E_Z(Z)}{m^{7/3}}\longrightarrow\frac37q^{-4/3}.
\end{equation}
\end{proposition}

\begin{proof}
At each fixed $t>0$ choose any $n_t$ minimizing $E_n+tm^{4/3}n$ and put
\[
 d_{Z,m}(t)=\frac{Z-n_t}{m},\qquad
 G_{Z,m}(t)=m^{-7/3}\bigl[P(tm^{4/3})-tm^{4/3}Z-E_Z(Z)\bigr].
\]
The choice $s=t^{-1/4}m^{-1/3}$ satisfies \eqref{eq:scaling-regime}
($Zs^3=t^{-3/4}Z/m\to\infty$), so Theorem~\ref{thm:priced-deficit} gives
\begin{equation}\label{eq:priced-scaled-deficit}
 d_{Z,m}(t)\longrightarrow qt^{3/4}\quad\text{for every fixed }t>0,
\end{equation}
for every choice of minimizing indices.

For $0<v<t$ the two endpoint minimizers give
\begin{equation}\label{eq:quantum-secants}
 -d_{Z,m}(t)\le\frac{G_{Z,m}(t)-G_{Z,m}(v)}{t-v}\le-d_{Z,m}(v),
\end{equation}
evaluating the objective at $n_v$ for the upper bound and at $n_t$ at price $v$
for the lower; this needs no differentiability at a multi-minimizing price. For a
fixed partition $v=t_0<\dots<t_r=t$, summing \eqref{eq:quantum-secants} bounds
$G_{Z,m}(t)-G_{Z,m}(v)$ by endpoint Riemann sums; using
\eqref{eq:priced-scaled-deficit} at the finitely many points and refining gives,
since $qt^{3/4}$ is continuous,
\begin{equation}\label{eq:quantum-price-increments}
 G_{Z,m}(t)-G_{Z,m}(v)\longrightarrow-\tfrac47q\bigl(t^{7/4}-v^{7/4}\bigr).
\end{equation}
The zero-price end fixes the constant: the neutral sector competes and
$E_{n_v}\ge E\ge E_Z(Z)-C$ by \eqref{eq:neutral-offset}, so
\begin{equation}\label{eq:quantum-zero-endpoint}
 -Cm^{-7/3}-v\,d_{Z,m}(v)\le G_{Z,m}(v)\le0.
\end{equation}
For fixed $v>0$ this gives $\limsup|G_{Z,m}(v)|\le qv^{7/4}$; with
\eqref{eq:quantum-price-increments} and then $v\downarrow0$,
\begin{equation}\label{eq:quantum-price-energy}
 G_{Z,m}(t)\longrightarrow-\tfrac47qt^{7/4},\qquad
 \frac{E_{n_t}-E_Z(Z)}{m^{7/3}}=G_{Z,m}(t)+t\,d_{Z,m}(t)
 \longrightarrow\tfrac37qt^{7/4}.
\end{equation}
No assertion about the minimizers as $t\downarrow0$ is needed.

Choose fixed $0<t_-<q^{-4/3}<t_+$. By \eqref{eq:priced-scaled-deficit} their
deficits are eventually below and above $m$, so $n_{t_-}>Z-m>n_{t_+}$, and
monotonicity gives
$E_{n_{t_-}}-E_Z(Z)\le E_Z(Z-m)-E_Z(Z)\le E_{n_{t_+}}-E_Z(Z)$. Apply
\eqref{eq:quantum-price-energy} and let both prices approach $q^{-4/3}$: the
bounds converge to $\tfrac37q(q^{-4/3})^{7/4}=\tfrac37q^{-4/3}$, proving
\eqref{eq:fixed-sector-limit}.
\end{proof}

\medskip
\noindent\textit{The same charge in Thomas--Fermi theory.}\par\medskip
The weak-ionization law was established by P.~B\'enilan and H.~Brezis (J.\ Evol.\ Equ.\ \textbf{3} (2004), Section~6 Corollary~5), following the bounds and chemical-potential argument of E.~H.~Lieb and B.~Simon (Adv.\ Math.\ \textbf{23} (1977), Theorems~IV.11--IV.12). The argument is redone in the present normalization to
identify its coefficient with the charge from the conditional quantum limit. Real nuclear charges and real electron masses are allowed; the
admissible class is
$\mathcal X=\{\rho\ge0:\rho\in L^1(\R^3)\cap L^{5/3}(\R^3)\}$. All terms of
$\mathcal T_Z$ are finite on $\mathcal X$: interpolation gives $\rho\in L^{6/5}$
and the Sobolev embedding $L^{6/5}\subset(\dot H^1)^*$ gives finite Coulomb
energy; the nuclear term uses $|x|^{-1}\in L^{5/2}_{\mathrm{loc}}$ near zero and
finite mass away. This is exactly the finite-mass class of $E_Z^{\TF}(M)$.

\begin{proposition}\label{prop:tf-identification}
With the two-spin $T$ above,
\begin{equation}\label{eq:tf-unit-deficit}
 \lim_{Z\to\infty}\bigl[E_Z^{\TF}(Z-1)-E_Z^{\TF}(Z)\bigr]=\tfrac37q^{-4/3},
\end{equation}
through real charges $Z>1$. Consequently \eqref{eq:tf-constant} holds for every
fixed real $m>0$ with $a_{\TF}=\tfrac37q^{-4/3}>0$.
\end{proposition}

\begin{proof}
The charge in the unit-price problem is identified, then integrated in the price and
use exact density scaling.

\paragraph{Monotonicity in the electron mass.}
$E_Z^{\TF}(M)$ is nonincreasing in $M$. Truncate any admissible density of mass
$M_1$ to a large ball (kinetic, nuclear, direct terms converge). For $M_2\ge M_1$
put the missing mass in a nonnegative smooth packet of diameter $\sim R$ at
distance $R^2$; a packet of mass $a$ at that scale has kinetic and direct
energies $O(a^{5/3}R^{-2})$ and $O(a^2R^{-1})$, while its nuclear term and its
interaction with the fixed truncated density vanish as $R\to\infty$ (with $a$
bounded). Taking $R\to\infty$, then the truncation radius to infinity, then the
original energy down to its infimum proves monotonicity; the same shows mass
exactly $M$ or at most $M$ gives the same infimum.

\paragraph{The zero-price minimizer is neutral.}
Apply \FTFPair to the single positive point source $Z\delta_0$. The unconstrained
problem on $L^{5/3}\cap(\dot H^1)^*$ has a unique minimizer $\rho_0$, of finite
mass at most $2Z$, with field $\phi_0=V-K*\rho_0$ satisfying
\begin{equation}\label{eq:tf-neutral-field}
 \rho_0=k(\phi_0)_+^{3/2},\quad
 \Delta\phi_0=-4\pi Z\delta_0+4\pi\rho_0,\quad
 c\min\Bigl(\frac Z{|x|},|x|^{-4}\Bigr)\le\phi_0(x)\le C|x|^{-4}.
\end{equation}
The lemmas give the lower bound, finite mass, and decay. The upper bound is
independent of $Z$: on $B(x,R)$, $R=|x|/2$, compare $\phi_0$ with
$v(y)=CR^4/(R^2-|y-x|^2)^4$, which for large universal $C$ has
$\Delta v\le4\pi kv^{3/2}$ and diverges at the boundary, so the positive-part
comparison gives $\phi_0(x)\le CR^{-4}$. Uniqueness makes $\rho_0,\phi_0$ radial;
Newton's identity and finite mass give
$Z-\int\rho_0=\lim_{d\to\infty}d\phi_0(d)=0$, so $\int\rho_0=Z$ and
\begin{equation}\label{eq:tf-neutral-minimum}
 E_Z^{\TF}(Z)=\min_{\rho\in\mathcal X}\mathcal T_Z(\rho).
\end{equation}
There is no class discrepancy: every finite-mass admissible density is in the
Coulomb-dual class, and the minimizer has finite mass and energy, hence is in
$\mathcal X$.

\paragraph{The unit-price minimizer.}
Minimize $\mathcal T_Z(\rho)+\int\rho$ over $\mathcal X$. Removing density where
$V\le1$ cannot increase the functional (kinetic, $\int(1-V)\rho$, and removed
direct interactions there are nonnegative), so the analysis is restricted to $B(0,Z)$. Since
$V\in L^{5/2}(B(0,Z))$, coercivity and weak lower semicontinuity in $L^{5/3}$ give
a minimizer, and strict convexity of the kinetic term gives uniqueness. Denote it
$\rho_Z$, $\phi_Z=V-K*\rho_Z$; nonnegative variations, then two-sided
multiplicative variations on positive density, give
\begin{equation}\label{eq:tf-priced-euler}
 \rho_Z=k\bigl((\phi_Z-1)_+\bigr)^{3/2},\qquad
 \Delta\phi_Z=-4\pi Z\delta_0+4\pi\rho_Z,
\end{equation}
globally, since outside $B(0,Z)$ one has $\phi_Z\le V\le1$. The minimizer and
field are radial by uniqueness.

The unit-price chemical-potential comparison (Solovej 2003, Lemma~5.1) gives
\begin{equation}\label{eq:tf-price-comparison}
 \phi_0\le\phi_Z\le\phi_0+1.
\end{equation}
For the first: on a positive set of $\phi_0-\phi_Z-\eta$ ($\eta>0$) the
difference of the two reactions in \eqref{eq:tf-neutral-field},
\eqref{eq:tf-priced-euler} is nonnegative; testing with that positive part gives a
zero squared-gradient bound, the test compactly supported since both potentials
decay. For the second, $\phi_0+1$ solves the same unit-price equation off the
common point source and the same comparison applies; at the origin the
singularities cancel, the rest is continuous locally $H^1$ with locally $L^{5/3}$
sources, so compact tests are justified by $H^1$ approximation. Let
$\eta\downarrow0$. Continuity and decay hold for Coulomb potentials of finite
$L^1\cap L^{5/3}$ densities \FTFExistence.

Write $Q_Z=Z-\int\rho_Z$. By \eqref{eq:tf-price-comparison}, $\phi_Z\ge0$ and
$\rho_Z\le k\phi_0^{3/2}\le C|x|^{-6}$; Newton's identity gives
\begin{equation}\label{eq:tf-priced-mean}
 \phi_Z(d)=\frac{Q_Z}{d}+\int_{|y|>d}\Bigl(\frac1d-\frac1{|y|}\Bigr)\rho_Z(y)\,dy.
\end{equation}
Finite mass gives $Q_Z=\lim_{d\to\infty}d\phi_Z(d)\ge0$; at $d=1$ the nonnegative
second term and \eqref{eq:tf-price-comparison} give $Q_Z\le\phi_Z(1)\le C+1$. Using
the exterior integral of $C|y|^{-6}$ in \eqref{eq:tf-priced-mean},
\begin{equation}\label{eq:tf-uniform-exterior}
 0\le\phi_Z(d)\le\frac Cd+Cd^{-4}.
\end{equation}
For a fixed large $R$, $\phi_Z<1$ on $d>R$ for every $Z$, so
\eqref{eq:tf-priced-euler} gives $\rho_Z=0$ there and $\phi_Z(d)=Q_Z/d$.

\paragraph{Identification of the limiting charge.}
Let $Z\to\infty$ through real values. On every compact annulus
\eqref{eq:tf-neutral-field}, \eqref{eq:tf-price-comparison} bound $\phi_Z$ and
\eqref{eq:tf-priced-euler} bounds its source, so local Poisson estimates give a
locally uniformly convergent subsequence with limit $\phi_\infty$ satisfying
\eqref{eq:limit-pde} and $c|x|^{-4}\le\phi_\infty\le1+C|x|^{-4}$.
\eqref{eq:tf-uniform-exterior} gives decay at infinity. By
Lemma~\ref{lem:singular-profile} this is the same field as in the limit analysis,
with exterior charge $q$; at any radius beyond both exterior supports
$Q_Z=d\phi_Z(d)\to d\phi_\infty(d)=q$. Uniqueness of the subsequential limit gives
\begin{equation}\label{eq:tf-unit-charge}
 Q_Z\longrightarrow q\qquad(Z\longrightarrow\infty).
\end{equation}
Real charges are allowed by the arbitrary-positive-point-source hypotheses of
\FTFPair.

\paragraph{Other prices and their energies.}
For $t>0$ let $\rho_{Z,t}$ minimize $\mathcal T_Z(\rho)+t\int\rho$, with
$Q_Z(t)=Z-\int\rho_{Z,t}$; existence and uniqueness follow as above restricting to
$B(0,Z/t)$. With $b=t^{-1/4}$ the transformation
\[
 \widetilde\rho(x)=b^6\rho_{Z,t}(bx),\qquad
 \widetilde\phi(x)=b^4\bigl(V-K*\rho_{Z,t}\bigr)(bx)
\]
gives the unit-price minimizer at charge $b^3Z$: indeed
$\mathcal T_{b^3Z}(\widetilde\rho)+\int\widetilde\rho=b^7(\mathcal T_Z(\rho_{Z,t})+
t\int\rho_{Z,t})$ and the map is a bijection. Its mass is multiplied by $b^3$, so
\begin{equation}\label{eq:tf-price-scaling}
 Q_Z(t)=t^{3/4}Q_{t^{-3/4}Z}\longrightarrow qt^{3/4}\quad(t>0\text{ fixed}).
\end{equation}
Define $\Pi_Z(t)=\min_{\rho\in\mathcal X}(\mathcal T_Z(\rho)+t\int\rho)$ and
$G_Z^{\TF}(t)=\Pi_Z(t)-tZ-E_Z^{\TF}(Z)$. Endpoint minimizers give, for $0<v<t$,
\[
 -Q_Z(t)\le\frac{G_Z^{\TF}(t)-G_Z^{\TF}(v)}{t-v}\le-Q_Z(v),
\]
and \eqref{eq:tf-neutral-minimum} gives $-vQ_Z(v)\le G_Z^{\TF}(v)\le0$. The
fixed-partition integration and $v\downarrow0$ as in
\eqref{eq:quantum-price-increments}, \eqref{eq:quantum-zero-endpoint} give
\begin{equation}\label{eq:tf-priced-energy}
 G_Z^{\TF}(t)\longrightarrow-\tfrac47qt^{7/4},\qquad
 \mathcal T_Z(\rho_{Z,t})-E_Z^{\TF}(Z)\longrightarrow\tfrac37qt^{7/4}.
\end{equation}
Each priced minimizer also minimizes at its own mass (an improvement there would
improve the priced objective). For fixed $t_-<q^{-4/3}<t_+$,
\eqref{eq:tf-price-scaling} places the two masses on opposite sides of $Z-1$ for
large $Z$; mass monotonicity and \eqref{eq:tf-priced-energy}, followed by
$t_\pm\to q^{-4/3}$, give \eqref{eq:tf-unit-deficit}.

\paragraph{Every fixed removed mass.}
For $m>0$ the scaling $\rho(x)=m^2\sigma(m^{1/3}x)$ multiplies mass by $m$ and
gives $\mathcal T_Z(\rho)=m^{7/3}\mathcal T_{Z/m}(\sigma)$, a bijection between the
admissible classes. Hence
\begin{equation}\label{eq:tf-mass-scaling}
 E_Z^{\TF}(N)=m^{7/3}E_{Z/m}^{\TF}(N/m),\qquad
 I_m^{\TF}(Z)=m^{7/3}I_1^{\TF}(Z/m).
\end{equation}
For fixed $m>0$ the real charge $Z/m\to\infty$; applying
\eqref{eq:tf-unit-deficit} proves \eqref{eq:tf-constant} and the value of
$a_{\TF}$, completing the coefficient assertion of Theorem~\ref{thm:priced-deficit}.
\end{proof}

\medskip
\noindent\textit{The specified order of limits.}\par\medskip
\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
The joint conclusion \eqref{eq:joint-limit} follows from
Proposition~\ref{prop:fixed-sector-limit} and
Proposition~\ref{prop:tf-identification}. For the iterated limits, suppose first
that along unbounded integers $m_j$ the normalized inner upper limit exceeds
$a_{\TF}$ by a fixed $\eta>0$. By definition of limsup, for each $j$ one may pick
arbitrarily large integers $Z_j>m_j$, in particular $Z_j/m_j\ge j$, with
$I_{m_j}(Z_j)/m_j^{7/3}>a_{\TF}+\eta/2$, contradicting \eqref{eq:joint-limit}
(the same selection applies if the inner upper limits are infinite). Likewise a
normalized inner lower limit below $a_{\TF}-\eta$ along unbounded $m_j$ allows
$Z_j/m_j\ge j$ with energies below $a_{\TF}-\eta/2$, again a contradiction. Hence
\[
 \limsup_{m\to\infty}\frac{\limsup_{Z\to\infty}I_m(Z)}{m^{7/3}}\le a_{\TF},\qquad
 \liminf_{m\to\infty}\frac{\liminf_{Z\to\infty}I_m(Z)}{m^{7/3}}\ge a_{\TF}.
\]
Since an inner lower limit never exceeds the inner upper limit, both outer limits
exist and equal $a_{\TF}$. The construction always chooses $Z$ only after $m$ is
fixed; no convergence in $Z$ at fixed $m$ is assumed.
\end{proof}



\end{document}