\section{Introduction}\label{sec:01-introduction}

The atomic ionization problem asks for the maximum number $\AtomCount{Z}$
of electrons that a nucleus of charge $Z$ can bind~\cite{nam2020,simon2000}.
Experimentally, no isolated atom is known to bind more than one additional
electron in a stable state~\cite{andersen2004}. The ionization conjecture
asks for a bound $\AtomCount{Z}\leq Z+C$ with $C$ independent of $Z$.

Lieb's bound $\AtomCount{Z}<2Z+1$~\cite{lieb1984} is sharp for
hydrogen~\cite{bethe1929,hylleraas1930}, but permits four electrons for
helium and six for lithium. Later work established asymptotic neutrality,
$\AtomCount{Z}/Z\to1$ as $Z\to\infty$~\cite{lieb1988,fefferman1990,seco1990},
and improved explicit
general bounds~\cite{nam2012,hundertmark2025}, without reducing these
helium and lithium limits. Shelley and Tartakovsky recently excluded the
helium dianion, lowering the helium upper bound to three
electrons~\cite{shelley2026}. We prove that helium binds no additional
electron in its ground state and lithium binds exactly one.

We consider nonrelativistic electrons of spin $1/2$ and an infinitely heavy
point nucleus, in Hartree atomic units. For an integer $N\geq1$ and $Z>0$,
let $x_i\in\mathbb R^3$ be the position of electron $i$ relative to the
nucleus. On the fermionic space
$\FermionSpace{N}=\bigwedge^N L^2(\mathbb R^3;\mathbb C^2)$, set
\[
 \AtomH{N}{Z}=-\frac12\sum_{i=1}^N\Delta_{x_i}
 -Z\sum_{i=1}^N\frac1{|x_i|}
 +\sum_{i<j}\frac1{|x_i-x_j|},
\]
\[
 \AtomE{N}{Z}=\inf\sigma(\AtomH{N}{Z}),\qquad \AtomE{0}{Z}=0,
 \qquad
 \AtomCount{Z}=\sup\{N\geq1:\AtomE{N}{Z}<\AtomE{N-1}{Z}\}.
\]
The strict inequality defines binding. By the HVZ theorem
(Theorem~\ref{thm:hvz}), $\AtomE{N-1}{Z}$ is the detachment threshold,
the bottom of the essential spectrum for $N\geq2$.

\begin{theorem}\label{thm:main}
The maximum electron counts are
\[
 \AtomCount{2}=2,\qquad \AtomCount{3}=4.
\]
For $Z=2$ and $N\geq3$, and for $Z=3$ and $N\geq5$, the ground energy
$\AtomE{N}{Z}$ is not an eigenvalue of $\AtomH{N}{Z}$.
\end{theorem}

Equality of successive ground energies can still admit a square-integrable
threshold eigenfunction~\cite{bellazzini2014}. The theorem excludes this
possibility beyond the stated counts, so the counts also apply when
binding means existence of a ground state.

Lithium's positive electron affinity is established experimentally and
supported by accurate variational calculations~\cite{haeffler1996,sims2017}.
Lower-bound calculations for neutral lithium also have a substantial
history~\cite{king1995,ireland2022lower}. Our proof compares a variational
upper bound for the anion with a certified lower bound for the neutral
atom, controlling the spectral hypotheses, symmetry sectors and numerical
errors. Section~\ref{sec:06-lithium-binding} gives the resulting positive
lower bound for the affinity.

Nonbinding requires an operator lower bound at the exact detachment
threshold. For the dianions, we develop Lieb's and Nam's weighted
removal identities, using the square of an electron's distance from the
nucleus as the weight. We retain the weighted kinetic energy and use the
Pauli principle; geometric bounds on the repulsion and one-electron
spectral bounds then establish positivity of the required quadratic forms on the
fermionic space. This builds on the earlier helium
analysis~\cite{shelley2026}. Analytic estimates exclude the higher electron
counts allowed by Lieb's bound.

The helium monoanion requires comparison with the exact neutral energy.
We match estimates in a bounded region with estimates for an escaping
electron to establish nonbinding and exclude a threshold ground state.
This three-electron estimate also supplies a starting point for the lower
bound on neutral lithium used in the binding proof.

Sections~\ref{sec:02-setting}--\ref{sec:03-removal} give the setting and
common estimates; Sections~\ref{sec:04-helium-dianion}--\ref{sec:07-lithium-dianion}
treat the individual ions, and Section~\ref{sec:08-counts} completes the
proof of Theorem~\ref{thm:main}. The appendices contain the computational
certificates, including error bounds and case coverage, and the deferred
analytic proofs.