1 · The essentials
How many extra electrons can an atom or molecule hold, and how hard is it to remove them? These manuscripts study those questions in a model where nuclei stay fixed and electrons interact through electric forces. The first manuscript shows that the number of extra electrons a molecule can strictly hold is bounded by a universal constant times its number of nuclei. For a neutral atom, it also shows that the energy needed to remove the first electron stays between fixed positive bounds, independently of the nuclear charge. A similar pair of bounds controls the distance beyond which the expected amount of electron charge equals half an electron. The companion manuscripts give precise limiting formulas for the energy needed to remove many electrons and for distances enclosing all but a specified expected number of electrons. The energy result has a Lean formalization; the supplied source does not identify the other results as formalized.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine asking how many extra guests a venue can retain, how much effort it takes to persuade some to leave, and how far its outermost crowd spreads. The results place limits on all three quantities, but electrons are described through quantum states and expected amounts of charge, not individual guests standing at definite locations.
Why it matters
These results connect three questions about the outer electrons: how many can remain bound, how much energy their removal requires, and how far they extend. The bounds are universal within the stated model, rather than depending on a particular nuclear charge. The limiting formulas also identify the Thomas–Fermi energy constant and an explicit radius constant in the full Coulomb setting.
2 · Going further
The molecular problem concerns the full nonrelativistic Coulomb Hamiltonian with two electron spin states, fixed nuclei at arbitrary distinct positions, and real nuclear charges at least one. Writing for their total charge, Manuscript 1 shows that the molecule strictly binds at most electrons, with universal. For neutral atoms with integer , it also gives universal positive upper and lower bounds on the first ionization energy and on the radius outside which the expected electron count is one half; the radius statement holds for every ground state.
Manuscript 2 addresses the energy part of the generalized ionization conjecture. If denotes the energy needed to remove electrons, its result is whenever and , where is the Thomas–Fermi constant in atomic units. Its abstract also states that both conjectured iterated limits are obtained. This is the manuscript identified as Lean-formalized.
Manuscript 3 addresses the corresponding radius question for neutral atoms. Define the radius by requiring an expected exterior electron count of . For every choice of ground states, both its upper and lower large- limits are asymptotic to as . Thus the radius statement takes the large-nuclear-charge limit first, unlike the stated simultaneous regime for the energy formula.
Together, the manuscripts move from uniform bounds on excess charge and the outermost electrons to precise asymptotic energy and radius laws. The supplied abstracts state these conclusions but do not describe the argument or its intermediate estimates, so they do not support a more specific account of the proof strategy.
Useful notions
- Nuclear charge and neutral atoms
- Coulomb interactions
- Quantum ground states
- Expected electron counts
- Asymptotic formulas and order of limits
What was known before / what is new
Before
The source frames these questions as the ionization and generalized ionization conjectures. It identifies Thomas–Fermi asymptotics as the target for the removal energy and gives a corresponding explicit target for outer radii. It does not provide a history of earlier partial results.
New
Manuscript 1 shows a universal excess-charge bound for molecules and universal positive bounds for neutral-atom first ionization energies and half-electron exterior radii. The Lean-formalized energy manuscript establishes the stated many-electron removal asymptotics and reports both conjectured iterated limits. The radius companion shows the corresponding iterated-limit asymptotics for every choice of neutral-atom ground states.
Limits and assumptions
All results concern the full nonrelativistic Coulomb model with two electron spin states. The molecular bound assumes fixed, distinct nuclear positions and nuclear charges at least one; the neutral-atom bounds in Manuscript 1 assume integer nuclear charge Z ≥ 1. The energy asymptotic requires both m → ∞ and Z/m → ∞, while the radius asymptotic takes the large-Z limits before m → ∞. Exterior electron counts are expectations, not counts of particles at definite positions. Manuscript 2 is identified as Lean-formalized; the source does not identify Manuscripts 1 and 3 as formalized or specify whether every additional claim in Manuscript 2 is covered. No further open questions are specified in the supplied text.
Mini glossary
- First ionization energy —
- The energy needed to remove the first electron from a neutral atom.
- Excess charge —
- Here, the number of bound electrons beyond the total nuclear charge Z.
- Ground state —
- A quantum state of lowest energy for the system under consideration.
- Expected exterior electron count —
- The expected number of electrons outside a specified radius in a quantum state.
- Asymptotic equivalence —
- The relation expressed by ∼, meaning that the ratio of two quantities tends to one in the stated limit.
- Iterated limit —
- A limit taken in one parameter first and then in another, with that order forming part of the statement.
Keywords : Ionization · Coulomb atoms · Thomas–Fermi asymptotics
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as and ; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking first.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
24/09/2026We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.
Generalized ionization energies for full Coulomb atoms
24/09/2026We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to whenever and , with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.
Generalized outer-electron radii of neutral Coulomb atoms
24/09/2026We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to as m tends to infinity.