Simon 2000 #9: bounded excess charge

• Slug: simon-ionization-excess
• List: Simon 2000 #9 (Coulomb energies); 1984 10(a) if useful
• Solver: GPT-6 Astra (2026-09-07); earlier runs credited in README.
• Status: dent of the printed leading 1.1185 and the notebook leading 1.1006. Same HPS chain: $N_c(Z)<1.1005Z+3.933Z^{1/3}$ for $Z\ge4$. Exact claim and falsifier: q14/CLAIM.md. Ionization conjecture open.
• Area: Mathematical physics / many-body Schrödinger
• Sources: Simon 2000 #9; Simon 1984 10(a),(d); Lewin, charged quantum particles
• Started: 2026-08-27

In general

A nucleus of charge $Z$ and $N$ electrons has Hamiltonian

on the fermionic space. Write $E(N,Z)$ for the bottom of the spectrum. Binding means $E(N,Z)<E(N-1,Z)$. Let $N_0(Z)$ (Simon) or $N_c(Z)$ (Nam, HPS) be the largest such $N$. Zhislin proved binding whenever $N<Z+1$. Lieb proved $N_c<2Z+1$. The ionization conjecture, Simon 2000 problem 9, asks for a $Z$-independent bound on the excess $N_0(Z)-Z$.

Wikipedia is only a map. The published record is on arXiv.

Precise statement

Prove that $N_0(Z)-Z$ stays bounded as $Z\to\infty$, or give a verified finite improvement of a published non-asymptotic upper bound on $N_c(Z)$.

The 1984 companion 10(a) asks for monotonicity of the ionization energy $\Delta E(N,Z)=E(N-1,Z)-E(N,Z)$ in $N$. A certified range where that inequality holds is useful; it is not a substitute for a bound on the excess.

Published record (fetched 2026-08-27)

• Lieb, Phys. Rev. A 29 (1984): $N_c<2Z+1$ for all $Z>0$ (fermions or bosons).

• Nam, arXiv:1009.2367v3: $N_c<1.22Z+3Z^{1/3}$ for fermions; equivalently $\beta^{-1}Z+\cdots$ with $\beta\in[0.8218,0.8705)$.

• Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1: $N_c<1.1185Z+4Z^{1/3}$ for all $Z\ge4$, and the sharper

  $$N<b(3)Z+3.90\,Z^{1/3}+0.0134+0.184\,Z^{-1/3}+0.0196\,Z^{-2/3},$$

  with $1.1184<b(3)<1.1185$. Also $N_c<\frac12(\sqrt2+1)Z+2.96Z^{1/3}$ for $Z\ge2$.

• Lieb–Sigal–Simon–Thirring: $N_c(Z)/Z\to1$. Fefferman–Seco and Seco–Sigal–Solovej: $N_c\le Z+O(Z^{5/7})$.

• Solovej, arXiv:math-ph/0012026: the ionization conjecture holds in Hartree–Fock.

• Bosonic atoms: $N_c/Z\to t_c\approx1.21$ (Benguria–Lieb, Solovej, Baumgartner).

After 2026-08-27

The same HPS Section 7 chain, with Lieb’s $N/Z<2+1/Z\le9/4$ on the Prop. 2.5 range $Z\ge4$ and the exact $b(3)$ in $a_1$, gives

and the simplified

Also $N_c<b(2)Z+2.953Z^{1/3}$ for $Z\ge2$. This beats the printed remainders 3.90, 4, and 2.96. It does not beat the leading 1.1185. Replay: problems/simon-ionization-excess/compute/q1/run_all.sh.

$N_0(1)=2$ is replayed (Hylleraas $E=-815/1602<-1/2$, plus Lieb). That uniqueness is already in Lieb 1984.

A later search (same day) replayed HPS v1, Nam, Lieb, and Benguria–González-Brantes arXiv:2511.07582v1 ($N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge12$, bosonic / statistics- independent). That paper does not beat $1.1185$ for fermions. Two-shell dipoles make the HPS radialization form negative for every tested $s>3$, so $b(4)$ cannot be used. Lieb remains the best integer envelope at $Z=2,3,4,5$. A compact-aspect lower bound $Q\ge0.901924$ (aspect $\le4$) is certified and does not replace Theorem 2.2. A claimed $1.1168$ lift was withdrawn. Replay: problems/simon-ionization-excess/compute/q2/run_all.sh.

A third search the same day lifts the aspect-$12$ compact bound $Q\ge 0.899526$ to every radial probability in HPS (4.1): mass-stationarity on a used support of aspect $\ge 12$ forces $Q>12/13$, and every atomic measure reduces to one of those two classes. Truncation plus spherical-shell approximation extends the bound. The same Section 7 chain with this $\beta_3$ gives

and

This beats the printed leading $1.1185$. It does not use the aspect-$\le 4$ number $1.1087$ as an unrestricted bound, and it does not restore the withdrawn $1.1168$. Finite-$Z$ integers and $s>3$ stay residue. Replay: problems/simon-ionization-excess/compute/q3/run_all.sh.

A fourth search the same day keeps the mass-opt dichotomy and tightens the compact class: mid-radius faces at aspect $10$ with $26$ bins certify $Q\ge 0.904414$ (target $\varphi=0.9091$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q3's $1.1118$ and the printed $1.1185$. The unrestricted leading sits below the aspect-$\le 4$ class-only number $1.1087$; that class-only figure is not used as a bound. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q4/run_all.sh.

A fifth search the same day keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $30$. Faces certify $Q\ge 0.906238$ (target $\varphi=0.9103$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1057$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q5/run_all.sh.

A sixth search the same day keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $32$. Faces certify $Q\ge 0.906992$ (target $\varphi=0.9108$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1035$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q6/run_all.sh.

A seventh search the same day keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $33$. Faces certify $Q\ge 0.907407$ (target $\varphi=0.9111$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1026$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q7/run_all.sh.

An eighth search the same day keeps the mass-opt dichotomy at aspect $10$ and raises the face target on the $n=33$ mid-radius matrix from $0.9111$ to $0.9112$. Faces certify $Q\ge 0.907507$ (target $\varphi=0.9112$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q7's $1.1021$, q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1021$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q8/run_all.sh.

A ninth search the same day keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $34$. Faces certify $Q\ge 0.907716$ (target $\varphi=0.9113$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q8's $1.1020$, q7's $1.1021$, q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1020$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q9/run_all.sh.

A tenth search the same night keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $35$. Faces certify $Q\ge 0.908018$ (target $\varphi=0.9115$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q9's $1.1017$, q8's $1.1020$, q7's $1.1021$, q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1017$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q10/run_all.sh.

An eleventh search the next morning keeps the mass-opt dichotomy at aspect $10$ and raises the mid-radius bin count to $36$. Faces certify $Q\ge 0.908315$ (target $\varphi=0.9117$). The cut $Q>10/11$ still exceeds that $\gamma$. The same Section 7 chain gives

and

This beats q10's $1.1013$, q9's $1.1017$, q8's $1.1020$, q7's $1.1021$, q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1013$ and is residue. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay residue. Replay: problems/simon-ionization-excess/compute/q11/run_all.sh.

A twelfth search the same day keeps the mass-opt dichotomy at aspect $10$ and starts $n=37$ mid-radius bins, target $\varphi=0.9119$. SLSQP predicts printed $1.1006$ if faces certify. The dump of $2^{37}-1$ faces was stopped with the slowest shard at about $30\%$. Scanned faces stay copositive and $\min\varphi$ stays above $0.9119$; the rest of the mask space was not enumerated. Incomplete search is not a bound. Printed leading stays $1.1010$. $R\le 9$ with the mass-opt cut cannot beat $1.1010$ and is leftover. Finite-$Z$ integers and $N_0(Z)-Z$ stay leftover. Replay: problems/simon-ionization-excess/compute/q12/run_all.sh.

A thirteenth search finishes $n=37$ at the same split with a Gray-code enumerator in place of the incomplete mask dump. Faces $137{,}438{,}953{,}471$, copositive, $0$ residual skips, $\min\varphi=0.912085>0.9119$. Compact $\gamma=0.908606$, $1/\gamma=1.100587$, cut $10/11>\gamma$. The same Section 7 chain gives

and

This beats q11's $1.1010$, q10's $1.1013$, q9's $1.1017$, q8's $1.1020$, q7's $1.1021$, q6's $1.1026$, q5's $1.1035$, q4's $1.1057$, q3's $1.1118$, and the printed $1.1185$. $R\le 9$ with the mass-opt cut cannot beat $1.1006$ and is leftover. $1.1168$ stays withdrawn. Finite-$Z$ integers and $N_0(Z)-Z$ stay leftover. Replay: problems/simon-ionization-excess/compute/q13/run_all.sh.

2026-09-07 — rational matrix certificates

Two independent finite verifiers establish

This is the inequality in q14/CLAIM.md. A raised target $0.912$ on 37 rational bins proves $\beta_3\ge0.9087$. Python verifies a positive-definite matrix and its entrywise nonnegative remainder with exact rational arithmetic. Rust checks a separate Gram witness using outward integer intervals. The same mass-stationary cut at aspect ten extends the compact result, and each implementation independently checks the HPS constants. The proof and replay require no earlier face-enumeration certificate. The leading 1.1005 improves q13's 1.1006 and HPS's printed 1.1185; bounded excess remains open.

Replay: problems/simon-ionization-excess/compute/q14/run_all.sh.

What would count as a new bound

The 2026-09-12 attempt with 40 bins stopped before numerical search and produced no new certificate. The bound above remains unchanged; see q15/CLAIM.md for the unproved target.

A verified inequality that strictly improves a published record, with a replayable certificate. Examples: a leading coefficient below $1.1185$ for all large $Z$; a remainder strictly below the printed $3.90$ or $4$ on the HPS range; $N_c\le Z+C$ for a universal $C$; a unique exact $N_0(Z)$ for some $Z>1$ that the published inequalities do not already settle.

What does not count

An incomplete variational or DFT search is not a lower bound. Restating HPS Proposition 2.5 is not a new bound. Hartree–Fock or DFT tables are heuristics. Simon 2000 #9 remains open unless the excess is shown bounded.