These numbers are not a bound

The files certs/hf_table.json and certs/delta_e.json are a heuristic Hartree–Fock table for small $N$ and $Z$. They do not move the ionization conjecture, and they do not improve a published non-asymptotic bound on Nc. Status: residue.

Solovej already proved the ionization conjecture in Hartree–Fock (arXiv:math-ph/0012026). A numerical HF table cannot touch that theorem. The Schrödinger problem (Simon 2000 #9) is a different object; HF energies are not energies of $H(N,Z)$.

What was computed

Three series, all labeled HEURISTIC, in Hartree.

1. One-parameter Slater $1s$ Hartree–Fock for helium-like ions. For one electron the energy is the exact hydrogenic value $-Z^2/2$. For two electrons the restricted $1s^2$ ansatz with scale $\zeta=Z-5/16$ gives $E=-(Z-5/16)^2$.
2. Unrestricted HF in a small even-tempered Gaussian basis with 1s, 2s and 2p scales (tight exponents ~ $Z^2$, valence ~ $(Z/2)^2$, plus a short diffuse tail). Integrals are analytic, same-centre Cartesian $s$ and $p$. Occupations follow Hund’s rule. Closed shells stay spin-restricted if the core-Hamiltonian guess is.
3. The same UHF loop on $s$-Gaussians only, for $N=0,1,2$. Extra electrons in a spherical $s$ basis are not a model of B–Ne.

Replay:

python3 rhf_atoms.py --self-test
python3 rhf_atoms.py
python3 delta_e_table.py

Binding, and what would count

The table marks a pair as “binds” when the HF energy drops from $N-1$ to $N$. That comparison is not a proof that the true $E(N,Z)$ lies below $E(N-1,Z)$. Both numbers are variational upper bounds. A worse $N$-electron trial can sit above a better $(N-1)$-electron trial even if the $N$-electron atom is bound, and the other way around.

A variational energy becomes a binding certificate — a genuine lower bound on N0 — only if it lies strictly below a certified value of $E(N-1,Z)$. The only exact thresholds used here are the bare nucleus $E(0,Z)=0$ and the hydrogenic $E(1,Z)=-Z^2/2$.

For $Z\ge 2$ the Slater two-electron energy is below $-Z^2/2$, so N0 is at least 2. That is already in Zhislin’s theorem ($N<Z+1$ binds) and is not a new bound. For $Z=1$ the same ansatz gives $-(11/16)^2=-0.47265625$, which is above $-1/2$, so it does not certify that H- is bound. The Hylleraas certificate that N0(1) equals 2 is a different calculation.

Experimental ionization energies are leads, not certificates.

The 1984 10(a) check

From the same energies the script writes

and tests $\Delta E(N-1,Z)\ge\Delta E(N,Z)$ on consecutive pairs that both exist in the table. On this run the inequality held for every such pair (10 helium-like, 63 UHF $s/p$, 10 s-only). That is a property of these HF numbers, not a proof of monotonicity. A failed pair would not be a counterexample to 10(a) either: the inputs are HF numbers, not $E(N,Z)$.

Heuristic N0-hat in the UHF $s/p$ table is 1 for $Z=1$ and $Z$ for $Z=2$ through 10: no anion binds. H- is not below the exact $-1/2$, so there is no new binding certificate. For $Z\ge 2$ the two-electron energy is below $-Z^2/2$, which Zhislin already gives. Two lithium-anion SCF runs ($Z=3$, $N=4$ and $5$) hit the iteration cap; their energies still sit above Li.

Thomas–Fermi

A Thomas–Fermi or spherical LDA toy was not needed for the integrals and is not stored. Continuum TF energies do not decide a finite N0.