Small-Z residue — q2

Attempts at a certified finite-$Z$ bound on $N_0(Z)-Z$ for Simon ionization excess. Nothing here is a dent. Hydrogen uniqueness $N_0(1)=2$ is Lieb 1984 and is not claimed.

Replay: problems/simon-ionization-excess/compute/q2/run_all.sh.

Papers opened this session (record only):

• Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1 HTML
• Nam, arXiv:1009.2367v3 HTML

Published envelopes at $Z=2,3,4,5$

envelopes.py (mpmath) and envelopes_check.py (stdlib) replay the printed inequalities and the q1 remainder tightening. Integer exclusion is $N_c<U\Rightarrow N_c\le$ the largest integer strictly below $U$.

At every one of these four charges the best published envelope is Lieb $N_c<2Z+1$. Nam and both HPS forms (printed and q1) sit above $2Z+1$, so they exclude no extra integer $N$.

Zhislin binds for every $N<Z+1$, so $N_0(Z)\ge Z$. Combined with Lieb, the only uniqueness in the published record at these charges is still hydrogen. Precise decimals: certs/envelopes.json.

Lieb-style weights

Nam's write-up of Lieb (HTML §2.1): multiply the eigenvalue equation by $|x_N|\overline{\Psi}$, drop the non-negative $(H_{N-1}-E)$ and weighted kinetic terms, then use the triangle $(|x|+|y|)/|x-y|\ge 1$. That is $N<2Z+1$. The triangle is sharp along opposite rays, so the pair geometry of $\varphi(x)=|x|$ cannot be improved uniformly.

For a general weight $\varphi$, the same identity (kinetic dropped) excludes $N$ at charge $Z$ if

For $\varphi(x)=|x|$ one has $R_N\ge(N-1)/2$, hence $R_4\ge 3/2<2$ and $R_3\ge 1<2$. To exclude $N=4$ or $N=3$ at $Z=2$ one needs $R>2$.

lieb_weights.py minimises $R_3$ and $R_4$ for

• $\varphi=|x|^s$ ($s=1,1.25,1.5,2,3$),
• $\varphi=|x|/(1+\lambda|x|)$,
• $\varphi=1-e^{-\alpha|x|}$,
• $\varphi=|x|e^{-\mu|x|}$,
• $\varphi=\min(|x|,\rho)$,
• a $C^1$ compact cutoff $|x|\,\eta(|x|/\rho)$.

A search minimum is an upper bound on $\inf R$. The power family reproduces the equilateral / tetrahedron values ($R_3=2/\sqrt{3}\approx 1.155$, $R_4=3\sqrt{6}/4\approx 1.837$), both below $2$. The screened and compact families degenerate: the optimiser pushes mass where $\varphi$ vanishes and $R$ collapses toward $0$, which is still $R\le 2$. Those weights therefore cannot exclude $N=3$ or $N=4$ at $Z=2$. Kinetic positivity is proved in the literature only for $\varphi=|x|^b$, $b\in[0,1]$ (Lieb; Chen–Siedentop); it was not used as a bound for the other families.

Nam / HPS at small $N$

Nam Lemma 1: $\alpha_N(N-1)<Z(1+0.68\,N^{-2/3})$. Nam Proposition 1: $\alpha_N\ge\frac{N}{N-1}\bigl[\beta-3(\beta/6)^{1/3}N^{-2/3}\bigr]$ with $\beta\ge 0.8218$. At $N=4$ the remainder already exceeds $\beta$, so that lower bound is weaker than the triangle $1/2$.

Nam's $\sqrt5/4$ for $N\ge 3$ gives $\alpha_4\cdot 3\le 3\sqrt5/4\approx 1.677<2$. Even with the kinetic term dropped this does not exclude $N=4$ at $Z=2$. Excluding $N=3$ at $Z=2$ would need $\alpha_3>1$, which is impossible because $\alpha_N\le\beta<0.8705$.

HPS (7.10) at $Z=2$, $N=4$: $N\beta_2\approx 3.31$ versus a right-hand side $\approx 5$ (the $N^{1/3}$ remainder is $\sim 2.5$). No contradiction. Same at $Z=3,4,5$ for $N$ still allowed by Lieb. Dump: certs/nam_smallz.json.

The centred regular tetrahedron evaluates $\alpha_{4,2}$ exactly: all $|x|=\sqrt{3}$, all $|x-y|=\sqrt{8}$, so $\alpha_{4,2}\le\sqrt{6}/4$. Then $\alpha_{4,2}\cdot 3\le 3\sqrt{6}/4<2$ because $54<64$. Two independent checks: verify_tetra.py and verify_tetra.c. This is a certified obstruction to a dent by the $s=2$ pair geometry, not a dent. The same for $N=3$: the centred equilateral triangle gives $\alpha_{3,2}\le\sqrt{3}/3$, hence $\alpha_{3,2}\cdot 2=2/\sqrt{3}<2$.

alpha_n.py / alpha_n_check.py search $\alpha_{N,s}$. The search minimum matches those closed forms at $s\ge 1.5$. A search min is an upper bound on the infimum and is not used as a lower bound.

Temple / intermediate Hamiltonian

Hylleraas at $Z=2$, $\alpha=37/20$, $c=37/100$ gives $E(2,2)\le -54353/18800$ (q1, exact rational). That is a legal upper on the helium threshold.

Hydrogenic $1s^2 2s$ and $1s2s2p\,{}^4P$ trials on $H(3,2)$ sit above that upper bound by $0.204$ and $0.885$ (q1 three_electron_try.py, replayed). Temple requires $\mu<E_1$. Taking $E_1$ as the HVZ threshold $E(2,2)$ makes the hypothesis false. No Temple number is claimed.

Crude intermediate Hamiltonians:

• drop repulsion: $E(3,2)\ge 3E(1,2)=-6$, far below helium;
• replace $1/|x-y|$ by $1/(|x|+|y|)$: a product scan of that minorant reached about $-3.82$ at $\zeta=1.6$, still $0.93$ below $-54353/18800$.

Variational uppers on $E(3,2)$ were not used as non-binding evidence. HF/DFT were not used.

What would have been a dent

A certified $N_c(2)\le 3$ (i.e. $N_c(2)<4$) or $N_c(2)=2$, with an independent second path, would beat Lieb's integer bound at helium. Nothing in this folder certifies that.

Status

Residue. Best published integer bounds at the requested charges:

(Lieb $N_c<2Z+1$). $N_0(Z)\ge Z$ by Zhislin. No unique $N_0(Z)$ for $Z>1$. Certificate of the residue: certs/smallz.json.