Small-Z replay — q4

Nothing here is a dent. Best published integer bound at $Z=2,\ldots,6$ is still Lieb $N_c\le 2Z$.

The q3 leading dent $N_c<1.1118Z+3.966\,Z^{1/3}$ ($Z\ge 4$) tightens the printed HPS simplified envelope as reals but excludes no extra integer $N$ beyond Lieb, Nam, and the HPS printed lines.

Hydrogen uniqueness $N_0(1)=2$ is Lieb 1984 and is not claimed.

Replay:

python3 problems/simon-ionization-excess/compute/q4/work/smallz_replay.py

Certificate: compute/q4/certs/smallz.json.

Published envelopes at $Z=2,\ldots,6$

Same arithmetic as q3 work/smallz_replay.py, now with the q3 simplified column. Integer exclusion: $N_c<U$ means at most the largest integer strictly below $U$.

At every $Z\ge 4$, the q3 simplified $U$ sits below the printed HPS simplified $U$ and above $2Z+1$, so its integer cap matches Lieb ($N_c\le 2Z$). No extra integer is excluded by the q3 dent beyond what Lieb, Nam, and the HPS printed envelopes already leave open.

Nam states that $1.22Z+3Z^{1/3}<2Z+1$ when $Z\ge 6$. HPS states that Prop. 2.4 improves Lieb for $Z>5.3$. Both hold as real comparisons at $Z=6$, and both still have $U>12$.

Zhislin gives $N_0(Z)\ge Z$.

One finite-$Z$ try

Nam Lemma 1 at the Lieb edge. Plug $N=2Z$ (the largest integer still allowed by Lieb at each $Z$) into Nam Lemma 1, $\alpha_N(N-1)<Z(1+0.68\,N^{-2/3})$, using published lowers $\max\{1/2,\mathrm{Prop.\,1},\sqrt5/4\}$ on $\alpha_N$. At $Z=2,\ldots,6$ the lower side stays below the Lemma-1 right-hand side; same at the unsettled candidates $N=3,4$ for $Z=2$. No contradiction.

Four-electron Hylleraas-style comparison for $Z=2$, $N=4$. The separated trial He (exact Hylleraas $-54353/18800$) plus two hydrogen $1s$ electrons at infinity gives a legal variational upper bound $E(4,2)\le -54353/18800-1\approx -3.891$. The q1 hydrogenic $1s^22s$ three-electron upper is about $-7.289$. The separated four-electron upper sits above that three-electron upper, but comparing two variational uppers does not prove $E(4,2)\ge E(3,2)$ and therefore does not prove non-binding at $N=4$. A compact Hylleraas$\times$outer product omitting core–outer repulsion is invalid (energy too negative).

Pair geometry and Temple/tetrahedron routes are not repeated here (q2–q3).

$N_0(Z)-Z$

Cannot move this session. No verified integer envelope beats Lieb at a concrete $Z>1$. The q3 leading dent is asymptotic ($Z\ge 4$) and does not tighten integer caps. A leading coefficient $>1$ cannot prove a $Z$-independent excess bound on $N_0(Z)-Z$.

Status

Residue. No verifier-plus-certificate beats a published integer envelope at a concrete $Z>1$.