Small-Z residue — q3

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

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

Replay:

python3 problems/simon-ionization-excess/compute/q3/work/smallz_replay.py
python3 problems/simon-ionization-excess/compute/q3/work/smallz_check.py

Opened this session: Nam 1009.2367v3 HTML, HPS 2504.18487v1 HTML Prop. 2.4–2.5 and (7.10), BGB 2511.07582v1 HTML (only for $Z\ge 12$).

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

Same arithmetic as q2 envelopes.py, now including $Z=6$. Integer exclusion: $N_c<U$ means at most the largest integer strictly below $U$.

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 $U>12$, so they exclude no extra integer. BGB’s formula at $Z=6$ is about $14.61>13$ and is not applicable.

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

One finite-$Z$ try

HPS Prop. 2.4 maximises the remainder $a(x)$ on $N/Z\le 5/2$. At a fixed $Z$, Lieb is $N/Z<2+1/Z$. That $a(2+1/Z)$ is increasing for $x>0.81$, so the max is the right endpoint. The $Z$-local envelope $b(2)Z+a(2+1/Z)\,Z^{1/3}$ is $10.964878$ at $Z=5$ and $12.466731$ at $Z=6$: below $2Z+1$ as reals, still above $2Z$. Same integers as Lieb.

Direct HPS (7.10) and the $s=2$ form (7.6) at the Lieb edge $N=2Z$, using q1’s $\lambda\approx 0.628336$ for $s=2$ (printed $0.6284$ is weaker): no contradiction. Closest gap is $Z=6$, $N=12$: $N\beta_2\approx 9.941<10.269$.

Nam Lemma 1 / Prop. 1 likewise: published $\alpha$ lowers sit below the kinetic right-hand side (at $Z=6$, $N=12$ about $6.321<6.778$).

Pair geometry, kinetic dropped. Tetrahedron still blocks $N=4$ at $Z=2$ ($54<64$). Five vertices of the regular octahedron give

$\alpha_{5,2}\le(4\sqrt{2}+1)/10,\qquad \alpha_{5,2}\cdot 4<3$

because $128<169$: the $s=2$ pair ratio cannot exclude $N=5$ at $Z=3$.

The full octahedron has $5\alpha_{6,2}>3$ ($8>25/4$); a prism scan stays above $0.6$. Not a dent.

Status

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