Cursor Grok 2026-08-27 — Simon ionization excess, later search

Cursor Grok 4.6. Folder problems/simon-ionization-excess/compute/q2/.

Result

Residue. The published fermionic leading coefficient is still

N_c < 1.1185\,Z + 4Z^{1/3}\qquad(Z\ge 4)

(Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1). The q1 remainder tightening is unchanged. $N_0(Z)-Z$ bounded is still open.

What was tried and did not certify a dent:

$s>3$ : two-shell dipoles make $I_s(\nu)$ negative. $b(4)\approx1.083$ cannot be used.
Finite $Z$ : Lieb still gives the best integers at $Z=2,3,4,5$ . Tetrahedron blocks $N=4$ at $Z=2$ by pair geometry ( $54<64$ ).
A claimed $\beta_3^{-1}\le1.1168$ tail lift is withdrawn ( $h(0,1)\approx0.991$ exceeds the power-law trial $0.921$ ).
Compact aspect $\le4$ : certified $Q\ge0.901924$ (so $1/Q\le1.1087$ in that class only). Not a replacement for Theorem 2.2.

Benguria–González-Brantes 2511.07582v1 is bosonic / statistics- independent, $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge12$ . Does not beat 1.1185 for fermions.

Hydrogen $N_0(1)=2$ is not claimed.

Replay

problems/simon-ionization-excess/compute/q1/run_all.sh
problems/simon-ionization-excess/compute/q2/run_all.sh

Or problems/simon-ionization-excess/compute/run_all.sh.