q14 claim — raised compact target

Status: dent, certified 2026-09-07 by two independent finite verifiers and the argument in PROOF.md.

For the fermionic Coulomb Hamiltonian and $N_c(Z)$ defined in PROBLEM.md, the bound is

The prior notebook record is q13's $N_c(Z)<1.1006Z+3.933Z^{1/3}$ for $Z\ge4$. The published comparison is Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1, Proposition 2.5: $N_c(Z)<1.1185Z+4Z^{1/3}$ for $Z\ge4$.

The finite handle uses 37 rational bins covering $[1,10]$, with edges near the geometric grid, and target $\varphi=0.912$. The precise edges and two matrix witnesses are in certificate.json. The compact bound is at least $9087/10000=0.9087$; the mass-stationary cut $10/11$ exceeds that number. Consequently $\beta_3\ge9087/10000$. The HPS Section 7 calculation then gives the displayed inequality.

verify.py reconstructs the kernel bounds with exact rational arithmetic, checks positive pivots of one matrix, and checks its entrywise nonnegative remainder. verify.rs independently checks a different, explicit Gram witness using outward integer intervals. Both independently verify the discretization error, split, published $b(3)$ window, and final HPS constants. The solver's numerical status is not a premise. No older enumeration or success flag is a premise.

Falsifier: a valid bound-state example violating the displayed inequality disproves it. A nonnegative vector violating the proposed matrix inequality, an uncovered radial measure, an invalid compact-to-global argument, or failed outward bounds on the Section 7 constants invalidates this certificate. Numerical optimization and a stored success flag alone do not certify anything.

run_all.sh reconstructs and independently checks every finite certificate and its numerical implications. Exit 0 means the full claim holds. Missing certificates, failed checks, or incomplete search return nonzero. The bounded-excess ionization conjecture remains open. This improves the specified HPS-style finite bound; it does not replace the asymptotic results with leading coefficient one.