Ionization coefficient 1.1005

The certified inequality is

N_c(Z)<1.1005Z+3.933Z^{1/3}\qquad(Z\ge4).

This improves q13's leading coefficient 1.1006 and the published HPS bound with leading 1.1185. The bounded-excess conjecture remains open. CLAIM.md gives the precise scope and falsifier; PROOF.md supplies the argument from the matrices to the electron bound.

Run from any directory:

/path/to/repo/problems/simon-ionization-excess/compute/q14/run_all.sh

Only Python's standard library and rustc are required. The replay rebuilds all 37-bin kernel inequalities and checks two different rational witnesses. Python proves positive definiteness by exact Schur complements. Rust checks an explicit Gram matrix with outward integer intervals. Each independently verifies the discretization error and HPS constants. Exit 0 means CLAIM.md holds; missing or invalid certificates return nonzero. Earlier enumeration logs and solver status flags are not used.

certificate.json is the input. python-result.json records one exact replay; it is output, not a premise. check_rejections.py checks positive controls and deliberate corruptions, including the missing-certificate exit behavior of the shell driver.

The search used search.py, NumPy 2.5.3, CVXPY 1.9.2 and Clarabel 0.11.1, with one BLAS thread. Its optimal_inaccurate result supplied candidate entries only. The rational checks determine validity. To repeat that optional search in an environment with those packages:

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 search.py
./run_all.sh

The target remains $0.912$ on 37 bins. The edges are rational approximations to a geometric grid, and the verifiers treat the stored rational edges as the exact partition. The certified global floor is $\beta_3\ge0.9087$ .