Landau–Siegel zeros over varying number fields

Let $K$ be a number field of degree $n$ and absolute discriminant $D_K$. We give a direct arithmetic proof that every real zero $\beta\in(0,1)$ of $\zeta_K$ satisfies $(1-\beta)\{\log D_K+n\log(2n)+6n+1\}\geq\frac1{1370}.$ The bound is explicit for arbitrary varying fields, including nonnormal fields, and gives $(1-\beta)\log D_K\gg1/\log(2n)$ for K\ne\Q. The proof combines OpenAI's determinant method, Masser's zero estimate as used by Kerr, and Banks' selected-embedding calculation. The resulting bound is expressed directly in the degree and discriminant of $K$, without introducing those of its normal closure. Stronger nonvanishing conclusions follow from recent half-plane results and positivity. The contribution here is an arithmetic proof with explicit field dependence, using neither those half-plane results nor earlier zero-exclusion results for real Dirichlet $L$-functions.