Linear Hadwiger conjecture: graphs with no K_t minor are Ct-colourable

The preprint claims an absolute constant $C$ such that $\chi(G)\le C\,h(G)$ for every graph $G$, where $h(G)$ is the Hadwiger number, improving $O(t\log\log t)$ (Delcourt–Postle) and $O(t\log\log\log t)$ (Liu–Luo, preprint). The proof (i) shows $\chi(G)\le(4+\varepsilon)t$ for $K_t$-minor-free graphs on at most $t(\log t)^{1/2-\varepsilon}$ vertices, using the Reed–Seymour fractional bound, LP duality with pair-coverage constraints and a quantitative Pippenger–Frankl–Rödl matching theorem; (ii) bootstraps the admissible order exponent from $\eta$ to $4\eta/3$ at the cost of the constant; and (iii) applies the Delcourt–Postle reduction to graphs of order $t(\log t)^5$. The authors estimate $C\approx10^{100}$. It does not resolve the host problem: the $t=7$ case (6-colourability of $K_7$-minor-free graphs) is untouched.