The second Kahn-Kalai conjecture

Claims Theorem 1.1: for every $n\ge2$ and every finite simple graph $H$ with $h\ge1$ edges and at most $n$ vertices, $p_c(n,H)\le\min\{1,2048e^{50}p_E(n,H)(1+\log_2 h)\}$, hence $p_c(n,H)\le6144e^{50}p_E(n,H)\log_2 n$. The logarithm depends only on the number of edges of $H$, which is stronger than the conjectured $\log n$. The proof builds on Tran's spread-link cores and the Mossel-Niles-Weed-Sun-Zadik resampling, with a tree reduction replacing Tran's proposed coupling conjecture. Constants are explicit but huge; it says nothing about optimal constants or about hypergraph targets.

Claims Theorem 1.1: for every $n\ge2$ and every finite simple graph $H$ with $h\ge1$ edges and at most $n$ vertices, $p_c(n,H)\le\min\{1,2048e^{50}p_E(n,H)(1+\log_2 h)\}$, hence $p_c(n,H)\le6144e^{50}p_E(n,H)\log_2 n$. The logarithm depends only on the number of edges of $H$, which is stronger than the conjectured $\log n$. The proof builds on Tran's spread-link cores and the Mossel-Niles-Weed-Sun-Zadik resampling, with a tree reduction replacing Tran's proposed coupling conjecture. Constants are explicit but huge; it says nothing about optimal constants or about hypergraph targets.