Single-Variable Connectivity in Random $k$-SAT
We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.