A new framework is developed for studying phase transitions in CSPs. Motivated by phase transition problems in CSPs, we prove a more general concentration inequality that retains classical sub-Gaussian tails under a mild global linear-growth condition $|S_n| < Cn$, relaxing the bounded-increment assumption to finite exponential moments and requiring neither independence nor the martingale property. We further extend the concentration inequality to branching random walks (BRW), obtaining the first concentration inequality for BRW. As applications, we derive partial differential equations (PDEs) for the $K$-SAT and $q$-COL backbones, yielding new results, including \textbf{(a)} a resolution of the long-standing open question of where $(2+p)$-SAT transition changes from second to first order; \textbf{(b)} rigorous results for $α_d$ in $K$-SAT, which give new lower bounds on the phase transition for 3-SAT (4.0029 vs. 3.51) and 4-SAT (8.360 vs. 7.91); and \textbf{(c)} the prefactor of the 2-SAT critical window.

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