Sharp Cluster-Size Decay and Logarithmic Phase Boundaries in Spatial Random Graphs

We prove sharp finite-cluster decay and second-largest-component laws for Pareto-weighted spatial random graphs on Poisson and lattice vertex sets. Our backbone and reserved-connection argument resolves the Jorritsma–Komjáthy–Mitsche high–high exponent gap, improving the upper exponent from 4/9 to the matching value 2/3 in a representative regime. We remove the auxiliary local-seed hypothesis throughout the strictly supercritical weighted-lattice class with a positive long-edge exponent, including retention one and dimension one. We give a complete sharp classification of all positive maximal long-edge phase boundaries strictly above surface order. The four boundary rates and matching second-largest-component scales include a fractional logarithmic correction on the high–low/high–high boundary and a squared logarithm at the triple tie. An entropy-free maximum-decomposition theorem transfers a backbone estimate to finite-box component bounds for increasing pair rules, without regular variation of the mark law, local seeds or upper kernel-growth restrictions. For Pareto marks it removes the finite-box entropy condition from the strict high–high cutoff classification; rooted estimates retain an explicit weaker compatibility condition. This extends rapid-assortativity laws to superlinear and intermediate growth. Further results give sharp soft-profile corrections and high-density surface upper bounds uniform over finite marks and correlations.