One geometric barrier unifies melting, vitrification and jamming of hard spheres in all dimensions
The Lindemann criterion that a solid loses stability once atomic vibrations reach roughly a tenth of the interparticle spacing, has remained an empirical rule for over a century. The numerical value was reproduced by mode-coupling and replica theories but never isolated as the consequence of a simple, verifiable argument. Here we show that for hard spheres in $d$ dimensions the rule follows from three exact geometric ingredients. The contact theorem fixing the coordination number from the equation of state, an isotropy identity fixing how non touching neighbors project onto an escape direction, and a first-passage argument which is derived, in which the elementary hop spans one interparticle spacing rather than one particle diameter. The resulting parameter-free master equation locates the kinetic glass transition, random close packing, the Kauzmann point, glass close packing, and equilibrium crystal melting in $d=3$--$12$, each to within a few per cent of reported independent simulation and replica-theory values, and places all five on a single barrier surface. The theory makes two predictions that are verifiable, the Lindemann constant, $\c_L(3)=0.13$ per neighbor spacing in $3$ dimensions derived from the theory, which must fall systematically with increasing dimensions. The other being in two dimensions, the current theory predicts the arrest in the volume fraction $η_g=0.781$, the jamming at $η=0.832$, and both steps of the two-stage melting scenario, all of which are already corroborated by independent simulations and experiments.