Analytical Derivation of K_crit = 3/2 from the Spectral Geometry of the Triangular Lattice in GEP

This paper derives the critical stability threshold K_crit = 3/2 analytically from the spectral geometry of the triangular lattice in Granular Entropic Physics. The minimum eigenvalue of the normalized discrete Laplacian, evaluated at the high-symmetry K-point of the Brillouin zone, yields exactly -3/2 through elementary trigonometry. Linear stability analysis then identifies this value as the critical parameter, independent of all model parameters. The result is supported by numerical bisection simulations across three lattice geometries.