The Grid Carries the Dynamics - A Two-Sided Null Formulation in Which the Geometric Variables Themselves Carry the Evolution

We present a classical, two-sided characteristic organization of general relativity. The geometry of two intersecting families of null hypersurfaces and of their shared spacelike cuts is carried by the transverse metric, the two expansions, the two shears, the two inaffinities, the null normalization, and the twist of the normal bundle. The Einstein equations then fall into four classes: configuration equations, same-branch focusing and shear equations, cross-branch (mixed) equations, and normal-bundle constraints. We show that the expansions and shears are the null velocities of the transverse scale and shape, and that integrating the mixed equations over a null rectangle yields a finite cell relation that fixes the outgoing corner once the admissible data on both incoming links are given. The kinetic sector of the double-null Einstein–Hilbert action pairs the two null deformations through a cross contraction; we identify its structural role and leave a gauge-reduced canonical derivation open. In spherical symmetry the product of the two expansions reconstructs the Misner–Sharp mass, and for Schwarzschild perturbations the Regge–Wheeler and Zerilli master equations take an exact null-rectangle integral form. As a quantitative benchmark we study the even-parity mode in a finite domain with the outer condition at radius . The scaled eigenvalue converges as , where is the smallest positive root of a spherical Bessel relation and and follow from matched asymptotics; an independent high-precision integration of the Zerilli equation confirms all three coefficients. These results supply a classical consistency target for future quantum proposals; they do not establish a quantum cell.