Four Null Spacetime Theory V: Characteristic Dynamics of Free Gravity: The Spin Two Carrier, Null Evolution and the Stable Characteristic Network

This is the fifth paper of a series on Four Null Spacetime Theory, in which spacetime is built from finite cells spanned by four lightlike vectors in perfect balance. The earlier papers fixed the cell, its gluing, its propagation law and its fields; this one asks how free gravity evolves in time on the cells. The original spacelike Regge recurrence has a homogeneous scalar root at −2.890054, a mode that grows by that factor in magnitude at each step while alternating in sign. The three checks discussed here do not remove it; earlier modified updates are reported to cancel it in the homogeneous sector, without establishing stability at nonzero momentum. The full update that follows natively from the cell gluing has exactly the two light cone gravitons at long wavelength, but twelve pairs of modes and growth at every momentum tested; since a canonical reformulation reproduces the covariant solutions, a cure needs different dynamics, not a different slicing. Formulation A replaces that recurrence with characteristic constraint evolution along null rays, and removes the exponential root in the tested homogeneous sector; it leaves a repeated unit root whose drift must still be controlled. The shear free optical equation realizes the same repeated root, whose drift is the expansion of a light bundle and survives the gluing of rays and faces; the cell’s own screen has a fixed area, four over root three times the cell length squared, which removes the drift in the homogeneous sector without shear, if the native evolution carries one cell’s screen into the next. The spin two carrier of a cell is its five dimensional traceless strain, which transversality reduces to the two polarizations of general relativity; the six pair products of the generators reconstruct it, with a null preserving completion given by the third moment of the tetrahedron. Shared face transport preserves the products of the frame it carries, so it generates no imbalance in that frame; whether relaxation could damp gravitational waves in a coupled evolution is open, and that such an evolution keeps gravity and imbalance apart is an assumption, not a result. As benchmarks of discretized continuum general relativity, a perfect discretization keeps the reparametrization symmetry of a single ray exact, the constraints of nonlinear colliding plane waves propagate exactly and their evolution converges at second order, and the Damour constraint is conserved and feeds Raychaudhuri at a fixed rate. In the characteristic network built from the cell’s rays, each pairing’s transfer factor has unit modulus where it is defined; the network’s frequency is anisotropic at second order in the cell length, and averaging over the six pairings of generators reduces this anisotropy but does not remove it. The results support characteristic evolution as an alternative; whether a single coupled update of native cells is stable remains open. Every result is stated with its status.