Exact Kinematics of the Four Null Frame

Exact Kinematics of the Four Null Frame: Relational Clock, Hodge Exchange, and Incidence Rigidity This work develops the exact geometric and algebraic structure carried by a frame of four future directed null vectors whose spatial directions form a regular tetrahedron. Starting from the Gram matrix of the four null generators, the spacetime metric is reconstructed directly from their mutual products. The six pairwise bivector channels are then shown to split exactly into three gradient and three circulation degrees of freedom through the incidence structure of a tetrahedron. The Lorentzian Hodge star exchanges these two sectors, and the stronger Hodge incidence correspondence is shown to be rigid: it singles out the regular tetrahedral frame within a broader family of disphenoid configurations. The paper also introduces a relational clock determined uniquely by any four independent future null directions. The clock is Lorentz covariant, invariant under independent positive rescalings of the generators, and provides a canonical normalization of the frame. Equal clock times are shown to be equivalent to weak Hodge exchange. Finally, the four null directions generate tetrahedral spinor rays, nilpotent Clifford elements, and an exact directional reconstruction of the flat Dirac operator through the inverse Gram matrix. The results are kinematic and algebraic: no microscopic dynamics or gravitational field equation is assumed. The paper provides the mathematical foundation for the broader Four Null research program, in which the four null frame is investigated as a possible elementary causal structure of spacetime.