Exact Kinematics of the Four Null Frame Hodge Exchange, Incidence Rigidity and a Relational Clock

A frame of four real lightlike vectors whose spatial directions point to the corners of a regular tetrahe- dron, a symmetric null frame, is known to reconstruct the spacetime metric from its mutual products. This paper establishes the further exact structure such a frame carries. Its six pairs span the space of two forms, and the incidence of a tetrahedron’s vertices, edges and faces splits them into a gra- dient sector and a circulation sector of three dimensions each, with no remainder. The Hodge star exchanges the two sectors. For a general frame of four lightlike vectors this exchange holds exactly when opposite pairings agree, a family of frames whose spatial directions form a disphenoid, with two shape parameters and a scale; the exchange therefore does not force regularity. The full oriented correspondence between the Hodge star and the incidence structure, J⋆ DT = ±√3 CR with the sign fixed by orientation, holds if and only if the frame is regular. Any four independent future lightlike vectors also determine a unique timelike direction, their relational clock, characterized by a balance equation and constructed as a conformal barycenter; it is Lorentz covariant and unchanged when the vectors are rescaled. The two halves of the paper meet in one equivalence: the clock assigns all four rays equal clock times exactly when the Hodge star exchanges the two sectors. Dividing each ray by its clock time, which in the clock’s frame is the known conformal centering of the four light directions, therefore gives a representative of the four rays that depends only on their directions and satisfies weak exchange, its positive multiples being the only weak exchange representatives, and in that frame the spatial directions form a disphenoid, as Penrose and Rindler showed four null directions can be arranged to do: for fixed null directions weak exchange fixes the relative scales of the vectors, while regularity, which the full Hodge and incidence correspondence detects, fixes their directional shape. Finally, the frame realizes, through one chain of exact correspondences, tetrahedral spinor rays, the qubit measurement they define, nilpotent Clifford elements and a directional decomposition of the flat Dirac operator. All results are exact; a symbolic package machine checks every identity used in the proofs. The parallelotope the frame spans divides into 24 congruent simplices, one for each ordering of the four vectors, each with four lightlike edges and dihedral angles of 30◦, 60◦, 90◦ and 120◦.