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⋆ D^T = ±√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◦.

This is the third 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 first two papers established the geometry of one cell and how cells meet; this one asks how radiation and gravity act on them. On a light front gravity separates by spin into the Raychaudhuri and Damour constraints and two radiative polarizations, and the cell preserves the separation: its scalar sector has only a double root, its vector sector a positive dressing, and its finite step corrections have unique covariant forms. Symmetry and locality alone do not fix the finite cell law, as explicit counterexamples show; causality does, given one premise, that the law’s principal part depends on the reconstructed metric alone and not on the cell’s clock; observation bounds the speed difference such a coupling would produce at long wavelengths to about one part in 10¹⁵, but does not establish that the coupling vanishes. For a reversible two step law whose kernel stays within its causal reach and which is stable at every momentum, Bernstein’s inequality bounds every propagation speed by the reach, and equality forces Kirchhoff’s law on the sphere one cell away, with quartic coefficient one twelfth; the shell family that earlier work left open reaches beyond the light cone and is excluded, and scale invariance under blocking independently selects the same law. On the Regge lattice of the cell the Fierz Pauli graviton appears with the correct normalization. Requiring the propagation law on curved backgrounds to come from one action fixes the Einstein coupling to one and, through order of the cell size squared in the transverse traceless sector, a unique completion whose coefficients are exact rationals, with the extrinsic coefficient two thirds, all certified in exact a rithmetic. The constraints remain first class, their lattice forms are preserved to fourth order along rays, and the ray network radiates by the quadrupole formula. Every result is stated with its status, and every verified result is reproduced by the archived package.

Add the senteces in crete the text for me to copy Four Null Spacetime Theory is a candidate microscopic framework for spacetime in which finite cells are spanned by four balanced lightlike generators. Building on the geometry of a single cell established in Part I, this paper derives how neighboring cells join and how curvature is encoded in their gluing. An oriented shared face uniquely determines the Lorentz transport and fixes the neighboring balanced frame up to reflection. Across the reflected branch, the face transport remains a metric isometry, while the neighboring cell’s own relational clock differs from the transported clock. Primitive hinge deficits occur in integer multiples of (\pi/3). Under exact balance a cell has no local shape freedom, its tangent space being only Lorentz transformations and scale, and the Regge action reduces to a count of hinges and cells. For the admissible hinges of two to six cells examined here, composing the face transports reproduces the deficit rotation. Under the minimum-angle convention, reflection gluings admit two equally minimal spin lifts. Global spin consistency is formulated as a (\mathbb Z_2) cochain equation. We construct closed balanced complexes with the topologies of (S^4) and (S^3\times S^1). For the latter, conjectured counting formulas give an asymptotic averaged curvature-to-volume ratio within 2.9% of the corresponding value for a round static universe. This comparison concerns one averaged scalar. The results establish the geometry of balanced gluing; smooth Lorentzian convergence and a dynamical rule that selects gluings remain open.

What is the minimum structure needed to describe spacetime? In general relativity spacetime is described by a metric, and light reveals its most fundamental property, the causal boundary between events that can influence one another and events that cannot. Four Null spacetime theory reverses this approach: rather than positing a smooth metric manifold through which light moves, it takes balanced null connections as the structural primitives, from which spacetime geometry, clocks, field strengths and the spinors of matter are reconstructed. It proposes that spacetime is built from microscopic cells, each spanned by four balanced lightlike vectors, the null generators, whose spatial directions point to the corners of a regular tetrahedron. The four generators and the six bivectors formed by their pairs, \(q_a\wedge q_b\), are the foundational structure of the theory. The generators carry the geometry: the metric is reconstructed exactly from their mutual products, and each cell has a unique relational clock. The six bivectors are the channels in which field strengths and curvature live: they split exactly three plus three into electric and magnetic sectors, exchanged by the Hodge star, just as the six edges of the tetrahedron join its four vertices. With the four triples and the four volume, a cell carries the complete graded structure \(1,4,6,4,1\) of spacetime forms, and the same generators carry the spinors and the Dirac operator of spin \(\tfrac{1}{2}\) matter. The dynamics rests on this kinematic core. On null fronts the scalar and vector parts of gravity act as constraints, the Raychaudhuri and Damour equations, while its two shear modes propagate. Under the composition postulate a single Huygens law is selected as the unique fixed point of free propagation: Kirchhoff's exact formula over one cell tick, with free waves on the light cone and one finite cell coefficient, \(a_4=1/12\). The Maxwell and Dirac equations, each with its own gauge or Clifford structure, propagate their physical solutions on the same shell, with \(g=2\) as a consistency result. Under the integrability postulate, the response of this law to curvature gives the Einstein coupling \(\kappa=1\) and, within the stated transverse traceless basis through order \(\mathcal{L}_c^2\), a unique finite cell completion with exact rational coefficients. On the F4 lattice, used as a Euclidean quantum regulator, the quadratic kinetic terms studied are protected against Lorentz violation, a chiral fermion without doublers exists, the photon is transverse and kept massless by an exact Ward identity, and two healthy low energy graviton modes are found; a candidate improvement \(c_{\rm imp}=1/18\) removes their order \(\mathcal{L}_c^2\) birefringence but loses positivity at the cutoff. For massless fields carried by the selected law, and provided the cone is protected quantum mechanically, the theory predicts no energy dependent speed of light or gravity at order \((k\mathcal{L}_c)^2\), and horizon entropy may be counted in cells. For neighboring cells sharing a face the transport is fixed by geometry, and exact balance makes their gluing discrete, with primitive hinge deficits in units of \(\pi/3\). The principal open problems are the rule selecting and evolving the discrete face identifications, causal adjacency and topology of the balanced cell complex, and whether this dynamics fixes the last coefficient of the gravitational interaction beyond the transverse traceless sector; nonlinear constraint preservation in 3D; a physical real time realization of the cells that keeps the regulator's Lorentz protection; the decoupling of cutoff states, unitarity and interacting quantum gravity; the dynamical origin of balance; and the size of the cell.

This is the fourth 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 first three papers built the cell, the gluing of cells, and the propagation of radiation and gravity; this one places the other fields on the same cells. Fields of form degree p live on the faces of dimension p, so potentials sit on links, field strengths on the six pair bivectors and currents on three dimensional faces, and the vanishing of the boundary of a boundary gives gauge invariance and the Bianchi identity. The scalar and the photon, each given its own most general principal structure, independently select the same light cone, and the Maxwell ratio of electric to magnetic channels is exactly the ratio of the cell’s own channel metric. Matter is native: the Dirac operator is built from the generators, and a massive law follows from one Clifford structure with exact blocking, while an older additive rule is excluded by its exact blocking defect. On the fabric, quantum fields behave as a sound regulator requires: transport preserves entanglement, the nonuniversal area term carries the lattice fingerprint while the universal coefficients lie near the continuum, and every reconnection mesh carries the Einstein kernel. But a fermion kept exact up to a sharp band edge has a nonlocal vacuum, so the realization of the exact law near the cell scale must be local and analytic. Lorentz protection is established for a Euclidean regulator with the symmetry of the F4 lattice, not for a physical crystal of cells, which no finite range operator can make exactly Lorentz invariant. Every result is stated with its status; results computed in the program but not reproducible here are reported as possibilities, never as proved.