Four Null Spacetime Theory I

Introduction General relativity begins with a metric and derives the light cone from it. Four Null Spacetime Theory reverses that order. It begins with four light rays, lightlike or null vectors, the directions along which massless signals travel, all making the same angle with one another, and it asks how much of physics follows from them. Each pair of the four rays spans a bivector, so every cell has six bivector channels, and these six channels are the foundation on which the whole theory is built. Spacetime becomes a fabric of finite cells, each spanned by one such set of rays. Why begin from four light rays. Theories of spacetime usually take a metric as given and then add, one at a time, the structures physics needs: a time direction for observers, a split of fields into electric and magnetic parts, spinors for matter, and, in discrete approaches, a lattice whose cell shapes are chosen and whose curvature is a continuous variable. Four Null takes the opposite route and asks for the smallest causal object from which all of these can be read. Its answer is rigid. Four lightlike rays with equal mutual products leave only one free number, the size of the cell; the metric, a clock, the electric and magnetic split, the spinors and the operator of matter then follow from the same four rays, with nothing added. The rigidity continues between cells: a shared face fixes how one cell joins the next up to a single mirror choice, identical cells close flat space exactly, and curvature comes in fixed quanta that can be counted. The result is a description in which causal structure is primary, the number of free choices is small, and every structure physics needs is carried by one object. This is what Four Null offers in comparison with descriptions that begin from a metric or from a lattice of chosen shapes: not a replacement of general relativity, which it recovers at large scales, but a reason for its structures, read from the light rays themselves. A structure that survives its tests. The series states every result with its status: exact when proved, verified when reproduced numerically with convergence shown, conditional when it rests on a named premise, and open when no claim is made. Exact results are machine checked, with the scripts archived, and several have been reproduced in independent audits. Where the structure meets a limit, the series says so rather than hiding it: the native time update of the cells is unstable, an exactly Lorentz invariant finite lattice is impossible, and the equality of the speeds of light and gravity at the quantum level rests on a stated premise. The kinematic and geometric core has survived every test applied to it. The series of six papers, with the kinematics preprint and the companion paper, establishes the first fourteen of the following points, each stated there with its proof and its status; the fifteenth records what remains open. Light comes first. Four rays in perfect balance fix the light cone completely, up to an overall size, and rebuild the metric from their mutual products. Four null conditions alone leave five free parameters of the cone; equality of the six products removes all five (Paper I; Exact Kinematics). Time is derived. The four rays define one unique clock, the relational clock, which does not depend on how the rays are scaled, transforms covariantly under Lorentz transformations, and in whose frame every ray takes the same time to cross the cell (Exact Kinematics; Paper I). One cell, four levels: 1D, 2D, 3D and 4D. The same four rays build compatible structures at every dimension: in 1D, the same clock time along each ray; in 2D, a screen across each ray; in 3D, ordinary space, rebuilt from the four directions; and in 4D, the volume of the cell (Exact Kinematics; Paper I). The six bivector channels split into electric and magnetic sectors. The tetrahedron splits the six bivector channels into two sectors of three, and the Hodge star exchanges them: this is the split of every field strength into its electric and magnetic parts, read directly from the cell. The exchange of the sectors alone allows a family of shapes; the full correspondence between the Hodge star and the incidence of the tetrahedron holds exactly when the tetrahedron is regular, so requiring that correspondence forces the regular shape. The clock assigns equal times to the four rays exactly when the exchange holds (Exact Kinematics; Paper I). Gravity, light and matter share one cell. The cell carries the metric and its curvature; the Maxwell field lives on its six bivector channels, with the balance of electric and magnetic parts fixed by the cell itself; each ray is the square of a spinor, the four spinors form the tetrahedral informationally complete measurement of a qubit, and the four rays build the Dirac operator of matter (Papers I, IV). Cells join rigidly. A shared face fixes the transport from one cell to the next, an isometry that carries vectors, bivectors and the clock, and for balanced cells it fixes the neighbor up to one mirror choice. An exactly balanced cell has no local shape mode and a fixed balanced complex has no continuous link mode, so building spacetime is a discrete choice, not a continuous adjustment (Paper II). Curvature can be counted. Every hinge has corner angles of 60° and 120°, so cells close flat space exactly, curvature appears only where cells meet, in steps of sixty degrees, and the gravitational action of Regge becomes a count of cells and of hinges, the two dimensional faces around which cells meet. Composed face transports reproduce each deficit exactly (Paper II). Closed universes and their spin structures. Five lightlike vectors with equal products exist in five dimensions, and the boundary of their parallelotope is a closed curved complex of ten balanced cells; a closed balanced complex with the topology of S³ × S¹ also exists. On it the holonomy equals the deficit at every hinge, spin structures are the solutions of a ℤ₂ cochain equation, and there are exactly two (Paper II). Causality chooses the wave law. Among reversible laws whose reach in one tick stays inside the light cone, stability at every wavelength and propagation at the full speed of that reach single out Kirchhoff’s law on the sphere one cell away, with a healthy spectrum and the same form at every scale. Light and gravity then travel on the same cone, provided the law contains no preferred clock (Paper III). Einstein’s gravity returns at large scales. The lattice of cells carries the Fierz–Pauli graviton, its first nonlinear interaction is exactly Einstein’s, and integrability fixes the Einstein coupling; on a light front only the shear propagates and the constraints are preserved along rays. The static finite cell corrections are contact terms that leave the exterior potential unchanged at their order (Paper III). Fields obey their laws on the fabric. A field of form degree p lives on the faces of dimension p, and the boundary of a boundary gives gauge invariance and the Bianchi identity. Maxwell’s field propagated by the Huygens law keeps Gauss’s law; matter couples to curvature by one quarter of the scalar curvature; a unitary fermion step has a single species; and quantum electrodynamics on the cell has the Ward identity and g equal to two (Paper IV). Lorentz invariance: its protection and its boundary. On the regulator lattice of the 24-cell, symmetry forbids an electric and magnetic split of the photon, and averaging over the 24 clock directions removes the preferred frame and cancels the curvature squared terms exactly at the value selected by the wave law. No operator of finite range can depend exactly on the Lorentz quadratic form, so exact Lorentz invariance of a finite lattice is impossible, and a rigid clock lattice has been reported to split the electric and magnetic normalizations; the series states this boundary rather than hiding it (Paper IV). The quantum cell reads curvature. The relational clock selects the group SU(2), and a quantized ray is null as an operator. Quantized, each corner of a cell is four spin one half legs with two states of opposite chirality, the eigenstates of its oriented volume, and a full cell carries four past and four future legs. Carried around a hinge whose curvature is k steps of sixty degrees, the chirality of the cell is multiplied exactly by (−1)ᵏ: the quantum cell records the parity of the curvature, consistently across the whole spacetime, and glued cells form a spin network with one bit of entanglement per cut face (Paper VI; Kinematic Quantum Transport). Horizons can be counted, and vacuum energy drops out. Every null plane of the lattice carries lattice null lines of one universal density, one line per screen area of 4L꜀²/√3, with L꜀ the size of the cell, so horizons can be counted by the cells that cross them; with one bit per crossing this fixes the cell size relative to the Planck length, conditionally. With a fixed volume per cell, the energy of the vacuum drops out of gravity, and one cell measures the null energy of a perfect fluid (Papers I, VI). What remains open. A stable time evolution native to the cells, which Paper V tests and finds that the update following from the gluing grows while evolution along null rays turns scale into a constraint; a quantum dynamics of gravity, including why light and gravity share one cone at the quantum level, the proviso in point 9; whether the counting of cells produces smooth spacetime on large scales; and the value of the cell size independently of horizons (Papers V, VI). What this offers is one structure that carries geometry, time, curvature, light, gravity and matter at once, with a light cone fixed by symmetry rather than by tuning, curvature that can be counted, and a home that every field and its quantum states can share. How to read the series. Paper I and the kinematics preprint establish the single cell; Paper II joins cells and counts their curvatu