The Theory of Everything Must Be Discrete: Gravity, Space and Forward Time as Causal Steps

Everything that happens must be discrete, in any universe, because a cause produces its effect as the next state, and a continuum has no next state for a cause to produce: like Zeno's runner, who must reach every halfway mark first, a continuum has no first step to take. The argument is an a priori proof in the logician's sense. Our physics agrees. Gravity and quantum theory together let a region of finite size hold only finitely many distinguishable states: the covariant entropy bound caps a region's entropy by the area of its boundary. A finite description singles out only countably many states, and continuous amplitudes, like every continuum, are analysis, a description of many steps at once. Gravity, space and time are then causal transitions. The relations they change exist as mathematics, with zero-width points for nodes, so discreteness gives nothing a size; what it supplies is a next step. Each transition is a rule applied to rules, and nothing outside the mathematics applies it: time is that application, and space is how far it spreads at the base rate, the speed of light. General relativity's smooth spacetime is their large-scale behavior. The result cuts off the infinities of continuum quantum gravity at their source, makes black-hole entropy a count of states, and sets aside a continuum base and, with observed Lorentz symmetry, a regular lattice. That leaves discreteness without a grid, as in causal sets, whose random count estimated the cosmological constant before it was measured. A theory of everything is a rule for transitions, a rewriting system.