Box-space sprinklings, sequential growth and the Lorentz cone

Abstract

Preprint, not peer reviewed. This record contains the article, the Python scripts and logs that produce every number and table in it (including the interval-arithmetic certificates), and the LaTeX source. Plain-language summary Causal set theory models spacetime as a discrete set of events ordered by cause and effect. The standard way to obtain such a set from a spacetime is to scatter points at random and record which pairs can influence each other. Most possible orders look nothing like spacetime, so the theory needs a principle that picks out the spacetime-like ones. This paper starts from a symmetry of the scattered points in flat spacetime. If one point lies below all the others, the rest behave exactly like a smaller copy of the whole sample. This property was known to mathematicians under the name box-space. We use it as a test. Growth models in which each new event looks only at its own past fail the test already with three events. The test does not single out ordinary spacetime by itself: any order defined by a homogeneous convex cone passes it. Among the most symmetric of these cones, a simple counting rule picks out the light cone of relativity. The rule compares the number of events with nothing to their future with the number of related pairs. The light cone is the only cone for which the two counts report the same dimension, with the boundary events forming a two-dimensional-lower "area". The rule does not explain why spacetime has four dimensions. Why it matters A dynamics for causal sets has to explain why spacetime-like orders are selected. The results turn this question into concrete tests that can be computed from the order alone. They show which local growth rules cannot produce flat spacetime, already with three events. They also show that the light cone can be recognised among a large class of alternatives by counting. The main theorem is proved for all symmetric cones; part of it is computer-assisted, with two independent interval-arithmetic certificates included in this record. Contents of this record Kaneko_2026_box_space_sprinklings_Lorentz_cone.pdf: the article (CC BY 4.0). paper_code.zip: Python scripts (MIT License) and logs. The README maps each theorem, proposition, table and numerical statement to its script, log and running time. The scripts include the exact rational computations, the two interval-arithmetic certificates (mpmath.iv and Arb) and the Monte Carlo runs, with their random seeds. latex_source.zip: LaTeX source.