Causal sets that locate and grow themselves Sub-discreteness self-localization and order-intrinsic sequential growth of 3 + 1 Minkowski orders

Abstract In causal set theory, a Poisson sprinkling of Minkowski spacetime is defined relative to the sampled spacetime, yet no known growth dynamics generate such orders beyond two dimensions. We perform a numerical analysis to determine the geometric information contained within a 3+1 sprinkled causal set and assess whether this data is sufficient to drive order growth autonomously. Interval counts establish the squared proper time of related pairs at approximately 1.4 ℓ² for all separations, where ℓ = ρ⁻¹/⁴. By fitting each element to its links, we achieve a positional accuracy of approximately 11 ℓ/K, where K represents the links per element—a scaling consistent with the information bound derived from the count law. Jointly fitting all causal relations improves this precision by a factor of two overall, and by four to seven in the bulk, subject to a conformal map; interval counts constrain this map to the precision of their inherent Poisson fluctuations. Starting from an initial embedding with an accuracy of approximately ℓ, and employing positions derived in this manner, a sequential growth process—relying solely on the order and defining each new element’s relations at inception—generates orders whose ten test statistics remain within approximately 2.5 standard deviations of the reference sprinkling ensemble. This alignment is maintained across five independent realizations and two orders excluded from the rule’s design. Utilizing positions derived strictly from interval counts, the same rule yields a link deficit of a few percent, which we attribute to the precision limits of the most recent elements. The exact sequential law acts as the posterior predictive of the Poisson process given the order, requiring no refit; furthermore, growth based on positions from a configuration consistent with all relations, calculated once, demonstrates comparable alignment with the sprinklings. This growth represents an order-intrinsic algorithm rather than a local dynamical law. A candidate local rule, which interprets a new element's relations from a neighborhood and its reaching light cones, achieves a link count within 5–7% at a range of 4.5 ℓ. With a perfect frame, it correctly identifies all neighborhood relations, though, net of false links, it loses 20–25% of the links derived from distant light cones; the birth measure remains global.