Counting Causal Orderings and Volume Time: Typical Linear Extensions of Sprinkled Causal Sets

A causal set fixes the order of its events but not a time. This work asks whether the order alone, weighted by counting, singles out a family of time slices. For a causal set obtained by Poisson sprinkling into a globally hyperbolic region, we consider the uniform measure on its linear extensions, which is the conditional law of every discrete-covariant sequential growth dynamics given the final causal set, and we study where the typical birth order places its stages. Using Markov-chain sampling with sprinklings of up to 4000 elements, we find that the typical leaves are described by volume slicing, N√h = C(x) in adapted coordinates, which in a homogeneous cosmology is unimodular (four-volume) time. In flat 1+1-dimensional strips with deformed boundaries, constant-mean-curvature (CMC), harmonic and coordinate-Laplacian slicings are rejected; in a conformally curved region, harmonic slicing is rejected and CMC is rejected at the highest density; in a causal diamond the Pittel–Romik lattice shape is rejected and the leaves lie close to the logarithmic functional; and in 2+1 dimensions, on a metric whose sprinkling density carries no information, CMC and harmonic slicing are rejected while volume slicing is accepted. In 3+1 dimensions CMC and harmonic slicing are rejected and volume slicing is by far the closest, but a residual of 10–29% of the separation from CMC remains; under an assumed systematic floor the result is marginal at the lower density and volume slicing is not rejected at the higher one. The leaves are reproduced by the profile-entropy functional S[τ] = γρ ∫ √(−g) ln q, with q = −g^{μν} ∂_μτ ∂_ντ. A power-law family σ′(q) ∝ q^p fitted to nonlinear data gives p = −1.01 ± 0.03, the decay rate of boundary memory equals the wavenumber, and harmonics generated by the nonlinearity are predicted in advance. Relabelling invariance restricts local first-derivative functionals to ln q and √q. The functional is derived from the chain rule for the entropy of linear extensions under three stated assumptions; the coefficient γ = (1 − α)/2 = 0.505 ± 0.030 from the scaling of the birth step agrees with γ = 0.500 ± 0.005 (stat) from ordering fluctuations in 1+1 dimensions, with a sensitivity of ±0.012 to the extrapolation form. The origin of the 3+1 residual and an analytic proof of concentration, formulated as a conjecture, remain open. Files: the article (PDF, 54 pages, with full derivations, statistical procedures and the main simulation programs in the appendices) and a reproducibility archive (ZIP) containing all simulation programs, analysis scripts, run command lines, random seeds and raw outputs.