A closed form for the past light-cone four-volume in power-law cosmology, with an application to volume-based Everpresent Lambda

I derive a closed form for the past light-cone four-volume V(a) in an arbitrary spatially flat power-law FLRW background, V(a) = (4π/3) a^(4p) I(p) / [H₀⁴ (p−1)³] with p = 3(1+w)/2 and an elementary expression for I(p), together with its limits at the acceleration threshold p = 1 and in de Sitter space. Since the expected number of causal set elements in a region is the sprinkling density times its four-volume, the same formula gives that count in closed form. I apply the result to the causal-set-inspired Everpresent Lambda scenario, in which the dark-energy density scales as 1/sqrt(V). For a power-law background with constant w and vanishing initial action, the smooth branch obeys w_eff = w exactly, independently of w; three independent derivations agree to a few parts in 10^12. The acceleration criterion dlnV/dlna < 4 is therefore self-referential — satisfied only if the background is already accelerating — and on a LambdaCDM background the tracker lags by 0.40 Hubble times, the residual after a sharp transition decaying as a^(−6(1+w2)). The theorem does not cover branches carrying a term rho ∝ 1/V, where acceleration is possible and has been reported. Note on priority: that this component tracks the ambient density is not new. It is stated in the original proposal by Ahmed, Dodelson, Greene and Sorkin (2004) and confirmed by Das, Nasiri and Yazdi (2023). What is contributed here is the closed form, the exact statement valid for every constant w and through the acceleration threshold, and the explicit no-go consequence. This record contains the English version (primary) and a German translation of the same manuscript, together with the LaTeX sources.