Chaos expansion of the smeared Benincasa-Dowker action: an exact series in the smearing parameter, and second-order correlations that are not boost-finite in four dimensions

Echeverría-Arrondo recently found that fluctuations of the local Benincasa-Dowker action grow as ΔS ∝ (ρV)3/4 in four dimensions (Zenodo 10.5281/zenodo.23930122). He supported the extension to strong smearing by arguing that the measure of pairs correlated with a given nearly null pair is finite. Using the Wiener-Itô chaos (Fock space) representation of Poisson functionals together with the Poisson-thinning reading of the smeared weights, we show that at fixed non-locality density ρ_k = ερ the variance is an exact power series in ε: Var S = ε4/d−1 Σ_n εⁿ⁻¹ Φ_n(ρ_k V), with each Φ_n independent of ε and non-negative. In two dimensions: the series is confirmed against direct sprinklings; the second-chaos kernels have closed forms; the coefficient of ρ_kV ln ρ_kV in Φ_2 is 7/3 times that of the same-pair term. In four dimensions: the second-chaos kernel of a timelike pair is boost-finite; the kernel of a spacelike pair grows linearly with the region size, because the two points share thin nearly null intervals of every rapidity in the Lorentzian complement of their separation. In a 4D diamond, ρ_kV = 400 to 25600, nested Monte Carlo gives: a timelike second-chaos ratio of 4.82 ± 0.07; a spacelike ratio rising from 0.84 to 1.55, a slope of 0.19 ± 0.06 per ln ρ_kV; a cutoff estimate for that rise that is logarithmic, with slope 0.28. The finiteness assumption therefore fails at second order. Whether the exponent 3/4 survives strong smearing depends on the first and higher chaoses, which are not computed in 4D. Code and all numbers are in the deposit.