A closed-form leading-order persistence-diagram differential entropy for a smooth Gaussian process

We compute the leading-order per-area differential entropy of the sublevel-set persistence diagram of a smooth one-dimensional stationary Gaussian process. The persistence diagram is treated as a marked point process whose intensity is the Rice critical-pair density μ=(2π)⁻¹λ₄/λ₂, whose marks are paired (birth, death) heights at consecutive minimum-maximum extrema, and whose differential entropy admits the standard marked-Poisson decomposition a=μ·[1−ln μ+H(marks)]+R, H(marks)=Hb+Hd−I(b;d). Two ingredients close in published form. The pair density μ closes through the Rice extrema formula. The birth and death marginals close through the Cartwright–Longuet-Higgins (1956) density for the heights of local maxima of a Gaussian process, with single regularity parameter γ=λ₂/√(λ₀λ₄)∈(0,1]. Combining these gives the leading-order Dai–Claude (DC) closed form a₀(γ)=μ·[1−ln μ+2H_CLH(γ)], which for the worked-example Gaussian spectrum S(k)=e^(−(kℓ)²/2) at correlation length ℓ=2 evaluates to a₀=0.7724. Two corrections quantify the leading deviations from this Poisson-marked baseline. The mark mutual information I(b;d)≈0.43 captures the joint law of consecutive minimum–maximum heights (Lindgren successive-extrema theory). The positional correction R≈−0.053 captures the sub-Poisson repulsion between extrema as a two-body excess entropy in the liquid-state theory sense (Ladgham–Lachièze-Rey 2023 planar analog). The assembled leading-order value is a≈a₀−μI+R=0.660, against a direct simulation a₁=0.69 for the plug-in Poisson-positions + empirical-mark-joint estimator at finite size. We explicitly do not claim a closed form for the full a: only a₀ is closed in this sense; I and R are quantified leading corrections. We also document a UV divergence on the K side of the K–TDA edge: the per-sample differential entropy rate of a smooth Gaussian process diverges to −∞ as the grid refines, which requires that the algorithmic-complexity probe K be defined at a fixed reference resolution. A separate appendix records why the analogous two-dimensional massive Gaussian free field, which had been considered in an earlier draft, has no continuum persistence diagram: as the UV cutoff Λ→∞, the field variance diverges logarithmically and the maximum density diverges quadratically, so the lattice "constants" of the two-dimensional version are not universals.