The Maintenance Floor on the Heavy-Tailed Branch

Abstract

Abstract Classical proofs of dissipation lower bounds rest on spectral gaps or logarithmic Sobolev inequalities, and both tools are natively suited to light-tailed steady states. In multiplicative open systems whose tail index lies in the Pareto range the spectral gap disappears, the modified logarithmic Sobolev constant can only be taken as zero, and the whole apparatus fails; worse, the Kullback-Leibler divergence to the passive baseline may itself diverge, so even the carried quantity is in question. This paper does not proceed from functional inequalities. First, the Legendre-transform cost of tilting the allocation rule, D(θ) = θ·M′(θ) − M(θ), is introduced; it has a closed form for Pareto laws, is finite over the whole range κ★ > 1, and the concentration coordinate already used in the framework is shown to be exactly its value at θ = 1. This is a cost measure native to the heavy-tailed side, but it is an information-side quantity and is not automatically equal to thermodynamic dissipation. Second, of the two routes connecting it to power, the steady-state accounting route is shown to be an identity and therefore unfalsifiable, and a way of converting it into an over-identification test is given. Third, the fluctuation-theorem route is taken: writing the maintenance turnover as a one-way cyclic current and applying the thermodynamic uncertainty relation yields the dimensionless floor σ/(N·Σ_gross) ≥ 2·A²·R(κ★), where A is the irreversibility asymmetry of the turnover and R(κ★) = (E[s])²/E[s²], equal to κ★(κ★−2)/(κ★−1)² for Pareto tails. The floor contains no conversion coefficient and is therefore non-circular; it is non-trivial exactly when κ★ > 2 and vanishes linearly as 2(κ★−2) at κ★ → 2⁺. Fourth, every lower bound depending only on the first two moments of the maintenance current is shown to vanish identically for κ★ ≤ 2 — not a proof that no bound exists, but a localization of the failure to a single moment. Three observable predictions are given, one of which requires no prior estimate of the tail index. This paper takes up the gap left by its companion and does not repeat its critique of extremal unification.