The Floor Does Not Vanish; It Becomes a Function of Scale

Abstract The second paper of this series proved that the maintenance floor for multiplicative heavy-tailed systems, σ/(N·Σ_gross) ≥ 2A²R(κ★), is non-trivial exactly when κ★ > 2, and localized the failure on 1 < κ★ < 2 to the divergence of the second moment. The third showed that self-referential systems actively drift into that band along a trajectory of falling tail index. This leaves a gap precisely where real systems sit: what does thermodynamics still guarantee there? The answer given here is that the floor does not vanish; it turns from a scale-free constant into a function of the cutoff. For a Pareto law truncated at b, the precision factor has the closed form R_b(κ★, b), whose behaviour splits into three bands: for κ★ > 2, R_b converges as b grows to the cutoff-free value R_∞(κ★); at κ★ = 2 it decays as 2/ln b; and for 1 < κ★ < 2 it decays as K(κ★)·b^{−(2−κ★)} with K(κ★) = κ★(2−κ★)/(κ★−1)². The identity of κ★ = 2 is thereby confirmed a second time and from a different side: it is the point at which the floor turns from scale-free into scale-dependent, the transition exponent being exactly 2 − κ★ and passing continuously through zero there. Three falsifiable predictions follow, the first of which ties three independently measurable quantities — the observed upper cutoff, the estimated tail index, and the dimensionless dissipation ratio — to a single power law. The floor is then accumulated along the self-referential trajectory of the third paper, giving the dissipation budget that thermodynamics guarantees will be paid over one runaway; the portion accruing after the crossing of κ★ = 2 is only 0.08 per cent, so that beyond that point thermodynamics guarantees essentially nothing about the remaining descent. Finally two obstructions are identified in precise form: the point at which the fractional-moment route fails is the Hölder conjugate exponent (p < 2 forces q > 2, and only the q = 2 moment of the score is bounded by entropy production), and the ultimate open conjecture is whether self-reference can alter the distributional family of the multiplier and so escape the Kesten basin. All numerical results are deterministic algebra and ODE integration, with no stochastic content.