The Fragmentation Hysteresis Theorem: Restoration Debt, a Latent Heat of Habitat Loss, and the Information Price of Putting a Landscape Back

Working paper, not peer reviewed. Results are conditional on the stated assumptions; the Lean sources have not been re-checked by our current certification pipeline.

Biodiversity offset policy assumes that habitat destroyed can be compensated by habitat restored at some multiplier. This paper studies a model of that cycle. Destruction needs no knowledge of where a landscape's connective backbone lies; restoration does, because habitat placed off the backbone adds area without connectivity. The model encodes this as a siting efficiency q in (0, 1] and uses the percolation giant-component fraction S(a) as the connectivity measure.

Within the model: (i) the restoration branch lies below the degradation branch for every q < 1, so a degrade-and-restore cycle encloses a non-negative area, read here as dissipated connectivity work (Lean: the integral of a non-negative function is non-negative); (ii) restoring a lost area ΔA takes ΔA/q, so the offset multiplier is 1/q, which follows directly from the model's definition of q (Lean checks the algebra); (iii) if siting quality drops from q₁ to q₀ when the giant component breaks up, an extra restoration cost (1/q₀ − 1/q₁)·ΔA appears at the percolation threshold; (iv) if each bit of siting information doubles q up to q₁, the overshoot halves per bit up to log₂(q₁/q₀) bits, giving a closed-form optimal survey depth. If the Intelligence Bound conjecture holds, it caps how fast those bits can be acquired; that step is conditional on the conjecture.

Reported offset multipliers average about 2.9 (Bull et al. 2017), which within the model would imply q ≈ 0.35. An earlier version quoted an average of 4.9:1 with no source; that figure is withdrawn. Numerical checks (71 of 71) compare the model with Newman-Ziff percolation and finite-size scaling. The predictions are testable against offset registries; none has been tested.

Formal status: Definition-expansion for (ii), Conditional for (i), (iii) and (iv). Interpretive status: Empirical-hypothesis.

Authorship and AI use. Justin D. Hart is the accountable author. Lean proofs were generated with the Aristotle prover (Harmonic). Tools are credited as contributors, not authors.