Paper supported by machine-verified Lean 4. Landscape fuel management and habitat-connectivity planning are usually treated as competing claims on the same land. We show the tension between them is not primarily an optimisation trade-off but a topological one, governed by a single integer. Represent a conserved corridor network as a compact set C in a landscape Ω, and the untreated fuel matrix as Ω \ C. Habitat connectivity is a property of C; fuel discontinuity is a property of its complement. By Alexander duality the number of connected components of Ω \ C is exactly rank H1 (C ∪ ∂Ω) — for a corridor strictly interior to the landscape, μ (C) + 1, where μ is the circuit rank. Five consequences follow. (1) An interior tree cannot cut. Any corridor that is a spanning tree in the landscape interior — including the minimum-spanning-tree and space-colonisation dendrites now standard in corridor design — has μ = 0 and therefore leaves the fuel matrix in exactly one connected component, no matter how much area it occupies. Every hectare of such a corridor is, from the standpoint of fuel continuity, spent on nothing. (2) Compartment count is exactly μ + 1: containment is bought in integer units of circuit rank, not in hectares. (3) Achieving n compartments and spanning m terminals cost the same order of corridor length, so a dual-function network is a constant-factor surcharge, not a second land budget. (4) Disjoint support. Connectivity criticality and fuel-cut capacity are carried by disjoint edge sets — removing a bridge provably cannot merge fuel compartments — so the two objectives never compete for the same edge. (5) Boundary anchoring. A tree meeting the landscape boundary in b arcs partitions the landscape into exactly b compartments while its circuit rank stays zero; cycles closed through the boundary or through any existing non-fuel feature cost a fraction of interior cycles. Numerical support. Lattice experiments (140×140 cells, 30 m grain, up to 12 landscapes per test) support four of the five; Proposition 3 is asymptotic and a single lattice size cannot test it. For interior tree corridors the largest fuel component equals 1 − φ to machine precision in 12 of 12 landscapes. Holding conserved fraction at φ ≈ 0. 25 and connectivity at 18/18 terminals, raising μ from 0 to 34 cuts the largest fuel component from 0. 74 to 0. 40 and mean simulated burned area from 0. 39 to 0. 18 — a 54% reduction bought with no additional land and no loss of connectivity. Across 316 corridor edges, all 89 bridges had area-controlled cut capacity of exactly zero. Boundary anchoring dominated interior augmentation: six anchors on an otherwise unchanged μ = 0 dendrite took the largest fuel component from 0. 74 to 0. 22 for a conserved-fraction increase of 0. 26 to 0. 29. The practical implication inverts current practice. Existing joint approaches treat the habitat corridor as an exclusion zone to be routed around; we show the corridor should be routed as the fuel break, and that doing so requires closing cycles in a network that connectivity optimisation alone would leave as a tree — cheapest first through the landscape boundary and existing non-fuel inclusions, and only then through interior links. Verification status. All twenty-one named results of the combinatorial core are machine-verified in Lean 4 against Mathlib, with zero sorry and a clean per-theorem axiom audit (propext, Classical. choice, Quot. sound). The standing hypothesis is exhibited as inhabited, so no result is vacuously true. Two things are assumed and stated as such: the topological step (Alexander duality) is carried as an explicit hypothesis and is not proved — the exhibited witness is the tautological model and establishes non-vacuity only, not that the geometric compartment count satisfies the relation; and Proposition 3 is not formalised. The deposit includes the complete Lean development, the axiom-audit log, the simulation code and seeds, and the raw output of all three Aristotle runs — including the two that failed — so the provenance is checkable rather than asserted. Theorems co-authored with Aristotle (Harmonic). Part of the Viridis Canon; derives from the Intelligence Bound spine (10. 5281/zenodo. 19317982) and belongs to Series S5 — Corridors & Spatial (10. 5281/zenodo. 20777068). Correspondence: Justin@viridisconservation. com
Hart et al. (Wed,) studied this question.