[DUPLICATE RECORD - DO NOT CITE] Trees Cannot Cut: Circuit Rank as the Design Variable Reconciling Habitat Connectivity and Fuel Discontinuity in Managed Forest Landscapes

Revised preprint v1.0.1, supported by machine-verified Lean 4. Landscape fuel management and habitat-connectivity planning are usually treated as competing claims on the same land. This paper argues that an important part of the tension is topological and is governed by circuit structure. 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 rank H1(C ∪ ∂Ω). For a corridor strictly interior to the landscape this is μ(C) + 1, where μ is circuit rank. For a boundary-anchored corridor the governing rank is the rank of the corridor together with the landscape boundary. (1) An interior tree cannot cut. An acyclic corridor in the landscape interior leaves the fuel matrix in one connected component, no matter how much area it occupies. (2) Compartmentalisation is quantised. For an interior corridor, each additional independent cycle adds one fuel compartment. (3) Constant-factor scaling, under explicit scope. In an idealised square-domain, zero-width network model that bounds the area of every complementary component, containment and terminal connectivity have the same asymptotic order of network length. (4) Disjoint support. Under binary terminal-connectivity criticality, bridge edges carry connectivity while non-bridge edges carry topological cut capacity. (5) Boundary anchoring. A tree meeting a genuine non-fuel boundary in b disjoint arcs partitions the fuel domain into b components while the corridor itself remains acyclic. Numerical support and limits. Seeded 140×140-cell lattice experiments support the zero-cut, bridge, cycle-augmentation, and boundary-anchoring mechanisms. These are synthetic experiments, not field-fire validation. The fire model omits real fuel classes, slope, fuel moisture, crown-fire transition, and spotting; wind results are confounded by the bounded domain. The paper reports unbalanced panels, unpaired ignitions, the abstract-graph versus raster-circuit-rank gap, and a counterexample showing that identical fuel-graph topology can still yield different simulated burned area. Verification status. Twenty-one named combinatorial results are machine-verified in Lean 4 against Mathlib with zero sorry and a clean per-theorem axiom audit within {propext, Classical.choice, Quot.sound}. The standing interface is exhibited as inhabited, so the downstream results are not vacuous. Two limits remain explicit: the geometric Alexander-duality step is carried as a hypothesis rather than formalised in Mathlib, and Proposition 3 is analytic rather than part of the Lean development. Lean certifies the stated combinatorial reasoning, not empirical magnitudes or field effectiveness. Version 1.0.1 correction. This revision fixes the maximum-compartment condition and perimeter lower bound in Proposition 3, distinguishes interior circuit rank from boundary-anchored rank in the abstract, narrows solver claims to realised acyclic geometry, replaces unfinished DOI/reference/acknowledgement placeholders, and adds explicit AI-use attribution. Justin D. Hart is the sole author and accountable human creator. Aristotle (Harmonic) and Claude are credited in the verification and editorial provenance, not as authors. Part of the Viridis Canon; derived from the Intelligence Bound spine (10.5281/zenodo.19317982) and included in Series S5 - Corridors & Spatial (10.5281/zenodo.20777068). Correspondence: [email protected]