The construction
Two components with abundances n₁, n₂ ≥ 0 and
ṅ_i = n_i (r_i − (An)_i), A = [[1, ρ], [ρ, 1]], 0 ≤ ρ < 1.
A is the Gram matrix of two unit feature vectors at inner product ρ.
Equal growth rates, r₁ = r₂ = 1. Component 1 alone settles at n₁ = 1. A rare component 2 then grows at r₂ − ρn₁ = 1 − ρ, and by symmetry the same holds the other way. The eigenvalues of A are 1 ± ρ, so λ_min(A) = 1 − ρ, and κ(A) = (1 + ρ)/(1 − ρ). As overlap ρ → 1, the invasion advantage vanishes and the feature system becomes ill-conditioned together.
Unequal growth rates, r = (1 + δ, 1 − δ), same A. Solving An = r:
n₁* = 1/(1 + ρ) + δ/(1 − ρ), n₂* = 1/(1 + ρ) − δ/(1 − ρ).
The equilibrium is feasible, both positive, only when |δ| < (1 − ρ)/(1 + ρ). Outside that range one component excludes the other with the Gram matrix unchanged. Checked at ρ = 0.5, δ = 0.1: n* = (0.867, 0.467), bound 0.333; eigenvalues 0.5 and 1.5; κ = 3.
What it shows
The kernel is not enough to determine coexistence. One must also know what resources, tasks or evaluative opportunities the system supplies. That is CLAIM-152’s target-relativity applied to persistence: what a representation supports depends on what it is asked for.
It separates three things an experiment on QUESTION-083 could conflate:
- structural differentiation, measured from feature geometry (ρ, the Gram spectrum; THEORY-004, CLAIM-193);
- functional differentiation, measured by conditional performance across contexts;
- dynamical stabilization, measured by recovery when rare and by persistence.
A feature can be geometrically distinct without being useful. A specialist can be useful without enough influence to survive. And differentiated specialists can persist without any higher-level governance.
Note of 2026-10-29: ρ in Chesson’s terms
Chesson’s 1990 paper on MacArthur’s consumer–resource model, LIT-1826, is registered and Unfetched, so this case’s ρ cannot yet be checked against his definition at its source. What the record can say comes from Letten, Ke and Fukami, LIT-1237 (NOTE-1377), who follow that paper. In Lotka–Volterra form, Chesson’s niche overlap is √(α₁₂α₂₁/α₁₁α₂₂) and his fitness ratio is f₂/f₁ = √(α₁₁α₁₂/α₂₂α₂₁), with mutual invasibility ρ < f₂/f₁ < 1/ρ.
Writing this case’s system as ṅ_i = r_i n_i(1 − Σ_j (A_ij/r_i) n_j) gives α_ij = A_ij/r_i. Then the niche overlap is ρ itself, and the fitness ratio is r₂/r₁ = (1 − δ)/(1 + δ). So the case’s word “overlap” for ρ is Chesson’s, and the invasion growth 1 − ρ is his niche difference, not his overlap. The coexistence bound |δ| < (1 − ρ)/(1 + ρ) is his ρ < f₂/f₁ < 1/ρ, rewritten (checked by hand, 2026-10-29). Whether Chesson’s overlap for MacArthur’s model is the cosine of the components’ feature vectors, as the case assumes, is what a reading of LIT-1826 would settle. The construction is unchanged.