Existence of the carrying simplex for a C1 retrotone map

The paper proves, under AS1–AS4, the existence of a weak carrying simplex Σ (and, under AS3′, an unordered carrying simplex) for Kolmogorov maps F(x)=diag[id] f(x), by: (i) selecting a bounded rectangle Λ via continuity of the spectral radius, (ii) showing F|Λ is a homeomorphism using the Gale–Nikaidô P-matrix criterion and the Kolmogorov Jacobian factorization DF=diag[f](I−Z), together with (DF)^{-1}≥0 from a Neumann series, (iii) deducing weak retrotonicity (or full retrotonicity under AS3′), (iv) identifying the compact attractor of bounded sets Γ, and (v) constructing Σ via a graph-transform in radial coordinates with Kuratowski/Harnack metrics, then establishing properties (I)–(IX). These steps are explicitly carried out in Theorem 2.1 and Lemma/Proposition chain including Lemma 2.5 and Proposition 2.6, the construction in §2.4, and the attractor results in §§2.3, 2.5–2.7 . The candidate solution verifies the same structural points: Gale–Nikaidô global univalence on Λ via P-matrix DF, positivity of (DF)^{-1} giving weak retrotonicity, and a graph-transform construction producing an invariant weakly unordered hypersurface homeomorphic to the probability simplex. Its outline matches the paper’s approach in substance. Minor issues: it incorrectly labels the radial inequality F(λx) ≥ λF(x) as “strict sublinearity,” and it cites a 2021 arXiv preprint as a “pre-2021” existence theorem. Nonetheless, the candidate’s reasoning tracks the paper’s main proof structure and conclusions.