LOCAL ENTROPY THEORY, COMBINATORICS, AND LOCAL THEORY OF BANACH SPACES

The paper proves, under amenability of Γ, the three identities relating IE_k(M(X)) and IE_k(X) (Theorem 1.1) and supplies a complete argument: convexity of IE_k(M(X)) (Lemma 3.2) and that (2)–(3) follow from (1) via Milman’s theorem and a barycenter lemma (Lemmas 3.3–3.5), with k=1 handled by a direct barycenter-support argument (Lemma 3.6) and k≥2 by new combinatorial techniques (Section 3.4). These steps, statements, and dependencies are explicit in the text . By contrast, the model crucially relies on two unsupported claims: (i) that IE_k(Y) = IE_k(Z) ∩ Y^k for every closed Γ-subsystem Y⊆Z (the paper only asserts inclusion; see Theorem 2.1(5) ), and (ii) a supposed “general barycentric theorem” (attributed to Kerr–Li) stating IE_k(K) equals the closed convex hull of IE_k(ext K) for any compact convex Γ-space K. The paper does not cite or use such a theorem and instead develops a dedicated proof for K = M(X). The model also omits the amenability hypothesis required for IE-tuples throughout. Hence the model’s proof is not valid, while the paper’s argument is correct and complete under its stated assumptions.