Moduli of Continuity in Metric Models and Extension of Liveability Indices.

The paper’s Theorem 3.5 and the approximation/error bound are essentially correct, but the proof implicitly assumes compactness of (D,d) via Proposition 3.3 without stating it in the theorem; this is a missing hypothesis. In addition, the text contains a K-factor mistake in the “moreover” consequences: after defining Ĩ = inf(I) + K lim dϕ(an,·), it claims both Ĩ(·) = I(a0) + dϕ(a0,·) and, in the normalized case, K Ĩ(·) = dϕ(a0,·), which are inconsistent—the correct normalized identity is Ĩ(·) = K dϕ(a0,·) when inf(I)=0. The candidate solution correctly derives the two-sided bounds and uniform error bound, gives a valid construction under separability (instead of compactness), and correctly fixes the K-factor in the “moreover” statement, while appropriately flagging the need for attainment/compactness for the single-point representation. See Theorem 3.5 and the surrounding discussion, including the compactness of standard indices used in the proof, and the inconsistent equalities after the theorem .