A practical guide to well roundedness

The paper’s Proposition 4.2 proves that an f-roundomorphism r:G→H pulls back an LWR family {B_T} in H to an LWR family {r^{-1}(B_T)} in G, via the two inclusions (r^{-1}(B_T))(+ε) ⊆ r^{-1}(B_T(+Fε)) and r^{-1}(B_T(−Fε)) ⊆ (r^{-1}(B_T))(−ε), and then the LWR inequality on H; the proof yields μ_G((r^{-1}(B_T))(+ε)) ≤ (1+F C_0 ε) μ_G((r^{-1}(B_T))(−ε)) for small ε, with parameters recorded as (T_0, F·max{C_0,1}) in the statement . The candidate solution reproduces the same outer and inner inclusions, the same measure inequalities, and the same final bound, differing only in two minor presentational points: (i) it briefly and unnecessarily “shrinks to symmetric neighborhoods,” which is not required in the paper’s argument and, as stated, does not by itself imply the inequality for the original neighborhoods; (ii) it records the Lipschitz constant as F·C_0 (a sharper constant) while choosing ε small enough to ensure both the local Lipschitz regime and the LWR regime, whereas the paper records a slightly larger constant F·max{C_0,1} to tie smallness to 1/C in one stroke. Aside from these cosmetic differences, the logical steps and proof structure match the paper’s proof .