Dyadic product estimate for group neighbourhoods (${\sum }_j{2}^{-n_j}\le 2^{-n}⟹{\prod }_j{g}_j\in U_{n-1}$)

Let $U_0=G$, and let the $U_n$ be decreasing symmetric sets containing the identity element with $U_n^3\subset U_{n-1}$. If $g_j\in U_{n_j}$ and their dyadic costs sum to at most $2^{-n}$ for $n\ge 1$, split the ordered word at an edge crossing its half-cost point. The words on either side each have cost at most $2^{-n-1}$. Induction on word length puts both side words in $U_n$, while the middle edge also belongs to $U_n$. Their threefold product lies in $U_{n-1}$. This controls the short-chain collapse that would otherwise make a path-distance construction degenerate.