Lemma 2 (Straddling Lemma), p. 42 — for ∅ ⊊ I ⊊ [k], ∏_{i∈I} α_i and ∏_{i∈I′} α′_i are distinct multilinear monomials in the γ's

Fix an integer $k$ and formal variables $\gamma_1,\dots,\gamma_{2k}$. Set

with index arithmetic modulo $2k$. For any subsets $I,I'\subseteq[k]$ with $\emptyset\subsetneq I\subsetneq[k]$, the products

are multilinear monomials in $\gamma_1,\dots,\gamma_{2k}$, and they are distinct.

The bundling of the scalars along a block of steps is what prevents an adversary from combining an inconsistent assignment of a block in the primal program with any part of the dummy program; this lemma is the combinatorial core of that argument.

Formalization Note. Monomials are represented by their exponent vectors in $\mathbb N^{2k}$ (finitely supported functions on $\mathrm{Fin}(2k)$), with 0-based indices: $\gamma_{2i-1},\gamma_{2i}$ for $i\in[k]$ become coordinates $2i$ and $2i+1$ for $i\in\{0,\dots,k-1\}$, and $\gamma_{2i+1}$ becomes coordinate $(2i+2)\bmod 2k$. "Multilinear" is the statement that every exponent is at most $1$. The page's random drawing of the $\gamma$'s "$\leftarrow\mathbb Z_q$" plays no role in the statement, which is about formal monomials, and is omitted.