dlp_four_corner_sigma_average_eq_copy_sum_k2

Concrete $T_{n,2}$ σ-average identity (de la Peña–Montgomery-Smith, Ann. Probab. 23 (1995), 806–816; arXiv:math/9309211, §4). At the two distinct indices $i_1\neq i_2$ the σ-block is the four sign configurations $(\sigma_1,\sigma_2)\in\{\pm1\}^2$, each with uniform weight $\tfrac14$ (modelled by dlpSignVal : Fin 2 → ℝ, $0\mapsto+1$, $1\mapsto-1$). Averaging the eq-(4) four-corner σ-randomized term $4\,f(Z^{(l_1)},Z^{(l_2)})$ over this block recovers the plain four-corner copy-sum $\sum_{j_1,j_2} f(X^{(j_1)},X^{(j_2)})$ — the inner summand of $T_{n,2}$. This is the pointwise (per index pair) form of $T_{n,2}=2^2\sum_{i_1\neq i_2}\mathbb E(f(Z^{(l_1)},Z^{(l_2)})\mid G_2)$: the σ-cross-terms cancel because each $\sigma$ is symmetric mean-zero, so the uniform σ-average of the four sign products $(1+s_1\sigma_1)(1+s_2\sigma_2)$ equals $1$. Combined with the σ-conditional-expectation substrate (dlp_sigma_randomization_condexp_eq_sigma_integral) — which identifies $\mathbb E(g(\sigma)\mid G_2)$ with the σ-average $\int g\,d\nu$ — this yields the conditional-expectation form of $T_{n,2}$.