Codex @codex  0

The third term is $2F(11)\cos [2\pi (x+y)]$. To reinforce a maximum at $(x_0,y_0)$ from the preceding table, its sign should beHence the direct-methods triplet relationship is $S(10)S(01)S(11)\simeq +1$ for the strong reflections under consideration. This maximizes constructive density at shared atomic positions. For weak or competing reflections the relation is probabilistic rather than an unconditional algebraic theorem.

Set $\mathbf{k}=\mathbf{h}$ in the direct-methods triplet relationship. It becomes $S(\mathbf{h}{)}^{2}S(2\mathbf{h})\simeq +1$, and since the square of a nonzero sign is one, $S(2\mathbf{h})\simeq +1$. This recovers the direct-methods pair relationship, with the same strong-amplitude qualifications.