Finite indexed geometric site sets and duplicate handling

Throughout this page, $\mathbb{R}$ is a commutative field with $1>0$, a total order compatible with addition and multiplication, and the least-upper-bound property. Let $\mathbb{D}=\{m{2}^e:m,e\in \mathbb{Z}\}$ be its dyadic subring. For $d\in \{2,3\}$, a nonempty finite indexed site list is a finite list $S=(({\iota }_1,p_1,w_1),\dots ,({\iota }_N,p_N,w_N)),$ where every ${\iota }_i\in \mathbb{Z}$ is a distinct stable identifier, every $p_i\in {\mathbb{D}}^d$, and each optional weight $w_i$ is either absent (meaning $0$) or belongs to $\mathbb{D}$. In unweighted mode, a duplicate means equal coordinates; collapse each such class to its least identifier. In weighted mode, a duplicate means an equal coordinate-weight pair; collapse only equal pairs to the least identifier. Equal coordinates with unequal weights remain distinct weighted sites.

The unweighted deduplicated set keeps, for each coordinate value $p$, the entry with least identifier among entries at $p$, and records each discarded identifier in a duplicate map. The weighted deduplicated set keeps the least identifier for each exact pair $(p,w)$ and records only repeated pairs in its duplicate map. A weighted computation retains different weights at the same coordinate; their power values are compared by the ordinary power-distance rule, so a dominated weight may be hidden.

For a finite point set $P\subseteq {\mathbb{R}}^d$, its convex hull is the set of all finite convex combinations of points of $P$. Its affine dimension is the dimension of its affine hull. A tuple $p_0,\dots ,p_d$ is affinely independent when the vectors $p_1-p_0,\dots ,p_d-p_0$ form a basis of ${\mathbb{R}}^d$; here a basis means every vector has a unique linear expansion in those vectors. This convention makes nondegenerate simplex coordinates unique.