The number of regions determined by all triangles spanned by a point set in R^3

Let $S$ be a set of $n$ points in general position in $\mathbb{R}^3$ and let $\Sigma$ be the union of the $\binom{n}{3}$ closed triangles spanned by $S$. We count the connected components $R(S)$ of $\operatorname{int}\operatorname{conv}(S)\setminus\Sigma$, the three-dimensional analogue of the classical count of the regions into which the chords of $n$ points on a circle divide the disc. Let $n_X$ be the number of points at which a spanned segment pierces the relative interior of a spanned triangle, $n_T$ the number of points lying in the relative interiors of three triangles, and $P$ the number of pairs of triangles meeting in a segment of positive length. For every configuration in the (measure-one) class of strongly general positions we prove the master formula $R = n - \binom{n}{2} + \binom{n}{3} - 1 + (P - n_X) + n_T$, whose underlying Euler relation $R = V - E + F - 1$ is not a formality: the regions are not a priori convex or simply connected, and we establish it by showing $\pi_1(\Sigma)=1$ and applying Alexander duality. We then prove the identity $P = \tfrac{n+1}{2} n_X$, which reduces the master formula to $R = n - \binom{n}{2} + \binom{n}{3} - 1 + \tfrac{n-1}{2} n_X + n_T$, exhibits $R$ as a function of $(n, n_X, n_T)$ alone, and reduces the $3$-dimensional rectilinear crossing number of $K_n^{(3)}$ to simplicial depth; in convex position it gives $R = B(n) + n_T$ with $B(n) = 3\binom{n}{6} + 2\binom{n}{5} + \binom{n-1}{3}$. Finally, $n_T$ is not a function of the order type for $n \ge 8$: two $8$-point integer configurations with identical chirotopes have $n_T = 159$ and $158$. A wall-jump theorem isolates the mechanism, computes the jump, and forces it to vanish for $n \le 7$, where an exhaustive enumeration of the $1, 2, 4, 246$ realizable uniform acyclic order types of rank $4$, each realized by explicit integer coordinates, certifies the complete spectrum of $R$: in particular $\min R = 45$ and $\max R = 109$ for $n = 7$, with the ten values $98, \dots, 107$ absent. MSC 2020: 52C35, 52C45, 05A15, 52B40. Reproducibility package (tr_zenodo_package.zip): the LaTeX source; the exact rational-arithmetic region counter with two independent re-verifications; the wall and witness verification scripts; the rank-4 order-type enumeration, realization and certification pipeline with the certified 253-row table for $n \le 7$; the triple-point density toolkit; and the data behind every table. Code MIT; all other content CC BY 4.0. Version 1.0.3 = paper v0.4.2, a text-only revision (no statement or formula changed; code and data unchanged); the sequel in $\mathbb{R}^d$ is doi:10.5281/zenodo.22317216. See README.md inside the archive.