The number of regions determined by all (d-1)-simplices spanned by a point set in R^d

Let S be a set of n points in strong general position in R^d, d ≥ 2, and let Σ_d be the union of the C(n,d) closed (d−1)-simplices spanned by S. We study R_d(S), the number of connected components of int conv S \ Σ_d, extending the exact theory available for d = 3. The cells of this decomposition were introduced in the theory of simplicial depth; counting them is new. We first prove that every region is convex (under the hypothesis that no d+1 points lie on a hyperplane), so that Σ_d is (d−2)-acyclic, simply connected for d ≥ 3, and homotopy equivalent to a wedge of R_d spheres S^{d−1}. Combining Alexander duality, which identifies R_d with the (d−1)-st Betti number b_{d−1}(Σ_d), with piecewise-linear Morse theory on Σ_d in the sense of Bestvina and Brady, we prove that R_d(S) equals the number N_d(S) of transversal d-fold points of Σ_d up to an explicit error polynomial of degree d^2−2; consequently R_d(S) = D_d(S) + O_d(n^{d^2−1}), where D_d counts vertex-disjoint d-fold points. This yields max_S R_d = Θ_d(n^{d^2}) for every d, and, when d is a prime power, min_S R_d ≥ C(n,d^2)/C(d^2+d−1,d−1) − O_d(n^{d^2−2}) via the constrained Tverberg theorem of Blagojević, Frick and Ziegler; for d that is not a prime power, the argument of this paper does not give the lower bound: the existence of the required point threshold d^2+d−1 is a special case of an open problem of Frick, and the optimal threshold remains a separate question. A nerve decomposition gives the exact formula R_d = C(n−1,d) + Φ_d, where Φ_d is a signed count of intersecting families of simplices without a common vertex, localized on the faces of the spanned hyperplane arrangement. Exact computations for d = 4 and n ≤ 9 (128 configurations in exact integer arithmetic, 124 of them with two independent region counts) show that R_4 is not determined by n alone in convex position — a negative answer, in dimension four, to a question raised in the R^3 paper — and confirm the convexity and acyclicity statements on every instance. MSC 2020: 52C35, 52C45, 52A35, 55U10, 57Q05, 05A16. Version 1.0.0 (32 pages). The deposit contains the paper (PDF and complete LaTeX source) and the reproducibility package zenodo_package_v100.zip: the exact region counter hd4.py for Σ_4 (pure Python, integer/rational arithmetic only) with its report generator and batch scripts; an independent from-scratch re-implementation of the region count (79/79 agreement); the exact region-convexity checkers for d = 3 and d = 4; the full-Gaussian-elimination cross-check of the GF(2) homology (79/79); the 128 frozen result files (integer coordinates, f-vectors, Betti numbers, both region counts, pattern and stratum counts, Morse and nerve data) with SHA-256 freeze list; the complete written proofs of Theorems A–D with their audit reports and the Phase-5 review ledgers; and a number-by-number cross-check of the paper against the data. This deposit continues the R^3 paper, concept DOI 10.5281/zenodo.21866448. Code: MIT. All other content: CC BY 4.0. See README.md inside the archive.