Lower bounds for the number of Tverberg partitions: an LP barrier, matchings, and amplification

We study lower bounds for the number of Tverberg partitions of (r-1)(d+1)+1 points in Rd into r=3 parts, the case of Sierksma's conjecture whose only settled non-trivial instances are d=2 (Hell) and d=3 (paper A). First, the classical weighted double-counting arguments of Vučić–Živaljević and Hell are shown to be dual feasible solutions of a linear program over the twisted pair-constraint system of paper A; its exact optimum for nine points in R³ is 665/216, so no weighted double count on that system certifies more than four partitions, and in general the fractional covering number is Θ((3/2)d+1) against Sierksma's 2d. Second, by hand, every nine points of R³ (a labelled 9-tuple, repetitions allowed) have at least six Tverberg partitions, via a seven-point hitting lemma for matchings; a computer-aided finite enumeration raises this to seven and shows that the matching covering number of nine labels is exactly seven. Third, amplification gives, for every d≥2, at least B_d Tverberg partitions of 2d+3 points in Rd by hand (6,9,13,19 for d=3,…,6), at least F_d≥ B_d by hand from partial matchings, and C_d computer-aided (7,10,14,20); asymptotically F_d and C_d improve the constant of the base-3/2 bound by the factors 32/27 and 112/81. Finally, for ten points in the plane and r=4 a scan of all 14,309,547 order types finds no set with fewer than 36 Tverberg 4-partitions; we explain why this is evidence and not a proof. MSC 2020: 52A35 (primary); 05A18, 52C45, 90C05 Reproducibility package: this record archives the manuscript (PDF, v0.1) and its sources; the linear-programming barrier code with the exact rational certificates (d = 1, …, 8) and their verifier; the checkers of the hand proofs of six and seven and their outputs, including two independent re-implementations of the finite enumeration behind the bound seven; the general-dimension amplification scripts, tables and referee checks; the tools and the witness family of the twist-necessity remark; and the exact planar Tverberg-partition counter for ten points with the digests of the scan of all 14,309,547 order types. Every reproduction command runs from a clean unpack of the archive. The companion paper, Sierksma's conjecture for nine points in R³ (Zenodo, concept DOI 10.5281/zenodo.22536949), proves the bound eight. Computations, proofs, and manuscript preparation were assisted by AI systems directed and verified by the author. The exact rational certificate of Theorem 3.1 (checked on the full program and by an independent floating-point solve), the finite enumeration behind Theorem 4.5 (a second, independent implementation over the full universe and a third check restricted to the universe of the 1,855 admissible partitions) and the exact partition counts of the two ten-point witnesses of Section 5 (independently recounted in exact rational arithmetic) were independently confirmed. The following are single computations, reported as such where they occur: the covering and twist counts of the families G_7 and H_7 in Remark 3.4; the machine checks of Proposition 3.3 for d≤4 and of Lemma 4.2; the LP values for 4≤ d≤8 and the support structure of the optima in Remark 3.2; the recursion table F_d > ⌈V_d⌉ for d≤100 in Appendix D; the order-type scan, the simulated-annealing search and the witness construction of Section 5; and the computational experiments mentioned in Section 6. All other results are human proofs.