Sierksma's conjecture for nine points in R³

Abstract

We prove Sierksma's conjecture for nine points in R³: every nine points of R³ have at least eight Tverberg partitions into three parts, to our knowledge the first case in dimension three. The proof combines equivariant topology (Dold's theorem on twisted pair-constraint complexes), a general position analysis showing that the Tverberg partitions of a generic configuration form a covering family of a system of 76,545 constraints, Hell's parity theorem for Birch partitions, and a certified computation: the covering number under the parity restriction equals 8, established by 42,373 unsatisfiable subproblems, each with a DRAT proof checked by drat-trim (34.8 TB of proofs, 9,122 hyperthread-hours). Without parity the covering number is 7, witnessed by an explicit family. By-products: at least six Tverberg partitions for any nine distinct points, lower bounds for 2d+3 points in R^d improving the Vučić–Živaljević bound by a constant factor, and the exact fractional covering number 665/216, capping counting arguments at 4. MSC 2020: 52A35 (primary); 05A18, 55M20, 68R05 Reproducibility package: this record archives the manuscript (PDF, v0.1.1) and its sources, the code (constraint generator, SAT encodings, symmetry reduction, audit scripts), the task ledgers with SHA-256 hashes of every CNF and DRAT proof, samples of the certificates, and the verification commands needed to regenerate any task and re-check its proof with drat-trim. Computations, proofs, and manuscript preparation were assisted by AI systems directed and verified by the author; all results were independently confirmed by exact rational arithmetic or by machine-checked proof certificates.