Sierksma's conjecture for nine points in R³

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 constraint complexes), a general position analysis (the Tverberg partitions of a generic configuration cover 76,545 constraints), Hell's parity theorem for Birch partitions, and a finite step: the covering number under the parity restriction is 8, established by an enumeration of matching-covering families, running in about twenty seconds and independently reimplemented; a 42,373-subproblem SAT certificate provides an independent cross-check. 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, v0.2.1) and its sources; the code and data of the finite enumeration (seeds, lifts, completion search, canonical forms and stabilizers, twist and parity filters, the maximum-multiplicity-two branch, and an independent definition-only reimplementation, with the 17 classes of matching-covering seven-families and the Sierksma control); the SAT cross-check (constraint generator, 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 re-run the enumeration in seconds and to regenerate any SAT 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.