Research · Papers · Exact answers in open problems · ML-099
Exclusion of 19- and 20-vector Kochen-Specker sets in C^6
No 19- or 20-vector Kochen-Specker set exists in C^6 (both excluded via MF-195, MF-199)
Published 2026-09-06
For everyone
Plain summary
Quantum mechanics shows that measurement outcomes cannot be fixed in advance without depending on context. A Kochen-Specker set proves this using a finite collection of measurement directions (vectors) where no consistent true-or-false assignment can satisfy every complete measurement basis. Finding the smallest set in a given dimension is a core problem in quantum foundations.
This entry settles the question for six-dimensional complex space (C^6). Neither 19 nor 20 vectors can form a Kochen-Specker set in C^6, setting the minimum size to at least 21 vectors.
Result
No 19- or 20-vector Kochen-Specker set exists in C^6. The minimum size m_6 of a Kochen-Specker set in six-dimensional complex space satisfies m_6 ≥ 21. Candidate sizes n = 19 and n = 20 are formally excluded under MF-195 and MF-199.
Setting and definitions
A Kochen-Specker set in C^6 is a set of n rays (unit vectors up to phase) admitting no non-contextual value assignment: no function from the ray set to {0, 1} assigns exactly one 1 and five 0s to every orthonormal basis of six rays.
Configurations are represented as 6-uniform hypergraphs whose vertices are rays and whose hyperedges are complete orthonormal bases in C^6:
- e_3, e_4: counts of basis pairs overlapping in 3 or 4 vertices.
- b': total number of participating bases.
- k: vertex degree (number of bases containing a given ray).
Method
Exclusion combined combinatorial constraints with automated SAT certificates:
- Combinatorial constraints at n = 19: An initial hand proof established that no vertex belongs to three distinct bases and that every basis intersects at most two others in disjoint 3-sets. This reduced feasibility to the Diophantine relation 8e_4 + 6b' - 3e_3 = 19, which has no integer solutions.
- Breakdown at n = 20: At n = 20, the degree bound relaxes to allow k ≥ 2, permitting three bases to meet at a single vertex and invalidating the direct counting argument.
- Automated refutation and realisability: Both sizes were refuted using a 5-cube symmetric case split with DRAT proof certificates per cube. Surviving combinatorial hypergraphs were tested for geometric realisability in C^6 using Groebner basis elimination and cvc5 SMT solving over the complex field.
Discussion
This entry was originally catalogued as a CONJECTURE for n = 19 and OPEN for n = 20. The hand proof for n = 19 stood as Theorem 2 in lane REPORT.md, but step D lacked independent verification and machine cover, leaving it unbanked.
Both candidate sizes were formally closed on 2026-09-06 under entries MF-195 and MF-199, establishing m_6 ≥ 21.
For everyone — the takeaway
What this means
You cannot prove quantum contextuality in six dimensions with 19 or 20 directions. Any vector-based proof of the Kochen-Specker theorem in C^6 needs at least 21 vectors.
Attribution and prior art
Sources: Xu, Chen, Guehne 2020 · Lisonek, Badziag, Portillo, Cabello 2014
Register references
- ML-099
- MF-195
- MF-199
- lanes/ks-d6/REPORT.md (Theorem 2)
- BANK-CANDIDATES.md (KS6-03)
- frontier/ks6/w2/cubes19/
- frontier/ks6/w2/cubes20/
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 2 of 2 receipt files bundled (20 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-06
- 2026-09-06Published on this site.
- 2026-09-06On 2026-09-06, machine verification for n = 19 completed and a proof gap was repaired, proving Theorem 2 (MF-195) that m_6 >= 20, so m_6 is 20 or 21. Only n = 20 remains open while an exhaustive computation runs to determine if m_6 = 21.
- 2026-09-06Added to ML-099 on 2026-09-06: verified checks across 143 cases resolve the n = 20 question, proving m_6 = 21 exactly (MF-199). The bound 18 <= m_5 <= 29 and the cited XCG lemma ("no GHZ-type proof on <= 9 vertices") remain unchanged.