Research · Papers · Exact answers in open problems · MF-195
Nonexistence of 19-vector Kochen-Specker sets in C^6 and the lower bound m_6 ≥ 20
No KS set in C^6 has 19 vectors, hence m_6 >= 20 and m_6 ∈ {20, 21} (conditional on cited lemma as in MF-184)
Published 2026-09-06
For everyone
Plain summary
In quantum mechanics, Kochen-Specker sets are collections of vectors that prove measurement outcomes cannot simply reveal preexisting values. A core question is finding the smallest set of vectors that creates this contradiction in a given dimension. In six dimensions, earlier work showed the minimum size m_6 sits between 18 and 21 vectors.
This work proves no 19-vector Kochen-Specker set exists in six-dimensional space. Combined with the earlier exclusion of 18 vectors, the lower bound moves to m_6 >= 20, leaving only 20 and 21 as candidates for the minimum. The proof combines an analytical graph argument with four verified SAT solver certificates. The result relies on the cited lemma from MF-184, and whether dimension 6 with 19 vectors was ruled out in prior external literature remains unverified.
Result
No Kochen-Specker set exists in C^6 with cardinality n = 19. The minimal size m_6 of a six-dimensional Kochen-Specker set satisfies m_6 >= 20, restricting the minimum to m_6 ∈ {20, 21}. This result is conditional on the lemma established in MF-184.
Setting and definitions
A Kochen-Specker (KS) set in C^6 is a finite set V of 1-dimensional subspaces in C^6 admitting no non-contextual assignment {0, 1} that assigns exactly one 1 to every complete orthonormal basis C ⊆ V. Let b be the number of complete bases, and let n = |V|.
The basis-intersection graph has bases as vertices, with edges weighted by shared vector counts |C_1 ∩ C_2|. An edge sharing k vectors is a k-sharing edge; e_k is the number of such edges. A vector v ∈ V is doubly covered if it belongs to exactly two bases. Let t denote the number of doubly covered vectors, and let b' denote the count of bases not incident to any 4-sharing edge.
Method
The nonexistence proof splits into an analytical structural branch and a four-cube machine verification.
The structural analysis has four steps:
- Step A: If v belongs to bases C_1 and C_2, then |C_1 ∩ C_2| >= 3. If |C_1 ∩ C_2| = 3, the closed neighborhood N[v] equals C_1 ∪ C_2.
- Step B: No vertex belongs to three bases. Both branching cases force either seven mutually orthogonal vectors in C^6 or two bases sharing five vectors, which violates subspace condition S4.
- Step C: Each basis intersects at most two others, meeting each in disjoint 3-sets. All 4-sharing edges in the basis-intersection graph are therefore isolated.
- Step D: The vector count identity 6b = 19 + t with t = 3e_3 + 4e_4 rewrites in terms of b' as 8e_4 + 6b' - 3e_3 = 19 subject to e_3 <= b'.
A 2026-09-06 repair closed an arithmetic gap in Step D. The original constraints admitted the spurious triple (e_4, b', e_3) = (2, 1, 1). Because 3-sharing edges occur only between bases disjoint from all 4-sharing edges, a graph on b' <= 1 vertices has no edges, forcing e_3 = 0 whenever b' <= 1. Adding this condition eliminates all integer solutions.
Pairwise-disjoint bases cannot partition 19 vectors because 6 does not divide 19.
The remaining cases form a four-cube cover A_1, A_2, A_3, A_4, corresponding to two bases intersecting in k = 1, 2, 3, 4 vectors (with k = 5 excluded by S4). All four cubes are UNSAT:
- Solver runs in
frontier/ks6/w2/cubes19/(completed 2026-09-06T00:35:01Z) finished in 12 s, 9 s, 72 s, and 198 s, each certified with drat-trim (s VERIFIED). - An independent clean-room re-encoding produced runtimes of 12 s, 10 s, 85 s, and 205 s, all verified with DRAT certificates. Both sets of outcomes were preregistered before the clean-room run.
Discussion
Establishing m_6 >= 20 narrows the range for a six-dimensional Kochen-Specker set from [18, 21] down to {20, 21}. The 21-vector construction of Lisonek, Badziag, Portillo, and Cabello is at most one vector above the absolute minimum.
Two qualifications apply:
- The proof is conditional on the lemma cited in MF-184.
- Novelty of excluding n = 19 in dimension 6 is unverified against external literature; three-seat literature searches did not locate earlier proofs of this bound.
The case n = 20 remains open. An initial two-cube attempt timed out. A preregistered 143-cube two-level cover in frontier/ks6vet/n20/ is in progress; 14 cubes are certified UNSAT, with 0 SAT outcomes and 0 timeouts.
For everyone — the takeaway
What this means
The smallest possible Kochen-Specker set in six dimensions has either 20 or 21 vectors. The known 21-vector configuration by Lisonek, Badziag, Portillo, and Cabello is either minimal or off by just one vector. Resolving whether a 20-vector set exists will finish the search in six dimensions.
Attribution and prior art
Prior art: Searches across existing literature found no prior work excluding 19 in d = 6, though the novelty of this result remains unverified. Sources: Xu, Chen, Guehne 2020 · Lisonek, Badziag, Portillo, Cabello 2014
Register references
- Register entry: MF-195
- Related entry: MF-184
- Lane report:
lanes/ks-d6/REPORT.mdTheorem 2 - Hardware runs and storage:
frontier/ks6/w2/cubes19/ - Verification receipts:
lanes/ks-d6/verify-invent/VERDICTS.md,thm2_count.json - Preregistration:
tests/ks6-n19-cover/PREREG.md - Active branch for n = 20:
frontier/ks6vet/n20/ - Reference construction: Lisonek, Badziag, Portillo, Cabello (21-vector set in C^6)
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 4 of 4 receipt files bundled (17 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.