The Algebraic Landscape of Kochen-Specker Sets
The Kochen-Specker (KS) theorem is one of the foundational results in quantum mechanics. It proves that quantum measurement outcomes cannot be predetermined independently of which other compatible measurements are performed — a property called contextuality.
Our research asks a new question: what algebraic invariant controls whether a set of quantum measurements exhibits contextuality?
Central thesis: Generator norm ≤ 2 is the controlling invariant for KS-uncolorability in dimension three. The integer identity 1+1=2, the Peres identity (√2)²=2, and the Eisenstein identity 1+ω+ω²=0 are all manifestations of the same algebraic constraint.
Six Constructions from Algebraic Rings
A systematic computational survey across algebraic number fields reveals that KS-uncolorable sets in dimension three arise from exactly six algebraic rings — discrete constructions separated by algebraic structure rather than continuous parameters.
| Construction | Ring | Min KS Size | Cancellation Identity |
|---|---|---|---|
| Integer (Conway-Kochen) | ℤ |
31 | 1 + 1 = 2 |
| Peres | ℤ[√2] |
33 | (√2)² = 2 |
| Eisenstein | ℤ[ω] |
33 | 1 + ω + ω² = 0 |
| ℤ[√-2] | ℤ[√-2] |
33 | |√-2|² = 2 |
| Heegner-7 | ℤ[(1+√-7)/2] |
43 | α·α̅ = 2 |
| Golden | ℤ[φ] |
52 | φ² = φ + 1 |
The Heegner-7 and Golden constructions are newly discovered — they do not appear in any existing KS catalogue. The Golden construction is particularly remarkable: invisible to direct alphabet search, it emerges only through cross-product completion of the coordinate set.
Graph Universality
A striking structural result: all known 31-vertex KS sets — whether constructed from integer coordinates, rational fields, mixed-field alphabets, or group-theoretic orbits (A4, S4) — are graph-isomorphic. The orthogonality graph appears to be a universal invariant at the minimum size.
Connection to Quantum Advantage
Every KS set defines a bipartite perfect quantum strategy (BPQS) — a nonlocal game where quantum players achieve perfect coordination that is impossible classically. The algebraic structure of the KS set directly determines the size and efficiency of the resulting quantum advantage.
| Construction | |SA| × |SB| | Product | Status |
|---|---|---|---|
| Eisenstein-33 | 5 × 9 | 45 | Exact |
| Peres-33 | 7 × 9 | 63 | Exact |
| Conway-Kochen-31 | 8 × 9 | ≤72 | Best found |
| ℤ[√-2]-33 | 7 × 9 | 63 | Exact (new) |
| Heegner-7-43 | 9 × 12 | ≤108 | Best found (new) |
| Golden-52 | 12 × 13 | ≤156 | Best found (new) |
Key Results
- 6|n Conjecture: n-th roots of unity produce KS-uncolorable sets if and only if 6 divides n (verified for n ≤ 30)
- Realizability gap: 49% of random abstract hypergraphs are KS-uncolorable, but 0% are realizable in ℝ³ — geometric embedding is the true bottleneck
- Fragility: The Conway-Kochen 31-vector set is destroyed by 1% coordinate perturbation, confirming the discrete algebraic nature of KS contextuality
- Norm-2 boundary: At generator norm ≥ 3, the cancellation identities required for orthogonality become impossible
Historical Context
The study of finite KS sets has a rich history stretching from the original Kochen-Specker theorem (1967) through the Peres 33-vector construction (1991), the Kernaghan 20-vector set in four dimensions (1994), the Kernaghan-Peres 40-vector construction connecting to GHZ paradoxes (1995), and the Cabello 18-vector minimal set in four dimensions (1996). Our work extends this tradition into the algebraic domain, classifying for the first time which number fields support contextuality in the smallest nontrivial dimension.
Publications
- M. Kernaghan, “The Algebraic Landscape of Kochen-Specker Sets in Dimension Three” (2026). In preparation for Physical Review A.
- M. Kernaghan, “New KS Sets from Algebraic Number Fields with Enhanced Contextual Advantage” (2026). In preparation for Physical Review Letters.
- M. Kernaghan, “Graph Universality of CK-31 and the Norm-2 Boundary” (2026). In preparation for Physical Review Letters.
Earlier Publications
- M. Kernaghan, “Bell-Kochen-Specker theorem for 20 vectors,” J. Phys. A 27, L829 (1994).
- M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).