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Z[sqrt(-2)] 33-Vector Kochen-Specker Set (3D)

Basic Metadata

Property Value
Name \(\mathbb{Z}[\sqrt{-2}]\) 33-Vector KS Set
Dimension 3 (spin-1 Hilbert space)
Number of Rays 33
Ring \(\mathbb{Z}[\sqrt{-2}]\)
Type State-independent KS set
Author Michael Kernaghan
Year 2026
Cancellation Identity $

Overview

The \(\mathbb{Z}[\sqrt{-2}]\) 33-vector KS set is constructed over the ring of Gaussian-like integers \(\mathbb{Z}[\sqrt{-2}]\). The cancellation identity \(|\sqrt{-2}|^2 = 2\) provides the same norm-2 mechanism as the Peres set's \((\sqrt{2})^2 = 2\), but using a complex quadratic extension rather than a real one.

This construction is significant as a demonstration that the flex belongs to the graph, not the algebra:

  • Graph-isomorphic to Peres: The orthogonality graph of the \(\mathbb{Z}[\sqrt{-2}]\) set is isomorphic to the Peres-33 graph
  • Same flex: Despite using complex coordinates, the set has exactly the same 1-dimensional infinitesimal flex as Peres — confirming that the deformation mode is a property of the graph structure
  • New algebraic realization: While not a "new island" in terms of graph type, it provides a genuinely different algebraic realization of the Peres graph

Key features:

  • 49-ray pool: The \(\mathbb{Z}[\sqrt{-2}]\) alphabet generates 49 rays (same count as the integer and Peres pools)
  • Norm-2 cancellation: \(|\sqrt{-2}|^2 = 2\) satisfies the controlling invariant
  • BPQS: \(7 \times 9 = 63\), identical to Peres (consistent with graph isomorphism)

Structure

The 33 rays live in \(\mathbb{C}^3\), using coordinates from \(\{0, \pm 1, \pm \sqrt{-2}\}\). Orthogonality between vectors relies on the identity \(|\sqrt{-2}|^2 = 1 \cdot 1 + (\sqrt{2})^2 = 2\) (more precisely, \(\sqrt{-2} \cdot \overline{\sqrt{-2}} = 2\)).

Orthogonality Structure

Property Value
Orthogonal pairs 72
Ray pool size 49
Minimum KS subset 33 vectors

Rigidity (Flex)

The \(\mathbb{Z}[\sqrt{-2}]\) set is infinitesimally flexible but finitely rigid — the same status as the Peres set:

Property \(\mathbb{Z}[\sqrt{-2}]\)-33 Peres-33
Orthogonal pairs 72 72
Null space dim 42 42
Symmetry dim 41 41
Deformation modes 1 (flex) 1 (flex)
Finitely rigid? Yes (2nd-order blocked) Yes (2nd-order blocked)

The single infinitesimal flex is blocked at second order (cokernel component 0.075), making both sets finitely rigid despite their infinitesimal flexibility. This confirms that the flex belongs to the graph: any embedding of this particular 33-vertex, 72-edge orthogonality graph — whether in \(\mathbb{Z}[\sqrt{2}]\), \(\mathbb{Z}[\sqrt{-2}]\), or any other ring — will exhibit the same 1-dimensional flex.

BPQS (Bipartite Perfect Quantum Strategy)

\[|S_A| \times |S_B| = 7 \times 9 = 63\]

Exact, verified. Identical to the Peres-33 BPQS, as expected from graph isomorphism.

Relation to Other KS Sets

Construction Dimension Vectors Pairs Rigid? Ring Graph type
Peres 3 33 72 Flex \(\mathbb{Z}[\sqrt{2}]\) Peres graph
\(\mathbb{Z}[\sqrt{-2}]\) 3 33 72 Flex \(\mathbb{Z}[\sqrt{-2}]\) Peres graph
Eisenstein 3 33 78 Rigid \(\mathbb{Z}[\omega]\) Eisenstein graph

The Peres and \(\mathbb{Z}[\sqrt{-2}]\) sets share the same graph (and therefore the same flex, BPQS, and coloring properties) but are algebraically distinct: one uses real coordinates, the other complex. Together with Gaussian integers \(\mathbb{Z}[i]\) (which also produce the Peres graph), these three rings demonstrate that the norm-2 cancellation \(x \cdot \bar{x} = 2\) can be realized in multiple algebraic settings while preserving graph structure.

References

  • M. Kernaghan, "The Algebraic Landscape of Kochen-Specker Sets in Dimension Three" (2026)