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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 flex as Peres — confirming that the deformation mode is a property of the graph structure
  • Same configuration, different parameter value: the two sets are not merely graph-isomorphic. They are the same 33-ray configuration at two points of one continuous family (see Rigidity below), so this is a new algebraic route to the Peres configuration rather than a geometrically distinct realization of it

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 flexible with one finite deformation dimension — 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? No — flex is finite No — flex is finite

The flex is finite: it integrates to the continuous one-parameter family of Gould and Aravind (Found. Phys. 40, 1096 (2010)), and the two sets sit at antipodal parameter values on that circle. 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. It also shows the two islands are one object seen twice.

Corrected 2026-07-27. This section previously stated the flex was blocked at second order with cokernel component 0.075, making both sets finitely rigid. That was an artifact of a constraint-ordering bug in ks_rigidity_finite.py; the true cokernel component is zero. Reported by Manuel Flores Gordillo, doi:10.5281/zenodo.21488474.

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)