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Heegner-7 43-Vector Kochen-Specker Set (3D)

Basic Metadata

Property Value
Name Heegner-7 43-Vector KS Set
Dimension 3 (spin-1 Hilbert space)
Number of Rays 43
Ring \(\mathbb{Z}[(1+\sqrt{-7})/2]\)
Type State-independent KS set
Author Michael Kernaghan
Year 2026
Cancellation Identity \(\alpha \cdot \bar{\alpha} = 2\)

Overview

The Heegner-7 43-vector KS set is constructed over the ring of integers of \(\mathbb{Q}(\sqrt{-7})\), a quadratic imaginary number field associated with the Heegner number 7. The algebraic integer \(\alpha = (1+\sqrt{-7})/2\) satisfies \(\alpha \cdot \bar{\alpha} = 2\), providing a norm-2 cancellation identity that enables exact orthogonalities in \(\mathbb{C}^3\).

This construction is significant for several reasons:

  • Genuinely new: Not present in any prior KS catalogue — the first KS set discovered from a Heegner number field
  • Rigid: With 107 orthogonal pairs, the Jacobian null space analysis shows zero deformation modes
  • Enhanced contextual advantage: The CSW contextuality ratio \(\theta/\alpha = 1.118\), the highest among all known 3D KS sets
  • Large ray pool: The full alphabet generates 145 rays, from which the minimal 43-vector set is extracted

Key features:

  • 145-ray pool: The Heegner-7 alphabet with bounded coefficients generates 145 rays
  • Norm-2 cancellation: \(\alpha \cdot \bar{\alpha} = 2\) satisfies the controlling invariant for KS-uncolorability
  • Smooth degradation: Ablation study shows the contextual advantage degrades smoothly from 1.118 to ~1.03 as rays are removed, with KS property preserved

Structure

The 43 rays live in \(\mathbb{C}^3\), using coordinates from the alphabet \(\{0, \pm 1, \pm \alpha, \pm \bar{\alpha}\}\) where \(\alpha = (1+\sqrt{-7})/2\). Orthogonality between vectors relies on the identity \(\alpha \cdot \bar{\alpha} = 2\) together with standard integer arithmetic.

Orthogonality Structure

Property Value
Orthogonal pairs 107
Ray pool size 145
Minimum KS subset 43 vectors

Rigidity

The Heegner-7 set is infinitesimally rigid: the Jacobian null space has dimension exactly equal to the symmetry dimension, leaving zero deformation modes.

Property Value
Null space dim 51
Symmetry dim 51
Deformation modes 0 (rigid)

This rigidity is a new result not covered by the Trandafir-Cabello analysis, which focused on previously known constructions.

Contextual Advantage

The Heegner-7 set exhibits the highest contextual advantage among all known 3D KS sets, measured via the Cabello-Severini-Winter (CSW) graph invariants on the full 145-ray pool orthogonality graph (not just the 43-ray minimal set):

Invariant Graph Value
Independence number \(\alpha(G)\) 145-ray pool 50
Lovász theta \(\vartheta(G)\) 145-ray pool 55.89
Contextuality ratio \(\theta/\alpha\) 145-ray pool 1.118

The CSW invariants are computed on the full pool graph because the contextual advantage measures the maximum quantum-over-classical violation achievable using any rays from the algebraic pool. The ratio \(\theta/\alpha > 1\) certifies quantum contextuality. The Heegner-7 value of 1.118 exceeds all other known 3D constructions, and the advantage degrades smoothly as rays are removed (ablation study).

BPQS (Bipartite Perfect Quantum Strategy) — Heuristic upper bound

The Heegner-7 set defines a bipartite perfect quantum strategy with:

\[|S_A| \times |S_B| \leq 9 \times 12 = 108\]

This is the best BPQS found by heuristic search (not proved optimal). The 23 bases of the Heegner-7 set define a new Bell scenario not previously studied.

Relation to Other KS Sets

Construction Dimension Vectors Pairs Rigid? Ring
Conway-Kochen (CK-31) 3 31 71 Yes \(\mathbb{Z}\)
Eisenstein 3 33 78 Yes \(\mathbb{Z}[\omega]\)
Peres 3 33 72 No (flex) \(\mathbb{Z}[\sqrt{2}]\)
Heegner-7 3 43 107 Yes \(\mathbb{Z}[(1+\sqrt{-7})/2]\)
Golden 3 52 124 Yes \(\mathbb{Z}[\varphi]\)

The Heegner-7 set occupies a middle position in the six-island hierarchy: larger than the 31-33 vector sets but smaller than the Golden-52, with the highest contextual advantage ratio.

Why "Heegner-7"?

The number 7 is one of the nine Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163) — the values of \(d\) for which \(\mathbb{Q}(\sqrt{-d})\) has class number one (unique factorization). All nine Heegner number fields were tested; only \(d=7\) produces a KS set. The key algebraic integer \(\alpha = (1+\sqrt{-7})/2\) satisfies \(\alpha \bar{\alpha} = 2\), providing the norm-2 cancellation needed for orthogonality. For larger Heegner numbers (\(d = 11, 19, \ldots, 163\)), the corresponding algebraic integers have norm too large to support the necessary cancellations.

References

  • M. Kernaghan, "The Algebraic Landscape of Kochen-Specker Sets in Dimension Three" (2026)
  • M. Kernaghan, "New KS Sets from Algebraic Number Fields with Enhanced Contextual Advantage" (2026, PRL letter)