Skip to content

Golden Ratio 52-Vector Kochen-Specker Set (3D)

Basic Metadata

Property Value
Name Golden Ratio 52-Vector KS Set
Dimension 3 (spin-1 Hilbert space)
Number of Rays 52
Ring \(\mathbb{Z}[\varphi]\), where \(\varphi = (1+\sqrt{5})/2\)
Type State-independent KS set
Author Michael Kernaghan
Year 2026
Cancellation Identity \(\varphi^2 = \varphi + 1\)

Overview

The Golden ratio 52-vector KS set is constructed over the ring \(\mathbb{Z}[\varphi]\) of integers in \(\mathbb{Q}(\sqrt{5})\), where \(\varphi = (1+\sqrt{5})/2\) is the golden ratio. The identity \(\varphi^2 = \varphi + 1\) provides a norm-2 cancellation mechanism (since \(\varphi \cdot \bar{\varphi} = -1\) and \(\varphi^2 - \varphi - 1 = 0\) imply cancellations producing exact orthogonalities).

This construction is remarkable for several reasons:

  • Invisible to raw search: The golden ratio set cannot be found by alphabet enumeration alone — it is revealed only by cross-product completion, where new rays are generated as cross products of existing alphabet rays
  • Genuinely new: Not present in any prior KS catalogue
  • Rigid: With 124 orthogonal pairs, the Jacobian analysis confirms zero deformation modes
  • Largest known minimal 3D KS set: At 52 vectors, this is the largest minimal KS set among the six algebraic islands

Key features:

  • 205-ray pool: The full golden ratio alphabet with cross-product closure generates 205 rays
  • Cross-product discovery: Raw alphabet search over \(\{0, \pm 1, \pm \varphi\}\) produces only colorable pools; KS-uncolorability emerges only after closing under cross products
  • No integer absorption: Unlike most other extended alphabets, the golden ratio pool does not collapse to the CK-31 integer set when integers are added

Structure

The 52 rays live in \(\mathbb{R}^3\), using coordinates from the golden ratio ring \(\mathbb{Z}[\varphi] = \{a + b\varphi : a, b \in \mathbb{Z}\}\). Unlike the Eisenstein and Heegner-7 sets which use complex coordinates, the golden ratio set lives entirely in real 3-space.

Orthogonality Structure

Property Value
Orthogonal pairs 124
Ray pool size 205
Minimum KS subset 52 vectors

Rigidity

The Golden-52 set is infinitesimally rigid: the Jacobian null space has dimension exactly equal to the symmetry dimension.

Property Value
Null space dim 60
Symmetry dim 60
Deformation modes 0 (rigid)

This rigidity is a new result not covered by any prior analysis.

Cross-Product Completion

The golden ratio island illustrates a key methodological point: not all KS sets can be found by simple alphabet enumeration.

  1. Start: Generate all rays with coordinates in \(\{0, \pm 1, \pm \varphi\}\)
  2. Problem: The resulting pool is KS-colorable — no subset proves the KS theorem
  3. Solution: Take cross products \(v_i \times v_j\) for all pairs of rays, normalize, and add new rays whose coordinates remain in \(\mathbb{Z}[\varphi]\)
  4. Result: The completed pool of 205 rays contains KS-uncolorable subsets, with minimum size 52

This "invisibility" to raw search makes the golden ratio island the most structurally subtle of the six islands.

BPQS (Bipartite Perfect Quantum Strategy) — Heuristic upper bound

The Golden-52 set defines a bipartite perfect quantum strategy with:

\[|S_A| \times |S_B| \leq 12 \times 13 = 156\]

This is the best BPQS found by heuristic search (not proved optimal). The 25 bases define a new Bell scenario.

Relation to Other KS Sets

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

Why Can't Other Real Quadratic Fields Produce KS Sets?

Extensive testing of other real algebraic numbers — the silver ratio \(1+\sqrt{2}\) (which IS the Peres set), the plastic ratio, tribonacci constant, and supersilver ratio — shows that only those with norm-2 cancellation identities support KS-uncolorable pools. The golden ratio satisfies \(\varphi^2 = \varphi + 1\), which enables the necessary cancellations; fields without such identities produce colorable pools regardless of pool size.

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)