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Papers

Overview

The algebraic islands research programme has produced one main paper and four PRL-style letters, each addressing a distinct aspect of the classification of Kochen-Specker sets in dimension three.

Author: Michael Kernaghan, Pacific Quantum Systems, Vancouver, Canada

Main Paper

The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

Length: ~39 pages Venue: Physical Review A (target) Status: Complete, pending arXiv submission Download PDF

The comprehensive paper presenting the full algebraic classification. Covers:

  • Systematic alphabet survey across number rings
  • The six algebraic islands and their cancellation identities
  • Cross-product closure methodology (revealing the Golden island)
  • Realizability gap analysis
  • Connection to bipartite perfect quantum strategies (BPQS)
  • CSW graph invariants and Bell inequalities
  • Rigidity classification (Jacobian null space analysis)
  • Three distinct 33-vector KS sets (Peres, Eisenstein, CK-33)
  • Merge saturation as a universal property
  • OCUS optimality proof for CK-31
  • Appendix with representative minimal KS sets for new islands

Key result: Low-complexity cancellation identities (modulus-2 or phase cancellation) are the controlling invariant for KS-uncolorability in dimension 3.

Revision notes (March 16, 2026): Fixed CSW α* numerical error in integer pool table (17.00 → 19.00); clarified ray-level MUS extraction methodology; added classification scope limitation; added falsifiability paragraph; strengthened novelty claim qualifications; added Cabello finite-dimensional qualifier. Reviewed by GPT-5.4-pro Oxford challenge method (32 issues assessed, 6 substantive fixes applied).

PRL Letters

1. New KS Sets from Algebraic Number Fields with Enhanced Contextual Advantage

Length: 4 pages Venue: Physical Review Letters (target) Status: Submitted to arXiv (March 2026), revision pending processing Download PDF

Introduces the Heegner-7 (43 vectors) and Golden ratio (52 vectors) KS sets — the first genuinely new 3D KS constructions in decades. Reports the enhanced contextual advantage of Heegner-7 (\(\vartheta/\alpha = 1.118\)) and the cross-product discovery mechanism for the Golden set.

Key result: Two genuinely new KS sets, neither in any prior catalogue, with the highest known contextual advantage in 3D.

Revision notes (March 16, 2026): Removed dependency on unpublished main paper ("in preparation" references); fixed Bell-scenario claim (necessity → candidates); added closure termination caveat; softened convention and Pavičić claims. Paper is now fully self-contained. GPT-5.4-pro Oxford review: 24 issues assessed, 5 substantive fixes applied.

2. Graph Isomorphism of 31-Vertex Kochen-Specker Sets Across Coordinate Alphabets

Length: 4 pages Venue: Physical Review Letters (target) Status: Submitted to arXiv (March 2026), revision pending processing Download PDF

Shows that all 31-vertex KS sets found across three alphabet-based searches share the same orthogonality graph (VF2-verified). Establishes the modulus-2 cancellation boundary as an empirical regularity. Honestly notes that the three searches may not be independent (rational and mixed alphabets contain the integer alphabet).

Key result: The CK-31 graph is recovered by every tested alphabet-based search that achieves 31 rays.

Revision notes (March 16, 2026): Removed dependency on unpublished papers ("in preparation" references); strengthened 109-ray projective equivalence verification; clarified Table 1 column header; added Schütte bibliography entry; anchored random hypergraph claim; defined rigidity parenthetically. Paper is now fully self-contained. GPT-5.4-pro Oxford review: 24 issues assessed, 6 substantive fixes applied.

3. Computational Evidence for the Optimality of CK-31

Length: 5 pages Venue: Physical Review Letters (target) Status: Complete, not submitted Download PDF

Presents six independent computational strategies that all fail to find a sub-31 KS set. Includes the OCUS exhaustive proof for the integer pool, 8-criticality of CK-31, cross-pool mixing results, and the merge saturation universal property.

Key result: No sub-31 KS set exists by any known method; CK-31 is 8-critical and merge-saturated.

4. Kochen-Specker Uncolorability in Cyclotomic Fields Requires Exactly 6|n

Length: 4 pages Venue: Physical Review Letters (target) Status: Submitted to arXiv (March 2026), revision pending processing Download PDF

The only letter with a complete algebraic proof (no computational evidence needed). Proves that the cyclotomic ray pool \(S_n\) is KS-uncolorable if and only if \(6|n\), via seven lemmas covering sufficiency and three necessity cases.

Key result: Complete characterization of KS-uncolorability across all cyclotomic fields, connecting to \(\mathbb{Z}[1/6]\) minimality.

Revision notes (March 16, 2026): Added footnote clarifying orthogonality table shows necessary divisibility conditions; marked Cabello 2025 as preprint. GPT-5.4-pro Oxford review: 15 issues assessed, 2 substantive fixes applied (proofs confirmed correct).

Computational Reproducibility

All results are reproducible via Python scripts in the contextuality/ repository. Key dependencies: Python 3.11, PySAT 1.8 (Glucose4), NumPy, SciPy. All scripts use random.seed(42) for reproducibility.