The Six Algebraic Islands

Classification

Systematic enumeration of algebraic number rings reveals exactly six "islands" — families of rays in $\mathbb{C}^3$ that produce KS-uncolorable pools with distinct graph types. Each island is defined by a number ring whose generators satisfy a norm-$\leq 2$ cancellation identity.

Note: "six islands" refers to six distinct graph families, not six rings. Some rings produce isomorphic orthogonality graphs — for instance, $\mathbb{Z}[\sqrt{-2}]$, Gaussian integers $\mathbb{Z}[i]$, and Peres' $\mathbb{Z}[\sqrt{2}]$ all produce the same "Peres graph." These are counted as one island (the Peres island) with multiple algebraic realizations.

#	Island	Ring	Min KS	Cancellation Identity	Pool Size
1	Integer (CK-31)	$\mathbb{Z}$	31	$1+1=2$	49
2	Peres	$\mathbb{Z}[\sqrt{2}]$	33	$(\sqrt{2})^2 = 2$	49
3	Eisenstein	$\mathbb{Z}[\omega]$	33	$1+\omega+\omega^2 = 0$	57
4	$\mathbb{Z}[\sqrt{-2}]$	$\mathbb{Z}[\sqrt{-2}]$	33	$	\sqrt{-2}
5	Heegner-7	$\mathbb{Z}[(1+\sqrt{-7})/2]$	43	$\alpha \cdot \bar{\alpha} = 2$	145
6	Golden	$\mathbb{Z}[\varphi]$	52	$\varphi^2 = \varphi + 1$	205

Islands 2, 4, and Gaussian ($\mathbb{Z}[i]$) are graph-isomorphic — they share the same orthogonality graph (the "Peres graph") despite using different algebraic coordinates. This means there are five distinct graph types among the six algebraic islands.

The Norm-2 Boundary

The central organizing principle is the generator norm: for a ring element $\alpha$ to participate in KS-producing orthogonalities, it must satisfy a cancellation identity with "norm" $\leq 2$ (where norm refers to $|\alpha|^2$ or the algebraic norm $\alpha \bar{\alpha}$).

What works (norm $\leq 2$): - $1 + 1 = 2$ (integers) - $(\sqrt{2})^2 = 2$ (Peres) - $1 + \omega + \omega^2 = 0$ (Eisenstein, norm 1) - $\alpha \bar{\alpha} = 2$ for $\alpha = (1+\sqrt{-7})/2$ (Heegner-7) - $\varphi^2 = \varphi + 1$ (Golden)

What doesn't work (norm $\geq 3$): - $\{0, \pm 1, \pm 3\}$ — no KS set (lacks $1+1=2$) - $\{0, \pm 1, \pm 4\}$ — no KS set - Plastic ratio, tribonacci, silver ratio, supersilver — all colorable - Heegner numbers $d = 11, 19, 43, 67, 163$ — all colorable (norm too large)

Sub-31 Optimality

Strong computational evidence that CK-31 is the minimum KS set in dimension 3:

OCUS Proof (Integer Pool) — Proved

The OCUS (Optimal Constrained Unsatisfiable Subset) algorithm exhaustively proved that no $\leq 30$-vector KS subset exists in the 49-ray integer pool $\{0, \pm 1, \pm 2\}$. Completed in 272 iterations (0.1 seconds).

Six Independent Strategies — Computational evidence

Strategy	Method	Result
1. SAT minimization	MUS + greedy + swap on all 6 pools	All pools: min = known minimum
2. Cross-pool mixing	All 15 pairwise + full 477-ray union	Cross-pool rays never help
3. Larger alphabets	30 expanded alphabets tested	Every alphabet with $\{0,\pm 1,\pm 2\}$ → 31
4. Numerical optimization	SA, perturbation, completion	Zero triads from random rays
5. CK-31 criticality	Exhaustive k=1..8, sampled k=9..12	8-critical (all subsets colorable)
6. Triad density bounds	Random rays in $\mathbb{R}^3$ and $\mathbb{C}^3$	Zero orthogonal pairs in random samples

Key finding: Every alphabet containing $\{0, \pm 1, \pm 2\}$ converges to CK-31 as its minimum. Alphabets lacking $\pm 2$ are either not KS-uncolorable or stay at $\geq 33$.

Graph Universality of CK-31 — Proved (VF2-verified)

All 31-vertex KS sets found by any method — integer alphabet, rational coordinates, mixed-field constructions, $A_4$ and $S_4$ group orbits — share the same orthogonality graph (verified by VF2 isomorphism, both graph and hypergraph).

This universality means the CK-31 graph is the unique minimal graph supporting KS-uncolorability in 3D with integer-type coordinates. Negative controls confirm: rings like $\mathbb{Z}[\sqrt{3}]$, $\mathbb{Z}[\sqrt{-5}]$ produce pools with many bases but all are KS-colorable.

MUS Landscape — Computational result

From 5000 MUS (Minimal Unsatisfiable Subset) trials on the 49-ray integer pool, exactly 6 distinct minimal 31-sets exist:

13 core rays ($\|v\|^2 \leq 3$) appear in all 6 sets — the $\{0, \pm 1\}$ skeleton
12 rays never used — all with $\|v\|^2 = 9$ (highest norm)
Norm stratification: $\|v\|^2 = 1,2,3 \to$ core (100%); $\|v\|^2 = 5 \to$ 83%; $\|v\|^2 = 6 \to$ 67%; $\|v\|^2 = 9 \to$ 0%
No swap connectivity — minimum Hamming distance 5 between any two sets
CK-31 is one of the 6 (rediscovered as set 5)

Merge Saturation — Computational result; conjecture for general case

All six minimal KS sets exhibit 100% merge preservation: merging any pair of non-orthogonal vertices preserves KS-uncolorability.

Island	Min	Non-orth pairs	Preserved	Rate
CK-31	31	394	394	100%
Peres-33	33	456	456	100%
Eisenstein-33	33	450	450	100%
$\mathbb{Z}[\sqrt{-2}]$-33	33	456	456	100%
Heegner-7	43	796	796	100%
Golden-52	52	1204	1204	100%

Total: 3,756 merges, all preserving KS-uncolorability. This motivates the merge-saturation conjecture (open): every minimal KS set in $\mathbb{R}^3$/$\mathbb{C}^3$ is merge-saturated.

Rigidity Classification — Proved (Jacobian null space)

Island	Vectors	Pairs	Null dim	Sym dim	Deformation	Status
CK-31	31	71	39	39	0	Rigid
Eisenstein-33	33	78	41	41	0	Rigid
Peres-33	33	72	42	41	1	Flex
$\mathbb{Z}[\sqrt{-2}]$-33	33	72	42	41	1	Flex
Heegner-7	43	107	51	51	0	Rigid
Golden-52	52	124	60	60	0	Rigid

The Peres and $\mathbb{Z}[\sqrt{-2}]$ flex is finite: it integrates to the continuous one-parameter family of Gould and Aravind (Found. Phys. 40, 1096 (2010)), on which the two islands are antipodal points, so they are one configuration seen at two parameter values. The flex belongs to their shared graph (72 pairs vs Eisenstein's 78), not to the choice of algebraic ring.

Corrected 2026-07-27. Previously stated as infinitesimal only, blocked at second order, with 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 / Bell Connections

Each KS set defines a bipartite perfect quantum strategy (BPQS), connecting the KS theorem to Bell nonlocality. Status varies per island — see the "Status" column:

| Island | $|S_A| \times |S_B|$ | Product | Status | |--------|---------------------|---------|--------| | Eisenstein-33 | $5 \times 9$ | 45 | Exact | | Peres-33 | $7 \times 9$ | 63 | Exact | | CK-31 | $8 \times 9$ | $\leq 72$ | Best found | | $\mathbb{Z}[\sqrt{-2}]$-33 | $7 \times 9$ | 63 | Exact (= Peres) | | Heegner-7 | $9 \times 12$ | $\leq 108$ | Best found | | Golden-52 | $12 \times 13$ | $\leq 156$ | Best found |

The Eisenstein-33 set achieves the smallest known BPQS (45) among all 3D KS sets.

The 6|n Cyclotomic Theorem — Proved (algebraic proof)

Theorem: The cyclotomic ray pool $S_n$ (rays with coordinates in $\mathbb{Q}(\zeta_n)$) is KS-uncolorable if and only if $6 | n$.

This is proved by complete algebraic argument (no computational evidence needed):

Sufficiency: When $6|n$, the Eisenstein 33-set embeds (since $\omega = \zeta_3 \in \mathbb{Q}(\zeta_n)$ when $3|n$)
Case 1 ($3 \nmid n$, $2 \nmid n$): Only the axis triad exists; trivially colorable
Case 2 ($3 \nmid n$, $2 | n$): Each non-axis ray has exactly one orthogonal partner; explicit coloring constructed
Case 3 ($3 | n$, $2 \nmid n$): Projective collapse reduces 6 permutations to 2 rays; each ray participates in exactly one triad; colorable by triad decomposition

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)
M. Kernaghan, "Graph Universality of CK-31 and the Norm-2 Boundary" (2026)
M. Kernaghan, "Computational Evidence for the Optimality of CK-31" (2026)
M. Kernaghan, "Kochen-Specker Uncolorability in Cyclotomic Fields Requires Exactly 6|n" (2026)
A. Cabello, "Simplest Kochen-Specker Set," Phys. Rev. Lett. (2025), arXiv:2508.07335
A. Cabello, "Simplest Bipartite Perfect Quantum Strategies," Phys. Rev. Lett. 134, 010201 (2024)
S. Trandafir and A. Cabello, rigidity of KS sets (2024)
Liu et al., "Equivalence between FNS, FN, AVN, PT," Phys. Rev. Research 6, L042035 (2024)
F. Li, A. Bright, and V. Ganesh, SAT-based KS search, IJCAI 2024, arXiv:2306.13319

#	Island	Ring	Min KS	Cancellation Identity	Pool Size
1	Integer (CK-31)	\(\mathbb{Z}\)	31	\(1+1=2\)	49
2	Peres	\(\mathbb{Z}[\sqrt{2}]\)	33	\((\sqrt{2})^2 = 2\)	49
3	Eisenstein	\(\mathbb{Z}[\omega]\)	33	\(1+\omega+\omega^2 = 0\)	57
4	\(\mathbb{Z}[\sqrt{-2}]\)	\(\mathbb{Z}[\sqrt{-2}]\)	33	$	\sqrt{-2}
5	Heegner-7	\(\mathbb{Z}[(1+\sqrt{-7})/2]\)	43	\(\alpha \cdot \bar{\alpha} = 2\)	145
6	Golden	\(\mathbb{Z}[\varphi]\)	52	\(\varphi^2 = \varphi + 1\)	205