Research
The Algebraic Islands Programme
This section documents original research (2026) on the algebraic classification of Kochen-Specker sets in dimension three. The central discovery is that KS-uncolorable ray pools in \(\mathbb{R}^3\) and \(\mathbb{C}^3\) cluster into exactly six algebraic islands (six distinct graph families), each characterized by number rings whose generators satisfy norm-\(\leq 2\) cancellation identities. Multiple rings can produce the same island when they yield isomorphic orthogonality graphs.
Central Thesis
Generator norm \(\leq 2\) is the controlling invariant for KS-uncolorability in dimension 3.
The integer identity \(1+1=2\), the Peres identity \((\sqrt{2})^2 = 2\), the Eisenstein identity \(1+\omega+\omega^2 = 0\), and the Heegner-7 identity \(\alpha \cdot \bar{\alpha} = 2\) are all manifestations of the same constraint: the alphabet must support cancellation identities that produce exact orthogonalities among triples of vectors. At generator norm \(\geq 3\), such cancellations become impossible.
Research Pages
The Six Algebraic Islands
The central classification result. Documents all six islands with their rings, minimal KS sets, cancellation identities, rigidity properties, and BPQS numbers. Includes the sub-31 optimality evidence, graph universality of CK-31, and the MUS landscape.
Papers
Listing of all papers produced by this research programme: one main paper (~36 pages) and four PRL-style letters, each addressing a distinct aspect of the classification.
New KS Set Constructions
The algebraic islands programme discovered four new KS sets:
| Construction | Vectors | Ring | Status |
|---|---|---|---|
| Eisenstein-33 | 33 | \(\mathbb{Z}[\omega]\) | Independently found by Cabello (2025) |
| Heegner-7 | 43 | \(\mathbb{Z}[(1+\sqrt{-7})/2]\) | Genuinely new |
| Golden-52 | 52 | \(\mathbb{Z}[\varphi]\) | Genuinely new |
| \(\mathbb{Z}[\sqrt{-2}]\)-33 | 33 | \(\mathbb{Z}[\sqrt{-2}]\) | New realization of Peres graph |
Key Results Summary
- Six algebraic islands — complete classification of norm-\(\leq 2\) KS-producing rings
- Sub-31 optimality — OCUS proof that no \(\leq 30\)-vector KS set exists in the integer pool; six independent strategies confirm CK-31 as minimal
- Graph universality — all 31-vertex KS sets (integer, rational, mixed-field, group-orbit) share the same orthogonality graph
- 6|n cyclotomic theorem — KS-uncolorability in cyclotomic fields \(\mathbb{Q}(\zeta_n)\) requires exactly \(6 | n\) (complete algebraic proof)
- Rigidity classification — CK-31, Eisenstein-33, Heegner-7, Golden-52 are rigid; Peres-33 and \(\mathbb{Z}[\sqrt{-2}]\)-33 are flex (the flex belongs to the graph)
- Merge saturation — all six minimal KS sets are merge-saturated (100% preservation under non-orthogonal pair merges)