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The largest noncontextual subset of the two-qubit 60-105 system has size ninety-three

Brent W. Jonah

Published Jul 7, 2026• 1 citation
Publisher

Abstract

The 60 complex rays and 105 orthonormal bases arising from the two-qubit Pauli group form the Kochen-Specker system of Waegell and Aravind (2011), recently named by Aravind, Burton, Nunez Ponasso and Richter (2026) as a natural target for symmetry-based analysis. This note determines exactly the largest number of the 105 bases satisfiable by a single {0,1} ray assignment: the answer is 93, so at least twelve bases must fail. The optimum has clean structure: an attaining assignment selects exactly one state from each of the fifteen maximal abelian subgroups of the Pauli group, so all fifteen stabilizer bases are satisfied and all twelve failures are hybrid bases. The bound is certified by two independent integer-programming solvers (CBC proven optimum; HiGHS optimality and infeasibility of 94), and the attaining assignment is re-verified in exact Gaussian-integer arithmetic. The tight inequality is 93 versus the quantum value 105 (ratio 35/27 in the +/-1 normalization). This is the exact two-qubit analog of the value 65 of 75 determined by the author for the 600-cell system. Verification: run python3 verify2q.py with rays60c.txt in the same directory; it rebuilds the system from the fixed data and re-verifies the anchors (60 rays, 105 bases, seven per ray), the certificate count of 93, the transversal property, and the twelve failing bases in seconds, without any solver.

DOI
10.5281/zenodo.21249916

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The largest noncontextual subset of the 600-cell's seventy-five bases has size sixty-five

Brent W. Jonah

The 60 rays and 75 orthonormal bases derived from the vertices of the 600-cell form the well-known Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 75 bases that a single {0,1} ray assignment can satisfy (exactly one ray valued 1 per satisfied basis): the answer is 65, so at least ten bases must fail. This improves the lower bound of nine failures obtained by Waegell, Aravind, Megill and Pavicic (2011) from parity-proof coverings, answers the question they left open, and settles the corresponding open problem restated by Aravind, Burton, Nunez Ponasso and Richter (2026) for the 600-cell. The lower bound is an explicit 17-ray assignment verified in exact Q(sqrt5) arithmetic; the upper bound is certified both by two independent integer-programming solvers and by an explicit family of 150 parity proofs (ten orbits under the triacontagonal C15 symmetry) admitting no nine-element hitting set. The tight noncontextuality inequality is 65 versus the quantum value 75 (ratio 15/11 in the +/-1 normalization). A transfer proposition brackets the analogous 120-cell maximum in [600, 625]. Verification: run python3 verify.py with proofs150.txt in the same directory. It rebuilds the system from the two stated generators and re-verifies the construction anchors, the certificate count of 65, the ten failing bases, and the validity of all 150 parity proofs in seconds, without any solver.

•Jul 7, 2026•1 citation

The largest noncontextual subset of the 120-cell's six hundred seventy-five bases has size six hundred

Brent W. Jonah

The 300 rays and 675 orthonormal bases of the 120-cell form the Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 675 bases satisfiable by a single {0,1} ray assignment (exactly one ray valued 1 per satisfied basis): the answer is 600, so at least seventy-five bases must fail. The proof avoids the intractable full-system optimization entirely: left and right multiplication by the binary icosahedral group partition the rays into five left and five right blocks, each an isometric copy of the 600-cell system; a single 60-ray integer program, proven optimal and transported to all ten blocks by the icosian symmetry, gives the per-block frontier 2*(block-only satisfied) + (doubly satisfied) <= 120; summing over each partition double-counts the doubly embedded bases and collapses the bound to 600. An explicit assignment attains 600. This improves the transfer bracket [600, 625] of DOI 10.5281/zenodo.21249805 to an exact value and closes the third member of the open family of Aravind, Burton, Nunez Ponasso and Richter. Whether parity proofs alone force the bound (as they provably do for the 600-cell and the two-qubit system) is posed as the sharpened open question: the argument gives hitting number h <= 75, and h = 75 remains to be decided. Verification: run python3 verify_max600.py with rays300.txt in the same directory. Structural checks are instant; the two 60-ray integer programs finish in minutes; no full-system optimization is used anywhere.

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The maximum exactly-one satisfaction of the E8 Kochen–Specker system is 1680

Brent W. Jonah

In the Kochen–Specker system of the E8 root system (120 rays, 2025 orthonormal bases), the largest number of bases containing exactly one selected ray is 1680, so every selection fails at least 345 bases. Every maximizer selects 14 rays whose orthogonality graph is seven disjoint edges; there are exactly 1080 maximizers, one for each antipodal pair of norm-4 vectors z of the E8 lattice (the rays r with |⟨r,z⟩| = 2). The proof combines a symmetry-reduced degree-4 Boolean moment relaxation with exact rational dual certificates, a certificate excluding induced three-vertex paths at the optimum, and a classification in the mod-2 quadratic space of E8. The upper bound, the classification, the certificates and two checking programs were produced with a GPT-based AI assistant in one session; these two programs are not independent review. An AI referee then checked that proof with its own code and reproduced every bound exactly. The certificate data are committed by SHA-256. Status: computer-assisted; exact rational certificates; AI-refereed (same model family as the coordinating assistant); not formally verified; not peer reviewed.

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The 120-cell Kochen-Specker system is not rigid

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Aravind, Burton, Nunez Ponasso and Richter (2026) proved that the 60-ray Kochen-Specker system of the 600-cell is not rigid and conjectured that the 300-ray/675-basis system of the 120-cell is rigid. This note shows it is not. The golden-ring lift phi from Z[alpha]^4 to Z^8 underlying their 600-cell construction preserves orthogonality for arbitrary golden-ring vectors, because the Z^8 inner product of two lifts equals the rational part of their R^4 inner product. Applied to the 300 rays of the 120-cell, the lift is an 8-dimensional rank-one realization satisfying all 4050 orthogonality relations while acquiring 7200 additional ones (exact integer census), so no unitary can map it into the original rays. An explicit integer witness pair is given, and the same computation reproduces the known 600-cell counts (450 orthogonal pairs, 360 extra). The rigidity of Gosset's polytope 4_21, and dimension-preserving rigidity of the 120-cell in R^4, remain open. Verification: run python3 verify_rigidity.py with rays300.txt in the same directory; it re-verifies the identity, the full census (4050 and 7200), and the witness in seconds. The bonus tool solve120.py is a self-contained long-run decider for the still-open exact 120-cell noncontextuality maximum (usage in its header).

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QUANTUM STATES ARISING FROM THE PAULI GROUPS: SYMMETRIES AND PARADOXES

Michel R. P. Planat

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice $BW_n$ ($n=2^m$). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs $v-l$ involving $v$ rays and $l$ $2n$-dimensional bases of $n$-qubits. Specifically, we look at the classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.

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QUANTUM STATES ARISING FROM THE PAULI GROUPS, SYMMETRIES AND PARADOXES

Michel R. P. Planat

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice $BW_n$ ($n=2^m$). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic {\it non-contextual} theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs $v-l$ involving $v$ rays and $l$ $2n$-dimensional bases of $n$-qubits. Specifically, we look at the classes of parity proofs $18-9$ with two qubits (A. Cabello, 1996), $36-11$ with three qubits (M. Kernaghan \& A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.

•Aug 20, 2012•0 citations
The 60 complex rays and 105 orthonormal bases arising from the two-qubit Pauli group form the Kochen-Specker system of Waegell and Aravind (2011), recently named by Aravind, Burton, Nunez Ponasso and Richter (2026) as a natural target for symmetry-based analysis. This note determines exactly the largest number of the 105 bases satisfiable by a single {0,1} ray assignment: the answer is 93, so at least twelve bases must fail. The optimum has clean structure: an attaining assignment selects exactly one state from each of the fifteen maximal abelian subgroups of the Pauli group, so all fifteen stabilizer bases are satisfied and all twelve failures are hybrid bases. The bound is certified by two independent integer-programming solvers (CBC proven optimum; HiGHS optimality and infeasibility of 94), and the attaining assignment is re-verified in exact Gaussian-integer arithmetic. The tight inequality is 93 versus the quantum value 105 (ratio 35/27 in the +/-1 normalization). This is the exact two-qubit analog of the value 65 of 75 determined by the author for the 600-cell system. Verification: run python3 verify2q.py with rays60c.txt in the same directory; it rebuilds the system from the fixed data and re-verifies the anchors (60 rays, 105 bases, seven per ray), the certificate count of 93, the transversal property, and the twelve failing bases in seconds, without any solver.
The 60 rays and 75 orthonormal bases derived from the vertices of the 600-cell form the well-known Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 75 bases that a single {0,1} ray assignment can satisfy (exactly one ray valued 1 per satisfied basis): the answer is 65, so at least ten bases must fail. This improves the lower bound of nine failures obtained by Waegell, Aravind, Megill and Pavicic (2011) from parity-proof coverings, answers the question they left open, and settles the corresponding open problem restated by Aravind, Burton, Nunez Ponasso and Richter (2026) for the 600-cell. The lower bound is an explicit 17-ray assignment verified in exact Q(sqrt5) arithmetic; the upper bound is certified both by two independent integer-programming solvers and by an explicit family of 150 parity proofs (ten orbits under the triacontagonal C15 symmetry) admitting no nine-element hitting set. The tight noncontextuality inequality is 65 versus the quantum value 75 (ratio 15/11 in the +/-1 normalization). A transfer proposition brackets the analogous 120-cell maximum in [600, 625]. Verification: run python3 verify.py with proofs150.txt in the same directory. It rebuilds the system from the two stated generators and re-verifies the construction anchors, the certificate count of 65, the ten failing bases, and the validity of all 150 parity proofs in seconds, without any solver.
The 300 rays and 675 orthonormal bases of the 120-cell form the Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 675 bases satisfiable by a single {0,1} ray assignment (exactly one ray valued 1 per satisfied basis): the answer is 600, so at least seventy-five bases must fail. The proof avoids the intractable full-system optimization entirely: left and right multiplication by the binary icosahedral group partition the rays into five left and five right blocks, each an isometric copy of the 600-cell system; a single 60-ray integer program, proven optimal and transported to all ten blocks by the icosian symmetry, gives the per-block frontier 2*(block-only satisfied) + (doubly satisfied) <= 120; summing over each partition double-counts the doubly embedded bases and collapses the bound to 600. An explicit assignment attains 600. This improves the transfer bracket [600, 625] of DOI 10.5281/zenodo.21249805 to an exact value and closes the third member of the open family of Aravind, Burton, Nunez Ponasso and Richter. Whether parity proofs alone force the bound (as they provably do for the 600-cell and the two-qubit system) is posed as the sharpened open question: the argument gives hitting number h <= 75, and h = 75 remains to be decided. Verification: run python3 verify_max600.py with rays300.txt in the same directory. Structural checks are instant; the two 60-ray integer programs finish in minutes; no full-system optimization is used anywhere.
In the Kochen–Specker system of the E8 root system (120 rays, 2025 orthonormal bases), the largest number of bases containing exactly one selected ray is 1680, so every selection fails at least 345 bases. Every maximizer selects 14 rays whose orthogonality graph is seven disjoint edges; there are exactly 1080 maximizers, one for each antipodal pair of norm-4 vectors z of the E8 lattice (the rays r with |⟨r,z⟩| = 2). The proof combines a symmetry-reduced degree-4 Boolean moment relaxation with exact rational dual certificates, a certificate excluding induced three-vertex paths at the optimum, and a classification in the mod-2 quadratic space of E8. The upper bound, the classification, the certificates and two checking programs were produced with a GPT-based AI assistant in one session; these two programs are not independent review. An AI referee then checked that proof with its own code and reproduced every bound exactly. The certificate data are committed by SHA-256. Status: computer-assisted; exact rational certificates; AI-refereed (same model family as the coordinating assistant); not formally verified; not peer reviewed.
Aravind, Burton, Nunez Ponasso and Richter (2026) proved that the 60-ray Kochen-Specker system of the 600-cell is not rigid and conjectured that the 300-ray/675-basis system of the 120-cell is rigid. This note shows it is not. The golden-ring lift phi from Z[alpha]^4 to Z^8 underlying their 600-cell construction preserves orthogonality for arbitrary golden-ring vectors, because the Z^8 inner product of two lifts equals the rational part of their R^4 inner product. Applied to the 300 rays of the 120-cell, the lift is an 8-dimensional rank-one realization satisfying all 4050 orthogonality relations while acquiring 7200 additional ones (exact integer census), so no unitary can map it into the original rays. An explicit integer witness pair is given, and the same computation reproduces the known 600-cell counts (450 orthogonal pairs, 360 extra). The rigidity of Gosset's polytope 4_21, and dimension-preserving rigidity of the 120-cell in R^4, remain open. Verification: run python3 verify_rigidity.py with rays300.txt in the same directory; it re-verifies the identity, the full census (4050 and 7200), and the witness in seconds. The bonus tool solve120.py is a self-contained long-run decider for the still-open exact 120-cell noncontextuality maximum (usage in its header).
We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice BWnBW_nBWn​ (n=2mn=2^mn=2m). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs v−lv-lv−l involving vvv rays and lll 2n2n2n-dimensional bases of nnn-qubits. Specifically, we look at the classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.
We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice BWnBW_nBWn​ (n=2mn=2^mn=2m). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic {\it non-contextual} theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs v−lv-lv−l involving vvv rays and lll 2n2n2n-dimensional bases of nnn-qubits. Specifically, we look at the classes of parity proofs 18−918-918−9 with two qubits (A. Cabello, 1996), 36−1136-1136−11 with three qubits (M. Kernaghan \& A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.