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A section–square law for Pauli quadrics, and an independently regenerated proof that the four-qubit elliptic quadric has contextuality degree 315

Brent W. Jonah

Published Oct 20, 2026• 4 citations
Publisher

Abstract

We prove, computer-assisted, that every elliptic quadric Q−(7,2) of the four-qubit symplectic polar space W(7,2) has contextuality degree 315. Hence d(W(7,2)) = 1575, closing the published bracket 1071 ≤ d ≤ 1575 (Muller et al., J. Phys. A 58 (2025) 215302), and every five-qubit hyperbolic quadric has d(Q+(9,2)) = 6975. These results settle the first open problem of Li, Gupta and Ganesh (arXiv:2610.26465) and answer their question whether d(Q+(9,2)) = 4743 in the negative. The optimal assignments of Q−(7,2) (391,680 gauge classes), W(7,2) (1,175,040) and Q+(9,2) (2,350,080) each form a single orbit of the symmetry group. For the hyperbolic quadric Q+(7,2), the degree 315 and the 240 optimal classes (one orbit) were first proved by Li, Gupta and Ganesh; our independent count agrees. The note also proves a section–square law: for a quadric Q of the n-qubit space and a point d off it, the completion problem of a section carrier on Q ∩ d⊥ ≅ W(2n−3,2) is governed by the anticommutator action of the (n−1)-qubit carrier operator on the exterior square (elliptic Q) or symmetric square (hyperbolic Q), with spectrum {(μi + μj)/2}. At four qubits this explains the two tight carriers of Q−(7,2): the Hoggar carrier (spectrum {21, −3^7}; its optimal completions are exactly the 144 sign vectors of one eigenspace) and a second carrier with characteristic polynomial (t+3)^5 (t^3 − 15t^2 − 117t + 579). The proof is regenerated here with new code in intrinsic three-qubit coordinates, and every input is shipped. The package contains the exhaustive census program (C), 42 exact lower-bound certificates checked by integer Bareiss elimination, a standard-library Python verifier that rebuilds the Pauli algebra from 2×2 matrices (ALL CHECKS PASS in seconds; 13 single-input mutations rejected), the classification programs, every run record, and the report of an independent re-implementation (a separate AI agent writing its own code from the definitions) that reproduced the census, every completion minimum and all 42 certificates. A proof log for the census step (LRAT, checked with the formally verified checker cake_lpr) was produced separately and is not part of this record. Computer-assisted; not formally verified; not peer reviewed. Computations and drafting were carried out with an AI assistant (Claude, Anthropic) under the author's direction.

DOI
10.5281/zenodo.23510633

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Answers to two open problems of Li, Gupta and Ganesh on multi-qubit contextuality degrees

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Li, Gupta and Ganesh (arXiv:2610.26465) determined the contextuality degree of the four-qubit hyperbolic quadric Q+(7,2): it is 315. They asked for the degrees of the elliptic quadric Q-(7,2) and of W(7,2), and whether the five-qubit hyperbolic quadric Q+(9,2) has degree 4743. This note proves, with computer assistance, that every elliptic quadric Q-(7,2) has degree 315, hence d(W(7,2)) = 1575. A short argument by hand shows that each hyperplane section of Q+(9,2) by a point off the quadric is a sign-consistent copy of W(7,2). Together with an explicit assignment this gives d(Q+(9,2)) = 6975, so the answer to the five-qubit question is negative. The resulting noncontextuality inequalities have classical-to-quantum ratio 7/17. The underlying regenerated proof, with code and certificates, is Zenodo record 10.5281/zenodo.23510634. Not peer reviewed; the computer-assisted parts and their checks are described in Section 4 of the note.

•Oct 20, 2026•2 citations

The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315

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The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual $\pm1$ assignment violates. For the four-qubit hyperbolic quadric $\mathcal{Q}^{+}(7,2)$, with $135$ points and $1575$ lines as contexts, Muller et al. conjectured from heuristic searches that the degree is $315$ and that the violated contexts of every optimal assignment form a dual polar space $DW(5,2)$. We prove both conjectures. The lower bound has a short proof by hand: every doily ($GQ(2,2)$) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives $1575/5=315$. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly $240$ optimal violated sets, one orbit of $\mathrm{O}^+(8,2)$, each isomorphic to $DW(5,2)$. The same argument gives the first lower bounds $\lceil L/5\rceil$ for all quadrics with $L$ lines in $N\ge4$ qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.

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The reach of the section-square law: hyperplane completions of symplectic and quadric Pauli configurations

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Let X be a configuration of n-qubit Pauli observables that is closed under products of commuting pairs, such as the symplectic polar space W(2n−1,2) or a nondegenerate quadric. A hyperplane d⊥ splits X into a section S and the remaining points O, and a ±1 assignment into a carrier σ on S and a completion on O. The section-square law of an earlier preprint by the author (doi:10.5281/zenodo.23510634) states that, for a quadric and d off it, the completion matrix Λσ has spectrum {(μi+μj)/2 : i<j} (elliptic) or {(μi+μj)/2 : i≤j} (hyperbolic), each value twice, where the μi are the eigenvalues of the (n−1)-qubit carrier operator. This note determines the reach of this law for W(2n−1,2), both quadric types and their hyperplane sections. For every such X and every d, Λσ is the matrix of Jordan multiplication by the carrier operator Aσ on the span of the outside Paulis. For n ≥ 3 the hyperplanes fall into 16 cases up to symmetry, besides degenerate ones. In these 16 cases, for real-valued carriers, the spectrum of Λσ is a function of the spectrum of Aσ if and only if the section has no vertex; seven cases qualify, three of them cones governed by the regular representation of Hermm(K), K = C, R, H. In the nine vertex cases the spectrum obeys a graded law of Kronecker-sum type. For ±1-valued carriers the vertex criterion fails at n = 3. Finally, for W(2n−1,2) and both quadric types with n ≥ 3, the product-of-spheres relaxation of a single hyperplane split has the same optimum as the global sphere relaxation. The spectral mechanism (Peirce decomposition of Jordan multiplication; Kronecker sums) is classical. The proofs were written with AI assistance, refereed by an AI agent in two passes and fact-checked in a separate AI pass; the note has not been peer reviewed. Supporting computational data are held by the author; their SHA-256 commitments are listed in Appendix A of the PDF and in SQREACH_COMMITMENTS_20261020.txt.

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Observable-based Kochen-Specker proofs are configurations of multi-qubit Pauli observables grouped into contexts whose products are plus or minus the indentity. Their robustness as state-independent contextuality tests can be measured by the tolerated error per context $\varepsilon = 2d/|H|$, where $d$ is the contextuality degree and $|H|$ the number of contexts. Since the degree depends only on an underlying abstract structure called the hypergram i.e. the pair formed by the context hypergraph and the anticommutation graph, the search for highly contextual proofs can be carried out on them instead, with no reference to qubits or to any particular Pauli labeling. We further exploit this by enumerating anticommutation graphs first, and then by associating to each graph $G$ the single hypergram carrying its entire hypergraph support $HS(G)$, so that exactly one candidate is examined per graph. Applied to the House of Graphs database and to censuses of vertex-transitive graphs on at most 24 vertices, this pipeline recovers the Peres-Mermin square, the doily and the Mermin pentagram, and yields configurations reaching $\varepsilon = 0.707$, against $0.424$ for the previous published record. The best configurations are predominantly those stemming from line graphs and unions of graphs; we explain the former by showing that every perfect matching of a graph is a context of its line graph, which exhibits the Peres-Mermin square and the doily as the first members of two infinite families. We close with finite geometric descriptions of the most striking configurations inside symplectic polar spaces, in terms of ovoids, hyperbolic quadrics and Fano planes.

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Structure theorems for hyperplane completions in Pauli contextuality

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Preprint v0.3, not peer reviewed; the proofs were written with AI assistance and are AI-refereed (same model family), not formally verified. For n-qubit Pauli observables closed under products of commuting pairs we prove the operator identity A^2 = NI + 2K - 4V. For W(2n-1,2) and both quadric types, the spectral lower bound on the contextuality degree is attained exactly when some assignment is violation-regular with a simple (Kramers-simple) top eigenvalue; for n >= 2 this happens exactly for W(5,2), Q+(7,2), Q-(3,2) and Q-(5,2) (three by exact computation). For the four-qubit elliptic quadric, a hyperplane completion violates 378 - <sigma, h(X)> outside lines, h(X) being read off a Gram operator. With computer assistance, an odd count forces Gram rank 8, and Gram rank 2 forces counts divisible by 4. That every completion minimum is a multiple of 4 is false (computer-certified counterexamples, not proof-checked). Data are committed by SHA-256.

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Results digest, 20 October 2026: multi-qubit contextuality degrees, Pauli-flat states and certification

Brent W. Jonah

A dated record of results from the author's research program on the contextuality degree of multi-qubit Pauli configurations, obtained on 20 October 2026. It gives statements and status labels only: lower bounds d(Q-(9,2)) ≥ 6,510 and d(W(9,2)) ≥ 28,830 (computer-assisted; not refereed); a characterization of the optimal assignments of Q+(7,2) as sign patterns of real Pauli-flat states, and a route to the Li-Gupta-Ganesh classification of its 240 optimal sets without computer enumeration (conditional on two cited classification results); a certification update for the four-qubit chain of record 10.5281/zenodo.23510634. Proofs, code and data are held privately. The record includes SHA-256 commitments to the evidence files and to a sealed document stating four further results. Not peer reviewed.

•Oct 20, 2026•0 citations
We prove, computer-assisted, that every elliptic quadric Q−(7,2) of the four-qubit symplectic polar space W(7,2) has contextuality degree 315. Hence d(W(7,2)) = 1575, closing the published bracket 1071 ≤ d ≤ 1575 (Muller et al., J. Phys. A 58 (2025) 215302), and every five-qubit hyperbolic quadric has d(Q+(9,2)) = 6975. These results settle the first open problem of Li, Gupta and Ganesh (arXiv:2610.26465) and answer their question whether d(Q+(9,2)) = 4743 in the negative. The optimal assignments of Q−(7,2) (391,680 gauge classes), W(7,2) (1,175,040) and Q+(9,2) (2,350,080) each form a single orbit of the symmetry group. For the hyperbolic quadric Q+(7,2), the degree 315 and the 240 optimal classes (one orbit) were first proved by Li, Gupta and Ganesh; our independent count agrees. The note also proves a section–square law: for a quadric Q of the n-qubit space and a point d off it, the completion problem of a section carrier on Q ∩ d⊥ ≅ W(2n−3,2) is governed by the anticommutator action of the (n−1)-qubit carrier operator on the exterior square (elliptic Q) or symmetric square (hyperbolic Q), with spectrum {(μi + μj)/2}. At four qubits this explains the two tight carriers of Q−(7,2): the Hoggar carrier (spectrum {21, −3^7}; its optimal completions are exactly the 144 sign vectors of one eigenspace) and a second carrier with characteristic polynomial (t+3)^5 (t^3 − 15t^2 − 117t + 579). The proof is regenerated here with new code in intrinsic three-qubit coordinates, and every input is shipped. The package contains the exhaustive census program (C), 42 exact lower-bound certificates checked by integer Bareiss elimination, a standard-library Python verifier that rebuilds the Pauli algebra from 2×2 matrices (ALL CHECKS PASS in seconds; 13 single-input mutations rejected), the classification programs, every run record, and the report of an independent re-implementation (a separate AI agent writing its own code from the definitions) that reproduced the census, every completion minimum and all 42 certificates. A proof log for the census step (LRAT, checked with the formally verified checker cake_lpr) was produced separately and is not part of this record. Computer-assisted; not formally verified; not peer reviewed. Computations and drafting were carried out with an AI assistant (Claude, Anthropic) under the author's direction.
Li, Gupta and Ganesh (arXiv:2610.26465) determined the contextuality degree of the four-qubit hyperbolic quadric Q+(7,2): it is 315. They asked for the degrees of the elliptic quadric Q-(7,2) and of W(7,2), and whether the five-qubit hyperbolic quadric Q+(9,2) has degree 4743. This note proves, with computer assistance, that every elliptic quadric Q-(7,2) has degree 315, hence d(W(7,2)) = 1575. A short argument by hand shows that each hyperplane section of Q+(9,2) by a point off the quadric is a sign-consistent copy of W(7,2). Together with an explicit assignment this gives d(Q+(9,2)) = 6975, so the answer to the five-qubit question is negative. The resulting noncontextuality inequalities have classical-to-quantum ratio 7/17. The underlying regenerated proof, with code and certificates, is Zenodo record 10.5281/zenodo.23510634. Not peer reviewed; the computer-assisted parts and their checks are described in Section 4 of the note.
The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual ±1\pm1±1 assignment violates. For the four-qubit hyperbolic quadric Q+(7,2)\mathcal{Q}^{+}(7,2)Q+(7,2), with 135135135 points and 157515751575 lines as contexts, Muller et al. conjectured from heuristic searches that the degree is 315315315 and that the violated contexts of every optimal assignment form a dual polar space DW(5,2)DW(5,2)DW(5,2). We prove both conjectures. The lower bound has a short proof by hand: every doily (GQ(2,2)GQ(2,2)GQ(2,2)) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives 1575/5=3151575/5=3151575/5=315. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly 240240240 optimal violated sets, one orbit of O+(8,2)\mathrm{O}^+(8,2)O+(8,2), each isomorphic to DW(5,2)DW(5,2)DW(5,2). The same argument gives the first lower bounds ⌈L/5⌉\lceil L/5\rceil⌈L/5⌉ for all quadrics with LLL lines in N≥4N\ge4N≥4 qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.
Let X be a configuration of n-qubit Pauli observables that is closed under products of commuting pairs, such as the symplectic polar space W(2n−1,2) or a nondegenerate quadric. A hyperplane d⊥ splits X into a section S and the remaining points O, and a ±1 assignment into a carrier σ on S and a completion on O. The section-square law of an earlier preprint by the author (doi:10.5281/zenodo.23510634) states that, for a quadric and d off it, the completion matrix Λσ has spectrum {(μi+μj)/2 : i<j} (elliptic) or {(μi+μj)/2 : i≤j} (hyperbolic), each value twice, where the μi are the eigenvalues of the (n−1)-qubit carrier operator. This note determines the reach of this law for W(2n−1,2), both quadric types and their hyperplane sections. For every such X and every d, Λσ is the matrix of Jordan multiplication by the carrier operator Aσ on the span of the outside Paulis. For n ≥ 3 the hyperplanes fall into 16 cases up to symmetry, besides degenerate ones. In these 16 cases, for real-valued carriers, the spectrum of Λσ is a function of the spectrum of Aσ if and only if the section has no vertex; seven cases qualify, three of them cones governed by the regular representation of Hermm(K), K = C, R, H. In the nine vertex cases the spectrum obeys a graded law of Kronecker-sum type. For ±1-valued carriers the vertex criterion fails at n = 3. Finally, for W(2n−1,2) and both quadric types with n ≥ 3, the product-of-spheres relaxation of a single hyperplane split has the same optimum as the global sphere relaxation. The spectral mechanism (Peirce decomposition of Jordan multiplication; Kronecker sums) is classical. The proofs were written with AI assistance, refereed by an AI agent in two passes and fact-checked in a separate AI pass; the note has not been peer reviewed. Supporting computational data are held by the author; their SHA-256 commitments are listed in Appendix A of the PDF and in SQREACH_COMMITMENTS_20261020.txt.
Observable-based Kochen-Specker proofs are configurations of multi-qubit Pauli observables grouped into contexts whose products are plus or minus the indentity. Their robustness as state-independent contextuality tests can be measured by the tolerated error per context ε=2d/∣H∣\varepsilon = 2d/|H|ε=2d/∣H∣, where ddd is the contextuality degree and ∣H∣|H|∣H∣ the number of contexts. Since the degree depends only on an underlying abstract structure called the hypergram i.e. the pair formed by the context hypergraph and the anticommutation graph, the search for highly contextual proofs can be carried out on them instead, with no reference to qubits or to any particular Pauli labeling. We further exploit this by enumerating anticommutation graphs first, and then by associating to each graph GGG the single hypergram carrying its entire hypergraph support HS(G)HS(G)HS(G), so that exactly one candidate is examined per graph. Applied to the House of Graphs database and to censuses of vertex-transitive graphs on at most 24 vertices, this pipeline recovers the Peres-Mermin square, the doily and the Mermin pentagram, and yields configurations reaching ε=0.707\varepsilon = 0.707ε=0.707, against 0.4240.4240.424 for the previous published record. The best configurations are predominantly those stemming from line graphs and unions of graphs; we explain the former by showing that every perfect matching of a graph is a context of its line graph, which exhibits the Peres-Mermin square and the doily as the first members of two infinite families. We close with finite geometric descriptions of the most striking configurations inside symplectic polar spaces, in terms of ovoids, hyperbolic quadrics and Fano planes.
Preprint v0.3, not peer reviewed; the proofs were written with AI assistance and are AI-refereed (same model family), not formally verified. For n-qubit Pauli observables closed under products of commuting pairs we prove the operator identity A^2 = NI + 2K - 4V. For W(2n-1,2) and both quadric types, the spectral lower bound on the contextuality degree is attained exactly when some assignment is violation-regular with a simple (Kramers-simple) top eigenvalue; for n >= 2 this happens exactly for W(5,2), Q+(7,2), Q-(3,2) and Q-(5,2) (three by exact computation). For the four-qubit elliptic quadric, a hyperplane completion violates 378 - <sigma, h(X)> outside lines, h(X) being read off a Gram operator. With computer assistance, an odd count forces Gram rank 8, and Gram rank 2 forces counts divisible by 4. That every completion minimum is a multiple of 4 is false (computer-certified counterexamples, not proof-checked). Data are committed by SHA-256.
A dated record of results from the author's research program on the contextuality degree of multi-qubit Pauli configurations, obtained on 20 October 2026. It gives statements and status labels only: lower bounds d(Q-(9,2)) ≥ 6,510 and d(W(9,2)) ≥ 28,830 (computer-assisted; not refereed); a characterization of the optimal assignments of Q+(7,2) as sign patterns of real Pauli-flat states, and a route to the Li-Gupta-Ganesh classification of its 240 optimal sets without computer enumeration (conditional on two cited classification results); a certification update for the four-qubit chain of record 10.5281/zenodo.23510634. Proofs, code and data are held privately. The record includes SHA-256 commitments to the evidence files and to a sealed document stating four further results. Not peer reviewed.