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.
The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual ±1 assignment violates. For the four-qubit hyperbolic quadric 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 O+(8,2), each isomorphic to DW(5,2). The same argument gives the first lower bounds ⌈L/5⌉ for all quadrics with L lines in N≥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.
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.
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∣, 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 ε=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.
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.
Abstract We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.
We develop a geometric framework for constructing and organizing Kochen--Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types F4 and H4, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.