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written with AI assistance, refereed by an AI agent in two passes and fact-checked in a separate AI pass; the note has not been peer reviewed.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.
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.
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.
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.
We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q , containing a square, into its factors. Illustrative low dimensional examples are the quartit ( q = 4) and two-qubit ( q = 2 2 ) systems, the octit ( q = 8), qubit/quartit ( q = 2 × 4) and three-qubit ( q = 2 3 ) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12,..., one defines a bijection between the σ ( q ) maximal commuting sets [with σ[ q )the sum of divisors of q ] of Pauli observables and the maximal submodules of the modular ring q 2 , that arrange into the projective line P 1 ( q ) and a independent set of size σ( q ) − ψ( q ) [with ψ( q ) the Dedekind psi function]. In the multiple qudit case, e.g. q = 2 2 , 2 3 , 3 2 ,..., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ (2,2) (if q = 2 2 ) and GQ (3,3) (if q = 3 2 ). More precisely, in dimension p n ( p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field F p . In this space, one makes use of the commutator to define a symplectic polar space W 2 n −1 (p) of cardinality σ ( p 2 n −1 ), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W 2 n −1 ( p ) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ ( p 2 N −1 ). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.
In the intriguing realm of quantum physics, the concept of contextuality raises interesting questions about the nature of quantum systems. Since the commutation relation in the N-qubits Pauli group corresponds to the colinearity relation in the symplectic polar space W(2N − 1, 2), we study subgeometries of this space in order to discover new contextual quantum configurations. We have developed algorithms and a corresponding software in C language, that can effectively determine quantum contextuality and quantify it, by charting these symplectic polar spaces of various ranks. This tool allowed us to uncover new insights that go beyond the achievements of earlier work, such as that of de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022). Our study spans across these geometric structures of ranks from two to seven, offering significant findings.In my talk I will explicitly illustrate the concept of quantum contextuality on certain wellknown finite-geometric configurations living in W(5,2), like the doily (and its two-spread), an elliptic quadric (and its spread of lines) and a hyperbolic quadric (and its associated Heawoodgraph); I will also consider configurations whose contexts are all the subspaces with a given dimension k < N.
Given a (2N-1)-dimensional projective space over GF(2), PG (2N-1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG (N-1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG (N-1, 4). Under such mapping, a nondegenerate quadric surface of the former space has for its image a nonsingular Hermitian variety in the latter space, this quadric being hyperbolic or elliptic in dependence on N being even or odd, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric Pauli operators into a new perspective. The N = 4 case is taken to illustrate the issue, due to its link with the so-called black-hole/qubit correspondence.
A ``magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space W(7,2) of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG(3,2)) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG(3,2)s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.