Finite Pauli Geometry and the Local Polytope: Exhaustive Enumeration of Contextuality Structures

Abstract

Exhaustive machine enumerations of the contextuality structures carried by the two- and three-qubit Pauli groups under their identification with the finite projective spaces PG(3,2) and PG(5,2), together with exact vertex and facet enumerations of the corresponding Bell local polytopes. Method. All counts come from exhaustive search in exact arithmetic: Gaussian-integer matrix representations for Pauli operators, F2 linear algebra for the geometry, and integer double-description for the polytopes. Zero floating-point tolerance in the combinatorial results. Commutation was verified against symplectic orthogonality on all 105 two-qubit pairs and all 1953 three-qubit pairs, with no exceptions. Principal results. PG(3,2): 15 isotropic lines, 3 negative; 720 ordered Mermin-Peres magic squares on 10 grids, forming a single orbit under the full symmetry group; 0 pentagrams. Best non-contextual model achieves 5 of 6 contexts. PG(5,2): 315 isotropic lines, 135 Lagrangians (7-point Fano planes), 12096 Mermin pentagrams, 3360 magic squares. Minimum contextuality witness size is exactly 6; all 945 weight-4 dependencies lie inside a single Lagrangian and are jointly measurable. Resolution of a recorded anomaly: the absence of four-dimensional totally isotropic subspaces of PG(5,2) is a theorem, not a search failure. Non-degeneracy gives dim W-perp = 6 - dim W, so total isotropy forces dim W <= 3. Lagrangians at three qubits are planes of 7 points, not the 15-point ambient space of the two-qubit case. The GHZ quadruple {XYY, YXY, YYX, XXX} spans a Lagrangian which is the GHZ stabilizer group, and is the complement of the line {ZZI, ZIZ, IZZ} within that Fano plane. 0 of 64 local assignments reproduce the quantum predictions. CHSH local polytope: 8 vertices, 16 facets (the 8 trivial facets are not redundant). Tsirelson's bound derived analytically and dimension-independently from S^2 = 4I - [A1,A2] tensor [B1,B2]. Strict nesting L ⊂ Q ⊂ NS with 2 < 2√2 < 4. I3322: 684 facets in exactly 3 orbits under a relabelling group of order 4608. Mermin n-party: local bound exactly 1, quantum value 2^((n-1)/2); fitted exponent slope 0.500000000000. Flagged openly. The 3360 three-qubit magic squares and the up-to-symmetry 720 -> 10 -> 1 reduction have no located published counterpart. Four independent optimisers return 0.250 for the I3322 quantum value against a literature supremum of 0.2508753, which reportedly requires unbounded dimension; the disagreement is one-sided and does not affect the qualitative conclusion. Errors found and corrected during the work, including a Gaussian-elimination bug that had falsely reported all three three-qubit context families as contextual, are documented rather than silently removed. An accompanying conceptual brief reviews EPR, Bell, and the loophole-closure experiments from primary sources, and corrects several common misstatements — in particular that Bell's theorem rules out local hidden variables, not hidden variables as such.