Hardy's Paradox and the Fano Associator: A Geometric Diagnosis of Quantum Contextuality

Abstract

Hardy's paradox is an all-versus-nothing logical contradiction for bipartite quantum systems: three jointly satisfiable conditions on measurement probabilities each force a fourth probability to zero by classical logic, yet quantum mechanics permits that fourth probability to be strictly positive. Unlike Bell inequalities, no statistical averaging is required — the contradiction is exact and logical. This paper presents a five-part numerical study connecting Hardy's paradox to the non-associative geometry of the Fano plane $PG(2,2)$ and the octonion associator $\mathcal{A}(e_i, e_j, e_k) = (e_i e_j)e_k - e_i(e_j e_k)$. Part 1 finds the Hardy state by constrained optimisation (Nelder-Mead, 100 initialisations), confirming all three zero-conditions to within $10^{-6}$ and the impossible event $P(Z^+, Z^+) = 0.10 > 0$ with residual $\sim 10^{-36}$. Part 2 shows that the Fano-SYK ground state restricted to the measurement node pair ${0,1}$ is entangled, with von Neumann entropy $S = 0.8762$ ebits, connecting the Fano dynamical structure to the Hardy contextual structure. Part 3 sweeps a one-parameter entangled state family and demonstrates that the Hardy impossible-event probability and the CHSH witness are correlated (Pearson $r = 0.982$), both vanishing at the separable limit. Part 4 verifies the full octonion associator table: $|\mathcal{A}(e_i,e_j,e_k)| = 0$ for all 7 Fano lines (associative triples) and $|\mathcal{A}| = 2$ for all 28 non-Fano triples, exact to $< 10^{-14}$. Part 5 maps the Hardy measurement settings $Z, X$ to octonion directions $e_0, e_1$, which lie on the unique Fano line ${0,1,3}$, and argues that entanglement forces the bipartite system to sample non-Fano directions in $\mathbb{R}^7 \otimes \mathbb{R}^7$ where no consistent classical truth-value assignment exists. The central claim — stated as the Fano-Contextuality Conjecture — is that a measurement triple admits a consistent joint classical value assignment if and only if the corresponding octonion directions lie on a Fano line ($|\mathcal{A}| = 0$). Hardy contextuality is diagnosed as the consequence of entanglement distributing quantum weight into the non-associative region of measurement space. A formal proof connecting this partition to the Kochen-Specker theorem is identified as the primary open problem. This is the first paper in the ASA Quantum Foundations series (Portfolio F), accompanied by the Spacelike Associator Paradox (doi:10.5281/zenodo.20058013) and the Fano Monogamy Paradox (doi:10.5281/zenodo.20058092). Working paper in the Adelic Simplicial Architecture (ASA) series.