$G_2$-Symmetric Stabilisation of Magnetic Hopfions: A Fano-Constraint Conjecture and Topological Ground State Computation

Magnetic Hopfions are genuinely three-dimensional topological solitons classified by the Hopf invariant $Q \in \mathbb{Z}$, the integer invariant of maps $S^3 \to S^2$. Their experimental stabilisation in conventional $SO(3)$-symmetric chiral magnets is notoriously difficult: the absence of a native algebraic obstruction to continuous deformation allows Hopfion strings to unwind under small perturbations.

This paper conjectures that elevating the magnetic substrate to one with $G_2$ symmetry — where the order parameter lives in the imaginary octonions and the energy functional must respect $G_2 = \mathrm{Aut}(\mathbb{O})$ — provides precisely such an obstruction through the Fano incidence structure. The Fano-Line Closure Theorem and the Fano-Fisher eigenvalue bound together rigidly constrain which field configurations are energetically accessible, forcing stable Hopfion solutions to crystallise at lower frustration thresholds than in associative substrates.

The paper develops the $G_2$-equivariant extension of the Skyrme-Faddeev energy functional, advancing through the Hopf fibration tower $S^1 \to S^3 \to S^7 \to S^{15}$ to the octonionic level. It further sketches how Topological Resonance Synthesis (TRS) reframes Hopfion formation as a constraint-satisfaction problem, bypassing time-domain PDE integration and computing topological ground states directly via the Maslov-Gibbs Einsum operator. A speculative appendix connects the $G_2$ Hopfion substrate to the 731-RPU hardware architecture.

The paper is part of the Adelic Simplicial Architecture (ASA) programme, Portfolio B (Mathematical Physics), and is a companion to the Self-Dual $G_2$ Architecture paper (Paper 271, doi:10.5281/zenodo.20101634) and the Fano-Fisher Decomposition Theorem (Paper 221, doi:10.5281/zenodo.20076498).