$G_2$-Symmetric Stabilisation of Magnetic Hopfions: A Fano-Constraint Conjecture and Topological Ground State Computation
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2026
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| author | Buckley, Ian R. C. |
| author_facet | Buckley, Ian R. C. |
| contents | <p><strong>Magnetic</strong> <strong>Hopfions</strong> 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.</p> <p>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 <strong>Fano-Line Closure Theorem</strong> and the <strong>Fano-Fisher eigenvalue bound </strong>together rigidly constrain which field configurations are energetically accessible, forcing stable Hopfion solutions to crystallise at lower frustration thresholds than in associative substrates.</p> <p>The paper develops the <strong>$G_2$-equivariant extension of the Skyrme-Faddeev energy functional,</strong> 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 <strong>Topological Resonance Synthesis (TRS)</strong> reframes Hopfion formation as a constraint-satisfaction problem, bypassing time-domain PDE integration and computing topological ground states directly via the <strong>Maslov-Gibbs Einsum</strong> operator. A speculative appendix connects the $G_2$ Hopfion substrate to the <strong>731-RPU </strong>hardware architecture.</p> <p>The paper is part of the <strong>Adelic Simplicial Architecture (ASA)</strong> programme, Portfolio B (Mathematical Physics), and is a companion to the <strong>Self-Dual $G_2$ Architecture</strong> paper (Paper 271, doi:10.5281/zenodo.20101634) and the <strong>Fano-Fisher Decomposition Theorem</strong> (Paper 221, doi:10.5281/zenodo.20076498).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20127777 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | $G_2$-Symmetric Stabilisation of Magnetic Hopfions: A Fano-Constraint Conjecture and Topological Ground State Computation Buckley, Ian R. C. <p><strong>Magnetic</strong> <strong>Hopfions</strong> 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.</p> <p>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 <strong>Fano-Line Closure Theorem</strong> and the <strong>Fano-Fisher eigenvalue bound </strong>together rigidly constrain which field configurations are energetically accessible, forcing stable Hopfion solutions to crystallise at lower frustration thresholds than in associative substrates.</p> <p>The paper develops the <strong>$G_2$-equivariant extension of the Skyrme-Faddeev energy functional,</strong> 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 <strong>Topological Resonance Synthesis (TRS)</strong> reframes Hopfion formation as a constraint-satisfaction problem, bypassing time-domain PDE integration and computing topological ground states directly via the <strong>Maslov-Gibbs Einsum</strong> operator. A speculative appendix connects the $G_2$ Hopfion substrate to the <strong>731-RPU </strong>hardware architecture.</p> <p>The paper is part of the <strong>Adelic Simplicial Architecture (ASA)</strong> programme, Portfolio B (Mathematical Physics), and is a companion to the <strong>Self-Dual $G_2$ Architecture</strong> paper (Paper 271, doi:10.5281/zenodo.20101634) and the <strong>Fano-Fisher Decomposition Theorem</strong> (Paper 221, doi:10.5281/zenodo.20076498).</p> |
| title | $G_2$-Symmetric Stabilisation of Magnetic Hopfions: A Fano-Constraint Conjecture and Topological Ground State Computation |
| url | https://doi.org/10.5281/zenodo.20127777 |