What question did this study set out to answer?

The aim is to derive hadronic observables using a new framework that treats nucleons as Hopf vortices without free parameters.

April 14, 2026Open Access

Hadronic Structure from Hopf Vortex Topology: Magnetic Moment Ratio, Deuteron Binding Energy, Effective Radius, and the Hadronic Shell as a Galactic-Scale Analogue

Key Points

The aim is to derive hadronic observables using a new framework that treats nucleons as Hopf vortices without free parameters.
Developed the Primordial Energy Wave framework to model nucleons as Hopf vortices.
Calculated nucleon magnetic moment ratio, binding energy, and effective radius from first principles.
Introduced a hadronic shell correction by interpreting residuals as a distributed amplitude field.
Magnetic moment ratio μn/μp = −2/3 shows 97.3% agreement, with a 2.7% residual.
Deuteron binding energy calculated as 2.2246 MeV has 99.987% agreement.
Effective radius reff(D) = 1.755 fm shows 99.97% agreement.

Abstract

The Primordial Energy Wave (PEW) framework models nucleons as Hopf vortices on a torus with R/r = φ (golden ratio), with ξp = 0.367 fm. This article derives three hadronic observables from first principles, with no free parameters, and introduces a fourth contribution (new in v2): (1) μn/μp = −2/3 A★★★ from SU(2)isospin. Agreement: 97.3%. Residual 2.7%. (2) BD = 3π/φ³ = 2.2246 MeV B★★ from Chern-Simons action on S³. Agreement: 99.987% (0.3 keV). (3) reff(D) = ξp·φ⁴ = 1.755 fm B★★ from Bethe-Peierls + OPEP tensor. Agreement: 99.97%. (4) The hadronic shell correction B★★: the three residuals are interpreted as the contribution of the distributed amplitude field Ã²(r) surrounding each nucleon vortex — the hadronic shell — in direct analogy with the galactic operculum shell of Article 7. The shell fraction δshell ~ 1/κ²p ≈ 4% provides a unified PEW explanation of the 2.7% magnetic moment discrepancy and the sub-per-cent deuteron residuals. (5) The universal shape hierarchy B★★: sphere (J→0, stars) → Hopf torus (hadronic J, nucleons) → spiral operculum (galactic J, galaxies), governed by R = J²/(GM³r). Numerical check: RSun ≈ 2×10⁻⁶ ≪ 1 (sphere ✓); Rproton ≈ 2×10¹⁹ ≫ 1 (torus ✓). MSC: 81V35 · 81T13 · 22E70 · 85A15 PACS: 21.10.Dr · 21.45.-v · 14.20.Dh · 21.30.-x · 98.35.Df

Hadronic Structure from Hopf Vortex Topology: Magnetic Moment Ratio, Deuteron Binding Energy, Effective Radius, and the Hadronic Shell as a Galactic-Scale Analogue

Key Points

Abstract

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