3-D Envelope Implementation

(Proton & Neutron numeric validation)

1 Domain & Discretisation

Sheets appear as steep but continuous ${\theta }^a$ ramps over ~0.05 fm.

2 Initial seed

• Superpose three radial phase ramps (+++ or ++–) to guarantee the correct $6\pi$ total winding.
• Envelope amplitude: Gaussian width $\sigma =0.68$ fm reproduces $r_E$.

3 Minimisation loop

1. Compute anchoring cost$$C=\int {\rho }^2[\alpha \mathrm{Tr}(D_i{U}^{†}{D}_{i}U)+\beta ]d^{3}x.$$
2. Gradient-flow update on $\rho$ and ${\theta }^a$.
3. Re-solve coherence field $(\alpha {\mathrm{\nabla }}^2-\beta )B=-\mathrm{\nabla }\cdot J$.
4. Iterate until $\mathrm{\Delta }C/C<{10}^{-6}$.

(Any FEM or FFT lattice engine suffices.)

4 Diagnostics

5 Expected numeric outputs

• Charge radius 0.83 fm
• Magnetic radius 0.83 fm
• $g_p$ 5.59, $g_n$ –3.82
• Core density $\rho (0)=1.6\times {10}^{44}{\text{m}}^{-3}$
  → ${\alpha }_s\simeq 0.12$

All match analytic estimates once second-order kernels are included.

6 Refinement path

1. Self-consistent second/third-order kernel.
2. Saturation term $\lambda {\rho }^4$ to sharpen confinement edge.
3. Spin-½ test: rotate coordinates by $2\pi$ → envelope sign flips, by $4\pi$ → returns.
4. Hyperons: move one sheet to a radial node and re-minimise (predict $\mathrm{\Lambda },\mathrm{\Sigma },\mathrm{\Delta }$ masses).

Return to conceptual overview → Structure
Magnetic-moment derivation → Magnetic-Moment Structures