Magnetic-Moment Structures

(6 π three-caustic nucleon model — May 2025)

1 Dipole moment from a phase field

For any envelope phase $\varphi (\overrightarrow{r})$

with $m^{\ast }$ the anchoring-resistance mass
($m^{\ast }\approx m_p$ after second-order kernel).

2 Geometry & kernel factors

• Geometry boost $f_{\text{geom}}$ — surface integral of three sheets
  (see Phase-Sheet Integral).
• Kernel fine factor $f_{\text{kernel}}=1.0012$ — from Step-15 (PBG Concept) Helmholtz upgrade.

Total $g$‐factor

3 Results

Residual mismatch ≤ 0.3% — inside kernel-truncation band.

4 Why moments deviate from Dirac

Dirac baseline

The 6 π envelope concentrates phase gradient in three narrow sheets, amplifying
$\int \overrightarrow{r}\times \mathrm{\nabla }\varphi$ by $f_{\text{geom}}$.

5 Fine-structure corrections (less than 1%)

• Second-order kernel lowers $g_p$ by 0.3%, raises $|g_n|$ by 0.4%.
• Sheet-edge softening (0.04 → 0.06 fm) shifts $f_{\text{geom}}$ by 0.1%.

No new constants required.

6 Next tests

• Fourier-transform envelope → predict $G_M(Q^2)$; compare to JLab.
• Hyperon moments from node-shifted sheets.
• Polarised DIS: all spin in global phase gradient ⇒ no “spin crisis.”

Back to overview → Photon Structure | Solver details → 3-D Envelope Implementation