Magnetic-Moment Structures

(6 π three-caustic nucleon model – rebuilt 2025-06-10)

Context – Uses the phase-sheet geometry introduced in
[[Coherence-Anchored Structural Model]] and the locked constants
from [[Foundations/Full-Derivations#§5.-Calibration]].

1 Dipole moment from a phase field

For any static envelope phase $\mathrm{\Phi }(\mathbf{r})$ the magnetic dipole is

• $Q_e=\pm e$ is the envelope charge (PDG $e=1.602\times {10}^{-19}$ C).
• $m^{\ast }$ is the anchoring-resistance mass; for nucleons we take
  $m^{\ast }=m_p$ (after second-order kernel, see Concept-v2 Stage D Step 15).
• Units: C·m gives A·m${}^2$ ⇒ J/T, correct for $\mu$.

2 Geometry & kernel factors

• Geometry boost $f_{\text{geom}}$ – area integral of the three
  $2\pi$ phase sheets; derivation in [[Phase-Sheet Integral]].
• Kernel fine factor
  $f_{\text{kernel}}=1.0012$
  (second-order Helmholtz upgrade, Concept-v2 Stage D Step 15).

Total nucleon $g$-factor:

(Factor 2 is the Dirac baseline.)

3 Numerical results

Residual mismatch ≤ 0.3 % — within kernel-truncation uncertainty.

4 Why moments deviate from Dirac

Dirac baseline (no sheets):

The 6 π three-sheet envelope compresses phase gradient into narrow
surfaces, amplifying the integral
$\int \mathbf{r}\times \mathrm{\nabla }\mathrm{\Phi }$ by the geometric factor
$f_{\text{geom}}$.

5 Fine-structure corrections (less than 1 %)

• Second-order kernel lowers $g_p$ by 0.3 % and raises $|g_n|$ by 0.4 %.
• Sheet-edge softening (0.04 → 0.06 fm) shifts $f_{\text{geom}}$ by 0.1 %.
• No new constants are introduced.

6 Next tests

• Fourier-transform the envelope → predict $G_M(Q^2)$; compare with JLab data.
• Extend to hyperons by node-shifted sheets.
• Polarised DIS: all spin resides in the global phase gradient ⇒ no “spin crisis”.

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