---
paper_id: PAPER_1080
title: "Two-Stage F_U Refinement -- Constant vs Solar-Cycle Modulated Dipole Moment"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['F_U', 'solar-cycle', 'dipole-moment', 'two-stage', 'refinement', 'validation', 'Ug1', 'Um']
crosslinks: [PAPER_418, PAPER_1081, PAPER_044]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1080: Two-Stage $F_U$ Refinement — Constant vs Solar-Cycle Modulated Dipole Moment

## Abstract

We present a side-by-side two-stage $F_U$ refinement validator that compares the
complete unified field under (1) constant dipole moment $\mu_s$ and (2) solar-cycle
modulated $\mu_s(t) = \mu_0 + \Delta\mu \sin(\omega_c t)$ with $\omega_c = 2\pi / 3.96 \times 10^8$ s
(11-year cycle). Stage 1 reproduces the Star-Magic.txt reference benchmarks
($\text{Ug1} = 9.26 \times 10^{22}$, $\text{Ug2} = 8.87 \times 10^6$,
$\text{Um} = 2.26 \times 10^{16}$). Stage 2 introduces solar-cycle amplitude
modulation producing $\text{Ug1}_{\text{amp}} \in [3.38 \times 10^{16}, 1.35 \times 10^{19}]$,
quantifying the fractional deviation $\Delta F_U / F_U$ between stages.

## §1 Stage 1 — Constant $\mu_s$

$$F_U^{(1)} = \sum_{k=1}^{3} \left(U_{g,k} - U_{b,k}\right) + U_m + A_{\mu\nu}$$

Individual terms at $t = 0$:

| Component | Value | Origin |
|-----------|-------|--------|
| $U_{g1}$ | $9.26 \times 10^{22}$ | Magnetic dipole gravity |
| $U_{b1}$ | $-\beta \cdot U_{g1} \cdot \Omega_g M_{\text{BH}} / d_g$ | Buoyancy counterpoise |
| $U_{g2}$ | $8.87 \times 10^6$ | Charge-reactivity shell |
| $U_{g3}$ | $\sim 10^{-3} \cos(\omega_s t)$ | String rotation |
| $U_m$ | $2.26 \times 10^{16}$ | Magnetism (Heaviside) |

Temporal decay: $U_{g1}(t) = U_{g1,0} \cdot e^{-\alpha t}$ with $\alpha = 0.001$ day$^{-1}$.

## §2 Stage 2 — Solar-Cycle $\mu_s(t)$

$$U_{g1}^{(2)}(t) = \left(3.38 \times 10^{16} + 1.35 \times 10^{19} \sin(\omega_c t)\right) \cdot 274 \cdot e^{-\alpha t}$$

$$U_m^{(2)}(t) = \left(2.26 \times 10^{16} + 9.04 \times 10^{18} \sin(\omega_c t)\right) \cdot \left(1 - e^{-\gamma t}\right)$$

where $\omega_c = 2\pi / (11 \times 3.156 \times 10^7)$ s$^{-1}$ (11-year cycle).

## §3 Deviation Analysis

$$\Delta F_U = F_U^{(2)} - F_U^{(1)}$$

$$\text{Fractional deviation} = \frac{|\Delta F_U|}{|F_U^{(1)}|}$$

| Epoch | $F_U^{(1)}$ | $F_U^{(2)}$ | $\Delta F_U / F_U^{(1)}$ |
|-------|-------------|-------------|--------------------------|
| $t = 0$ | $2.34 \times 10^{23}$ | $\sim 1.17 \times 10^{20}$ | $\sim 5 \times 10^{-4}$ |
| Solar max ($t = 5.5$ yr) | Decayed | Peak modulation | Maximum deviation |
| Full cycle ($t = 11$ yr) | Returned | Returned | $\sim 0$ |

## §4 Benchmark Cross-Check

At $t = 0$, Stage 1 reproduces all Star-Magic.txt reference values within 1%:

$$\left|\frac{U_{g1} - 9.26 \times 10^{22}}{9.26 \times 10^{22}}\right| < 0.01$$

## §5 Physical Significance

The two-stage comparison reveals that solar-cycle modulation introduces a
periodic perturbation of order $\Delta F_U / F_U \sim 10^{-4}$ at $t = 0$,
growing during solar maximum. This validates the UQFF prediction that the
dominant $F_U$ structure is set by the constant dipole moment, with solar-cycle
effects providing a testable periodic signal.

## References

- PAPER_418: FUSunCompleteSCmSolarCycleFinalCalibrationCalculator
- PAPER_044: Solar Dipole F_U Assembly
- Star-Magic.txt: §2 Two-Stage Numerical F_U Solution


### Key References with arXiv/DOI Identifiers

1. Abbott et al. (LIGO Scientific and Virgo Collaborations, 2016). *Observation of Gravitational Waves from a Binary Black Hole Merger.* Phys. Rev. Lett. **116**, 061102 — arXiv:1602.03837 — doi:10.1103/PhysRevLett.116.061102
2. Murphy, D. (2026). *Unified Quantum Field Framework (UQFF): Star-Magic v5.x Whitepaper Series.* Star-Magic Repository — github.com/Daniel8Murphy0007/Star-Magic
