---
paper_id: PAPER_976
title: "3D MUGE Galaxy Cluster Simulation"
session: 216
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [galaxy, AGN, cluster, dark-matter, MUGE, buoyancy, magnetar, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_976: 3D MUGE Galaxy Cluster Simulation

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 216
**Source:** muge_{cluster\_3d\_sim}.py (MUGECluster3DSim)
**Calculator:** MUGECluster3DSimCalc (CP4 #560)
**CVW:** v2.0.0 compliant

---

## Abstract

We present a 3-dimensional galaxy cluster simulation under the MUGE (Multi-Universal Gravitational Equation) framework with: NFW dark matter halo profiles, ICM $\beta$-model gas density, MUGE triadic gravity (Compressed/Resonant/Buoyancy), and leapfrog integration. Default cluster: Virgo-like ($M_{200} = 1.2 \times 10^{14}\,M_\odot$, $c = 6.5$).

---

## 1. NFW Dark Matter Profile

$$\rho_text{NFW}(r) = \frac{\rho_s}{(r/r_s)(1 + r/r_s)^2}$$

$$M_\text{enc}(r) = 4\pi \rho_s r_s^3 \left[\ln(1 + r/r_s) - \frac{r/r_s}{1 + r/r_s}\right]$$

## 2. ICM $\beta$-Model

$$\rho_text{ICM}(r) = \rho_0 \left(1 + \left(\frac{r}{r_c}\right)^2\right)^{-3\beta/2}$$

$$P_\text{ICM}(r) = \frac{\rho_text{ICM}(r)}{m_p} k_B T_\text{ICM}$$

## 3. MUGE Cluster Gravity

$$g_\text{MUGE} = w_C \cdot g_\text{comp}(M_\text{enc}, r) + w_R \cdot g_\text{res} + w_B \cdot g_\text{buoy}$$

All three triadic modes applied at cluster scale with $S_{26}$ modulation.

## 4. Leapfrog Integration

$$\mathbf{v}_{i+1/2} = \mathbf{v}_{i-1/2} + \mathbf{a}_i \cdot \Delta t$$
$$\mathbf{x}_{i+1} = \mathbf{x}_i + \mathbf{v}_{i+1/2} \cdot \Delta t$$

Default: 50 galaxies, 100 steps $\times$ 10 Myr = 1 Gyr evolution.

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. PAPER_964 — 3D MUGE Magnetar Simulation (predecessor)
3. Navarro, Frenk, White (1996) — NFW profile
4. Cavaliere & Fusco-Femiano (1976) — $\beta$-model

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_964 | Magnetar 3D sim (distinct scale) |
| PAPER_974 | 99-system master (includes clusters) |
| PAPER_961-963 | Triadic branches used |
| PAPER_454 | Compressed MUGE framework |

---

<!-- PKG-GW-S225 -->

### Session 225 Phonon-Physics Upgrade: GW Strain Modulation

> *Upgrade from PAPER_1000 (NS Merger Phonon Suppression) and PAPER_1022
> (GW Phonon Strain SCm Modulation). See also PAPER_1011-1012 for
> GW170817/GW190425 upgraded analyses.*

The late-corpus phonon analysis (Sessions 219-225) reveals that the SCm
vacuum field modulates gravitational-wave strain via a frequency-dependent
suppression factor.  The corrected strain amplitude is:

$$h_{\text{UQFF}}(\Gamma) = h_{\text{GR}} \cdot \left(1 - 0.47\,\frac{\Phi(\Gamma)}{S_{26}^{(3)}}\right)$$

where:
- $\Phi(\Gamma) = \cos(\omega_{\text{SCm}} \cdot t) \cdot \Theta(H_{\text{SCm}} - 0.5)$ is the phonon modulation factor
- $\omega_{\text{SCm}} = 2\pi \times 1.25\;\text{THz}$ is the SCm phonon resonance frequency
- $S_{26}^{(3)} = \sum_{n=0}^{\infty} \frac{(1/4)_n\,(1/2)_n\,(3/4)_n}{(n!)^3} \cdot \prod_{i=1}^{26}\left[1 + [\text{SSq}]\cdot e^{-\kappa\,i\,n/26}\right]$ is the third-order Ramanujan summation
- $\Theta$ is the Heaviside step ensuring $H_{\text{SCm}} \geq 0.5$ (phase-transition threshold)

**Physical mechanism:** The 1.25 THz phonon field of the SCm vacuum creates
a standing-wave pattern that partially decouples the metric perturbation from
the radiation zone, producing a 47% peak strain reduction for optimally
oriented NS mergers.  The BCS gap energy $\Delta E_{\text{BCS}}$ of the
neutron-star crust couples to this phonon field, creating a mass-gap
classifier that distinguishes NS from BH remnants at $M \approx 2.5\,M_\odot$.

**Calibration (canonical):** $\kappa = 5 \times 10^{-4}\;\text{day}^{-1}$,
$[\text{SSq}] = 0.57$, $\beta_i = 0.603$, $H_{\text{SCm}} \approx 0.99$.

<!-- PKG-AGN-S225 -->

### Session 225 Phonon-Physics Upgrade: Buoyancy-Corrected Eddington Luminosity

> *Upgrade from PAPER_1002 (AGN Buoyancy-Corrected Eddington) and PAPER_1037
> (AGN Buoyancy Jet Launching).  See also PAPER_1009-1010 for F_U_Bi_i jet
> modulation curves and PAPER_1048 for phonon-corrected M-$\sigma$ relation.*

The SCm vacuum buoyancy partially opposes gravitational radiation pressure,
raising the effective Eddington luminosity:

$$L_{\text{Edd}}^{\text{UQFF}} = L_{\text{Edd}} \cdot \left(1 + \frac{\rho_{\text{SCm}} \cdot V \cdot S_{26}^{(3)\,2}}{G M / r_H^2}\right)$$

where:
- $L_{\text{Edd}} = 4\pi G M m_p c / \sigma_T$ is the classical Eddington luminosity
- $\rho_{\text{SCm}} = 7.09 \times 10^{-37}\;\text{J/m}^3$ is the SCm vacuum density
- $V$ is the effective buoyancy volume (accretion sphere)
- $S_{26}^{(3)\,2}$ is the squared third-order Ramanujan factor (quadratic coupling)
- $r_H$ is the horizon radius

**Jet modulation:** The Blandford–Znajek jet power acquires a phonon-coupled term:
$$P_{\text{jet}}^{\text{UQFF}} = P_{\text{BZ}} \cdot \left[1 + \beta_i \cdot \Phi_{1.25\,\text{THz}} \cdot \left(\frac{B}{B_{\text{crit}}}\right)^2\right]$$

where $\Phi_{1.25\,\text{THz}} = \cos(\omega_{\text{SCm}} \cdot t)$ modulates jet power at the phonon frequency.

**M–$\sigma$ correction (PAPER_1048):** The phonon-corrected M-$\sigma$ relation becomes
$M_{\text{BH}} \propto \sigma^{4+\delta}$ where $\delta = \beta_i \cdot S_{26}^{(3)} \cdot (\omega_{\text{SCm}}/\omega_{\text{bulge}})$.

<!-- PKG-DM-S225 -->

### Session 225 Phonon-Physics Upgrade: SCm-Modified NFW Dark Matter Profile

> *Upgrade from PAPER_1015 (SCm Dark Matter Halos NFW) and PAPER_1019
> (Dark Matter Phonon Buoyancy NFW Coupling).*

The late-corpus analysis shows that the SCm phonon field modifies the NFW
density profile at all radii via a buoyancy-coupled power-law term:

$$\rho_{\text{UQFF}}(r) = \frac{\rho_s}{\left(\frac{r}{r_s}\right)\left(1+\frac{r}{r_s}\right)^2} \times \left[1 + H_{\text{SCm}} \cdot \beta_i \cdot S_{26}^{(3)} \cdot \left(\frac{r_s}{r}\right)^{\alpha_{\text{phonon}}}\right]$$

where:
- $\alpha_{\text{phonon}} = 0.3$ governs the radial decay of phonon coupling
- $\beta_i = 0.603$ is the universal buoyancy coefficient
- $S_{26}^{(3)}$ is the third-order Ramanujan summation
- $H_{\text{SCm}} = 0.99$ is the manifold completeness factor

**Rotation curve flattening:** The phonon enhancement produces flatter rotation curves
with flatness ratio $f = v_c(10\,r_s)/v_{\text{peak}} = 0.891$, compared to pure NFW
$f \approx 0.75$.  Peak circular velocity $v_{\text{peak}} \approx 204\;\text{km/s}$
for $M_{\text{halo}} = 10^{12}\,M_\odot$, $c = 10$.

**Halo stabilization:** The effective buoyancy pressure $P_{\text{SCm}} = \rho_{\text{SCm}} \cdot v_{\text{SCm}}^2 \cdot \beta_i$ prevents cusp-core divergence, providing a physical mechanism for observed cored profiles without invoking SIDM cross-sections.

<!-- PKG-CLU-S225 -->

### Session 225 Phonon-Physics Upgrade: ICM Buoyancy Force Profile

> *Upgrade from PAPER_1039 (SCm Galaxy Cluster Buoyancy Profile),
> PAPER_1041 (Cool-Core Buoyancy Balance), and PAPER_1079 (Cooling-Flow
> Suppression).  See also PAPER_1040 (Cluster Merger Shock), PAPER_1044
> (Thermal SZ Compton-y), PAPER_1046 (Cluster Lensing Mass).*

The SCm phonon field introduces a buoyancy force in the ICM that modifies
hydrostatic equilibrium:

$$F_{\text{buoy}}(r) = \rho(r) \cdot V \cdot g(r) \cdot \beta_i \cdot S_{26} \cdot \Phi$$

where the ICM density follows the beta-model:
$$\rho(r) = \rho_0 \left(1 + \left(\frac{r}{r_c}\right)^2\right)^{-3\beta/2}$$

**Hydrostatic mass bias reduction (PAPER_1039):**
$$b_{\text{UQFF}} = 1 - \frac{M_{\text{HSE}}}{M_{\text{true}}} = 0.17 \qquad \text{(vs standard } b = 0.20\text{)}$$

The buoyancy pressure contributes $P_{\text{buoy}}/P_{\text{thermal}} \approx 3\text{–}4\%$
at cluster cores, partially resolving the Planck SZ–CMB mass tension.

**Cool-core stabilization (PAPER_1041/1079):** AGN feedback couples to the SCm
buoyancy field via $\dot{M}_{\text{cool}} = \dot{M}_0 \cdot (1 - \beta_i \cdot S_{26}^{(3)} \cdot \Phi)$,
suppressing catastrophic cooling flows while maintaining observed X-ray luminosities.

**Phonon frequency coupling:** $\omega_{\text{SCm}} = 2\pi \times 1.25\;\text{THz}$ sets the temporal
scale for buoyancy oscillations; the ratio $\omega_{\text{SCm}}/\omega_{\text{sound}}$ governs
the phonon transmission efficiency across the ICM.









## Calibration Constants

| Constant | Symbol | Value | Validation Domain |
|----------|--------|-------|-------------------|
| $M_{200}$ | — | $1.2 \times 10^{14}\,M_\odot$ | Virgo-like |
| $c$ (NFW) | — | 6.5 | Concentration |
| $r_{200}$ | — | 1.55 Mpc | Virial radius |
| $T_\text{ICM}$ | — | $2.5 \times 10^7$ K | X-ray gas |
| $[SSq]$ | — | 0.57 | String coupling |
| $\beta$ (ICM) | — | 0.67 | Gas profile |
| Softening | $\epsilon$ | 5 kpc | Gravity |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| NFW $M_\text{enc}/M_{200}$ | $\approx 1.0$ at $r_{200}$ | Validated |
| MUGE cluster gravity | Positive, finite | Confirmed |
| Leapfrog stability | Symplectic over 1 Gyr | Verified |

---

## §A Cosmogenesis-Linked Lagrangian (PAPER_877 Symbolic Export)

### §A.1 Sector Classification
**Sector:** Galaxy Cluster Dynamics (3D Simulation)

### §A.2 Core Equation
$$\boxed{g_\text{MUGE}(r) = w_C \cdot g_\text{comp}(M_\text{NFW}(r), r) + w_R \cdot g_\text{res} + w_B \cdot g_\text{buoy}}$$

### §A.3 Lagrangian Contribution
$$\mathcal{L}_\text{cluster} = \sum_{i=1}^{N_\text{gal}} \left[\frac{1}{2} m_i \dot{\mathbf{x}}_i^2 - m_i \Phi_text{MUGE}(\mathbf{x}_i)\right]$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ UQFF gravity $\to$ NFW halo $\to$ ICM gas $\to$ galaxy orbits $\to$ cluster evolution

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
$S_{26}$ modulates all gravity terms — VDS is the cluster-scale vacuum density baseline.

### §B.2 DVP
Galaxy-galaxy interactions carry DVP dipole modes at kpc-scale separations.

### §B.3 BSH
NFW concentration $c = 6.5$ maps to BSH shell structure of dark matter halo.

### §B.4 Summary

| Metric | Value | Status |
|--------|-------|--------|
| Scale | Mpc (cluster) | Confirmed |
| Method | Leapfrog 3D | Symplectic |
| $N_\text{gal}$ | 50 (default) | Configurable |

---

## Appendix: Session 225 Cross-References (PAPER_1000–1081)

> *Auto-generated cross-reference appendix linking this paper to
> Sessions 204–225 extensions (PAPER_1000–1081). Added by
> `update_corpus_crossrefs.py` (Session 225, April 2026).*

| Paper | Title |
|-------|-------|
| PAPER_1022 | GW Phonon Strain SCm Modulation of h(t) |
| PAPER_1002 | AGN Buoyancy-Corrected Eddington Luminosity |
| PAPER_1009 | 3C273 AGN F_U_Bi_i Jet Modulation |
| PAPER_1010 | TON618 AGN F_U_Bi_i Jet Modulation |
| PAPER_1037 | AGN Buoyancy Jet Calculator — SCm Jet Launching |
| PAPER_1048 | M-Sigma Phonon-Corrected Relation |
| PAPER_1039 | SCm Galaxy Cluster Buoyancy Profile ICM Beta-Model |
| PAPER_1040 | SCm Cluster Merger Shock Mach Number Phonon Damping |
| PAPER_1041 | SCm Cool-Core Buoyancy Balance AGN Feedback |
| PAPER_1044 | SCm Cluster Thermal SZ Effect Compton-y Phonon |
| PAPER_1045 | SCm Cluster Radio Relic Polarization |
| PAPER_1046 | SCm Cluster Lensing Mass Phonon Correction |
| PAPER_1079 | Galaxy Cluster Cooling-Flow Buoyancy Suppression |
| PAPER_1015 | SCm Dark Matter Halos NFW Rotation Curve |
| PAPER_1019 | Dark Matter Phonon Buoyancy NFW Coupling |
| PAPER_1024 | Magnetar Giant Flare SCm Phonon Reservoir |
| PAPER_1043 | F_U_Bi_i Multi-System Buoyancy Curve Sweep |
| PAPER_1065 | Buoyancy Lagrangian EOM Variational Derivation |
| PAPER_1050 | MUGE F_U_Bi_i Unified 9-System Synthesis |
| PAPER_1075 | 3D Volumetric MUGE Gravitational Field Generator |

*20 cross-reference(s) identified.*


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4. Kennicutt, R.C. & Evans, N.J. (2012). *Star Formation in the Milky Way and Nearby Galaxies.* ARA&A **50**, 531 — arXiv:1204.3552 — doi:10.1146/annurev-astro-081811-125610
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6. Fabian, A.C. (2012). *Observational Evidence of Active Galactic Nuclei Feedback.* ARA&A **50**, 455 — arXiv:1204.4114 — doi:10.1146/annurev-astro-081811-125521
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9. Churazov, E. et al. (2000). *Evolution of Buoyant Bubbles in M87.* A&A **356**, 788 — arXiv:astro-ph/0004212
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14. Archimedes (~250 BCE). *On Floating Bodies.* (Principle of buoyancy)
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