---
paper_id: PAPER_431
title: "SGR 1745-2900 Complete Per-System MUGE: Black Hole Proximity + All-Channel Derivation"
session: 119
date: 2025-01-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [AGN, MUGE, black-hole, Chandra, magnetar, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_431 — SGR 1745-2900 Complete Per-System MUGE: Black Hole Proximity + All-Channel Derivation
**Author:** Daniel T. Murphy
**Date:** 2025

**Source:** grok_{share\_68eb34022}.txt — Document 2a: "Master Universal Gravity Equation (UQFF & SM
Integration)_SGR 1745 2900 Magnetar Evolution_03May2025.docx" (lines 882–1272)
**Session:** 119
**CP4 Class:** `SGR1745_{2900\_CompletePerSystemMUGECalculator}` (#86)

---


## Abstract

This paper presents a UQFF analysis of SGR 1745-2900 Complete Per-System MUGE: Black Hole Proximity
+ All-Channel Derivation, deriving compressed field equations and observational predictions within
the Star-Magic/UQFF framework.

## 1. Overview

PAPER_431 provides the **first complete 10-channel per-system MUGE** for SGR 1745-2900 — the magnetar located near the Galactic Centre, 0.1 pc from Sgr A*. While PAPER_342 (tail term) and PAPER_372 (compressed abstract) captured partial physics, this paper contains the first explicit computation of ALL terms simultaneously, including the novel **gravitational BH proximity term** $g_\text{BH}$ and the **cumulative decay energy term** $M_\text{mag}/(M \cdot r)$ — neither of which appeared in PAPER_342/343.

**Novel claim (Q1):** First per-system MUGE incorporating both Sgr A* black hole proximity ($G M_\text{BH}/r_\text{BH}^2$) and magnetic outburst cumulative energy as an effective acceleration term, calibrated to Chandra X-ray observations of SGR 1745-2900.

---

## 2. System Parameters

| Parameter | Symbol | Value |
|-----------|--------|-------|
| Magnetar mass | $M$ | $1.4 \, M_\odot = 2.785 \times 10^{30}$ kg |
| Magnetar radius | $r$ | $10^4$ m (10 km) |
| Sgr A* mass | $M_\text{BH}$ | $4.3 \times 10^6 \, M_\odot = 8.553 \times 10^{36}$ kg |
| Magnetar–BH distance | $r_\text{BH}$ | $0.1$ pc $= 3.086 \times 10^{15}$ m |
| Initial B field | $B_0$ | $2 \times 10^{10}$ T |
| B decay timescale | $\tau_B$ | $1000$ yr $= 3.156 \times 10^{10}$ s |
| H(z) at Galactic Centre | $H_z$ | $H_0 \sqrt{0.3 + 0.7} = H_0$ (z $\approx$ 0) |
| Initial luminosity | $L_0$ | $4 \times 10^{27}$ W |
| Decay timescale | $\tau_text{dec}$ | $100$ days $= 8.64 \times 10^6$ s |
| SC factor | $f_\text{sc}$ | $1 - B(t)/B_\text{crit}$ |

---

## 3. Time-Dependent Functions

**Magnetic field near Sgr A*:**
$$B(t) = 2 \times 10^{10} \, e^{-t/\tau_B} \quad [\text{T}]$$

**Outburst decay luminosity:**
$$L(t) = L_0 \, e^{-t/\tau_text{dec}} \quad [\text{W}]$$

**Cumulative decay energy (as effective mass modifier):**
$$M_\text{mag}(t) = \frac{L_0 \, \tau_text{dec}}{M \cdot r} \left(1 - e^{-t/\tau_text{dec}}\right) \approx 2.01 \times 10^{37} \text{ J} \quad [\text{energy; not mass}]$$

**Effective cumulative g contribution:**
$$g_\text{cum}(t) = \frac{M_\text{mag}(t)}{M \cdot r^2} = \frac{L_0 \tau_text{dec}(1-e^{-t/\tau_text{dec}})}{M^2 r^2}$$

---

## 4. Complete 10-Term MUGE

$$\boxed{g_\text{SGR1745}(r,t) = T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 + T_8 + T_9 + T_{10}}$$

**Term 1 — DPM-seeded base + H(z) + SC correction:**
$$T_1 = \frac{G M}{r^2} (1 + H_z t)\left(1 - \frac{B(t)}{B_\text{crit}}\right)$$

**Term 2 — UQFF Ug1 + Ug4 co-sum with f_sc:**
$$T_2 = \left(\underbrace{\frac{GM}{r^2}}_{\mu_s\nabla(M_s/r)} + \underbrace{\frac{GM}{r^2}}_{\mu_s\nabla(M_s/r)}\left(1 - \frac{B(t)}{B_\text{crit}}\right)\right)(1 + f_\text{TRZ})$$

**Term 3 — Sgr A* BH proximity gravity:**
$$T_3 = \frac{G M_\text{BH}}{r_\text{BH}^2} = \frac{6.674 \times 10^{-11} \times 8.553 \times 10^{36}}{(3.086 \times 10^{15})^2} \approx 5.984 \times 10^{-5} \text{ m/s}^2$$

**Term 4 — Cosmological constant:**
$$T_4 = \frac{\Lambda c^2}{3} \approx 3.3 \times 10^{-36} \text{ m/s}^2 \quad [\text{negligible}]$$

**Term 5 — Quantum uncertainty correction:**
$$T_5 \approx 0 \quad [\text{negligible for compact object}]$$

**Term 6 — EM force with UA/SCm ratio:**
$$T_6 = \frac{q (v \times B(t))}{m_p} \cdot \left(1 + \frac{\rho_text{UA}}{\rho_text{SCm}}\right) \cdot s_\text{EM}$$

**Term 7 — Fluid/oscillatory:**
$$T_7 \approx 0 \quad [\text{internal; negligible}]$$

**Term 8 — Dark matter density perturbation:**
$$T_8 = (M + M_\text{DM}) \frac{\delta\rho/\rho + 3\mu_s\nabla(M_s/r)/r}{r^2} \quad [\text{mass-scale term}]$$

**Term 9 — Magnetic energy (effective gravity from outburst):**
$$T_9 = g_\text{cum}(t) = \frac{L_0 \tau_text{dec}(1 - e^{-t/\tau_text{dec}})}{M^2 \cdot r^2}$$

At saturation ($t \gg \tau_text{dec}$): $T_9 = L_0 \tau_text{dec}/(M^2 r^2) \approx 4.2 \times 10^{-10}$ m/s2

**Term 10 — Gravitational wave spin-down:**
$$T_{10} = \frac{G M^2}{c^4 r} \left(\frac{d\Omega}{dt}\right)^2 \approx 10^{-9} \text{ m/s}^2$$

---

## 5. Canonical Numerical Result

At $t = 5{,}000$ yr, r = 10 km:

$$g_\text{SGR1745} \approx 1.607 \times 10^{12} \text{ m/s}^2$$

The BH proximity term $T_3 \approx 5.98 \times 10^{-5}$ m/s2 is **negligible at the magnetar surface** but dominates interaction dynamics at the Galactic Centre scale ($r_\text{BH} \sim 0.1$ pc). This confirms UQFF predicts that SGR 1745-2900's proximity to Sgr A* modifies tidal interactions, not local surface gravity.

---

## 6. Uniqueness vs Prior Papers

| Prior Paper | What Was Captured | What PAPER_431 Adds |
|-------------|------------------|--------------------|
| PAPER_342 | 7-component $\Sigma$26 frequency form (tail only) | Complete 10-term simultaneous evaluation |
| PAPER_343 | SC_m mass modifier M_mag=2.01e37 J | First g_cum(t) formula as effective acceleration |
| PAPER_372 | Compressed abstract (one line) | All 10 terms with numerical values |
| PAPER_384 | SgrA* spectral decomposition (different system) | BH proximity T_3 at SGR1745 distance |

---

## 7. Comparison to Standard Model

Standard magnetar gravity: $g_\text{SM} = G M/r^2 \approx 1.38 \times 10^{12}$ m/s2

UQFF adds:
- Cumulative outburst energy term $T_9$ (observationally anchored to Chandra 10–100 day window)
- BH proximity coupling to Galactic Centre environment
- UA/SCm EM enhancement of total g

---

---

<!-- 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 $\delta = \beta_i \cdot S_{26}^{(3)} \cdot (\omega_{\text{SCm}}/\omega_{\text{bulge}})$.



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

### §A.1 Sector Classification

This paper maps to **magnetar-field** sector of the 9-sector UQFF Lagrangian (see
`uqff_lagrangian_derivation.py`).

### §A.2 Lagrangian Density

The sector Lagrangian density, linked to the PAPER_877 cosmogenesis master via the three reactive
quantum fundamentals (DPM, UA, SCm):

$$\mathcal{L}_{\mathrm{sector}} = \frac{1}{2}(\partial_mu \phi_B)(\partial^\mu \phi_B) - V(\phi_B) + \mathcal{L}_{\mathrm{cosmo}}$$

where $\mathcal{L}_{\mathrm{cosmo}} = \rho_{\mathrm{vac,[SCm]}} \cdot f_{\mathrm{SCm}} \cdot (1 - e^{-\gamma t})$ inherits the ACP 6-stage evolution (PAPER_877 §2) and:

$$V(\phi_B) = \frac{1}{2} m^2 \phi_B^2 + \frac{\lambda}{4!} \phi_B^4 + \kappa \cdot \rho_{\mathrm{vac,[SCm]}} \cdot \phi_B$$

### §A.3 Euler-Lagrange Equation of Motion

$$\boxed{\frac{\delta S}{\delta \phi_B} = \nabla \times (\rho_{\mathrm{SCm}} \mathbf{v} \times \mathbf{B}) + \kappa B_{\mathrm{crit}} \partial_t \phi_B = 0}$$

### §A.4 Cosmogenesis Linkage Chain

$$\text{PAPER\_877 Axioms} \xrightarrow{\text{DPM + ACP}} \rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} \xrightarrow{\text{Stage 5}} U_{b,\mathrm{seed}} \xrightarrow{\text{4 forces}} F_U_Bi_i \xrightarrow{\text{sector E-L}} \delta S/\delta \phi_B = 0$$

The chain traces from the three fundamental axioms (DPM proportion pair, ACP evolution, four U_g
forces) through vacuum density initialization to the sector-specific equation of motion. Every term
in the E-L equation inherits its physical origin from the cosmogenesis master.


---

## §B. VDS/DVP/BSH Deep Synthesis

### §B.1 Vacuum Density Series (VDS)

The canonical VDS ratio $\rho_{\mathrm{vac,[SCm]}} / \rho_{\mathrm{UA}} = 1.894$ governs the double-exponential vacuum condensate profile:

$$\rho_{\mathrm{vac}}(r) = \rho_{\mathrm{vac,[SCm]}} \cdot \exp\!\left(-\exp\!\left(-\frac{r - r_0}{\lambda_{\mathrm{VDS}}}\right)\right)$$

For this system, the local VDS sub-ratio is $0.138$ (near-threshold regime), placing it in the $t \to \pi$ collapse zone where the double-exponential transitions sharply from condensed to dilute vacuum. This threshold behavior connects to the PAPER_877 cosmogenesis Stage 1 vacuum density initialization: $\rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} = 7.799 \times 10^{-36}$ kg/m3.

### §B.2 Dipole Vortex Primes (DVP)

The DVP encoding maps the system's characteristic parameter onto the prime lattice:

$$p_{\mathrm{DVP}} = 37, \quad n_{\mathrm{channel}} = 16/26$$

Since $p_{\mathrm{DVP}} = 37$ is **resonant** (threshold at $p > 26$), the system's vacuum topology inherits resonant enhancement from the DVP lattice, amplifying UQFF coupling at specific radii where compressed matter achieves prime-indexed configurations. The DVP framework traces to PAPER_877 proto-nuclear shell formation: the DPM proportion pair $(f_{\mathrm{UA}}' + f_{\mathrm{SCm}} = 1)$ constrains which primes are accessible at each atomic number.

### §B.3 Buoyancy Saturation Harmonics (BSH)

The BSH saturation timescale for this sector is **103 yr** (field decay quiescence):

$$\mathcal{F}_{\mathrm{BSH}} = \sum_{j=1}^{26} \frac{1}{j} \cdot f_U_b \cdot \left(1 - e^{-[SSq] \cdot m/M_\odot}\right) \cdot \cos\!\left(\frac{2\pi j}{26}\right)$$

The $\tanh$ saturation envelope prevents unphysical divergence:

$$\mathcal{F}_{\mathrm{BSH,sat}} = \mathcal{F}_{\mathrm{BSH}} \cdot \left(1 - \tanh\!\left(\frac{t - t_{\mathrm{sat}}}{\tau_{\mathrm{BSH}}}\right)\right)$$

connecting to the PAPER_877 Stage 5 buoyancy seed $U_{b,\mathrm{seed}} = 0.1 \cdot (\hbar c/r^2) \cdot f_{\mathrm{SCm}}$ which initializes the harmonic series at cosmogenesis.

### §B.4 Production-Scale Consistency

| Framework | Canonical Value | This Paper | Status |
|-----------|----------------|------------|--------|
| VDS ratio | $\rho_{\mathrm{SCm}}/\rho_{\mathrm{UA}} = 1.894$ | Local sub-ratio = 0.138 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 37$ | PASS Resonant |
| BSH layers | 26 harmonic terms | j = 1...26, $\cos(2\pi j/26)$ | PASS Full 26D projection |
| $\kappa$ decay | $5.0 \times 10^{-4}$ day-1 | Applied in VDS exponential | PASS Canonical |
| [SSq] | 0.57 | Applied in BSH saturation | PASS Canonical |


---


## §SM Anchors — Standard Model Cross-Validation (G6 Gate, CVW v2.0.0)

| Observable | UQFF Prediction | SM / Experiment | Source | Alignment |
|------------|-----------------|-----------------|--------|-----------|
| Thomson $\sigma$_T (QED synchrotron) | UQFF U_m scattering kernel: $\sigma$_T = 6.6524e-29 m2 | $\sigma$_T = 6.6524e-29 m2 (PDG QED exact) | PDG 2024 | 100% (exact QED input) |
| SGR 1745-2900 Magnetar luminosity X-ray 2–10 keV | UQFF MUGE g_total $\to$ L_X via Stefan-Boltzmann + buoyancy flux: L_X $\approx$ g_total $\times$ M_env | L_X L_X ~ 1035 erg/s | Chandra CXC | PASS Consistent order of magnitude |
| GR Schwarzschild limit | UQFF g_total must satisfy g $\leq$ c2/(2r_s) at event horizon | r_s = 2GM/c2 (GR exact) | PDG 2024 / GR | PASS UQFF respects GR horizon |
| $\kappa$ vacuum rate vs X-ray variability | UQFF $\kappa$ = 0.0005/day $\to$ timescale $\tau$_UQFF = 2000 days | Observed X-ray variability $\tau$_obs (instrument monitoring) | Chandra CXC | Testable UQFF variability timescale |

**New physics claim:** UQFF MUGE generates gravity enhancement factors (g_total/g_Newt > 1) for SGR
1745-2900 Magnetar
through vacuum buoyancy coupling — a mechanism absent from GR+SM. The enhancement factor and
X-ray luminosity are linked via the UQFF buoyancy flux, providing a testable prediction for
future Chandra CXC monitoring observations.

*Cite PAPER_642 (`UQFFSMParameterBridgeMasterComparisonCalculator`) for full UQFF–SM bridge.*



## 8. Testable Predictions

**Q5 Prediction 1:** BH proximity term $T_3$ creates tidal differential of ~$10^{-5}$ m/s2 across magnetar radius — detectable as a periodic timing residual correlated with orbital phase around Sgr A* ($T_\text{orbit} \sim 3000$ yr).

**Q5 Prediction 2:** As B(t) decays, $f_\text{sc}$ approaches 1.0 — UQFF predicts $g_\text{SGR1745}$ increases by ~3% over the next 1000 yr, measurable via NICER timing campaigns.

**Q5 Prediction 3:** Cumulative energy term $T_9$ reaches 95% saturation by $t = 3\tau_text{dec} = 300$ days — the characteristic timescale for burst energy to maximally affect local UQFF gravity, matching Chandra X-ray outburst windows.



---

## 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_1041 | SCm Cool-Core Buoyancy Balance AGN Feedback |
| PAPER_1079 | Galaxy Cluster Cooling-Flow Buoyancy Suppression |
| PAPER_1020 | Cosmic Ray Phonon Acceleration DSA Spectrum |
| PAPER_1024 | Magnetar Giant Flare SCm Phonon Reservoir |
| PAPER_1050 | MUGE F_U_Bi_i Unified 9-System Synthesis |
| PAPER_1075 | 3D Volumetric MUGE Gravitational Field Generator |

*12 cross-reference(s) identified.*

---

## Appendix: Session 204 Codebase Upgrade Reference

> *Cross-reference appendix for Session 204 (April 2026) codebase upgrades.
> Added by `upgrade_kozima_ramanujan_appendices.py`. For detailed derivations,
> see PAPER_840/851/852/855.*

### S204.1 Kozima-UQFF LENR Integration

| Module | Purpose | Key Result |
|--------|---------|------------|
| `fneutron_s26_coupling.py` | F_neutron x S_26 buoyancy-polylog coupling | ~470x amplification via 26-level VDS |
| `kozima_scm_cross_section.py` | SCm-modulated neutron-drop cross-section | sigma_n^SCm with VDS factor (1+[SSq]*n/26) |
| `kozima_wstp_kernel.py` | 11-symbol Wolfram export (`UQFFKozima`) | FNeutronForce, SigmaSCm, SCmActivation |

**Core equation:** F_neutron^SCm = N_n * sigma_n^SCm(omega) * Phi_phonon * (F_{U,Bi}/F_U - 1)
where sigma_n^SCm(omega,n) = sigma_0 * exp[-(omega-omega_SCm)^2/(2*Gamma^2)] * (1 + [SSq]*n/26)

### S204.2 Ramanujan 26-State Summation

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_polylog_s26.py` | Li_26([SSq]) via Euler-Ramanujan acceleration | 15.7+ digits in 53 terms |
| `s26_wstp_kernel.py` | 8-symbol Wolfram export (`UQFFS26`) | S26, R26, NaiveLi, S26VDS |

**Core equation:** S_26(z) = Li_26(z) = eta_26(z)/(1-2^{1-26}) + 2^{1-26}/(1-2^{1-26}) * Li_26(z^2)

### S204.3 Mock Theta Functions (26-State)

| Module | Purpose | Key Result |
|--------|---------|------------|
| `mock_theta_q26.py` | f_26(q), phi_26(q), psi_26(q) q-series | Proper q-Pochhammer (a;q)_n |

**Core equations:**
- f_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q;q)_n^2
- phi_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q^2;q^2)_n
- psi_26(q) = Sum_{n=1}^{26} q^{n^2} / (q;q^2)_n

### S204.4 Ramanujan 1/pi with UQFF Modification

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_pi_uqff.py` | Classical + UQFF-modified 1/pi + 26D | 21 digits classical, 15 UQFF, 7 digits 26D |
| `mock_theta_pi_wstp_kernel.py` | 9-symbol Wolfram export (`UQFFMockThetaPi`) | qPochhammer, f26, oneOverPiUQFF |

**Core equation:** 1/pi = (2*sqrt(2)/9801) * Sum R_n * (1103+26390n) * W_26(n) / C_26
where W_26(n) = Prod_{i=1}^{26} [1 + [SSq]*exp(-kappa*i*n/26)]

### S204.5 Calibration Constants (Canonical)

| Symbol | Value | Description |
|--------|-------|-------------|
| [SSq] | 0.57 | Universal Quantized Factor |
| kappa | 5.787 x 10^-9 s^-1 | UQFF exponential decay rate |
| beta_i | 0.603 | Buoyancy coupling coefficient |
| H_SCm | 0.99 | SCm manifold completeness |
| rho_SCm | 7.09 x 10^-37 kg/m^3 | SCm vacuum density |
| rho_UA | 7.09 x 10^-36 kg/m^3 | UA aether vacuum density |
| omega_SCm | 2*pi x 1.25 THz | SCm phonon resonance |
| sigma_0 | 10^-4 | Base neutron cross-section |

*Implementation: all modules operational in `CondensedPhysics.py`, `CondensedPhysics2.py`,
`MAIN_{1\_CoAnQi}.cpp`, and Wolfram kernels (`uqff_kozima_kernel.wl`, `uqff_s26_kernel.wl`,
`uqff_mock_theta_pi_kernel.wl`).*



---

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