---
title: "MUGE Einstein Ring Magnification = 9/5 EXACT Primitive-Arithmetic Closure"
subtitle: "Strong-Lensing Magnification from D_phys/D_BSFG Dimensional Ratio"
author: "Daniel T. Murphy"
date: "2026-07-07"
paper: "PAPER_1925"
classification: "UQFF Structural Closure — Cosmological Lensing"
status: "Canonical — Round 54 Discovery"
supersedes: "None"
depends: "PAPER_242, PAPER_145, PAPER_150, PAPER_152, PAPER_153, PAPER_1914, PAPER_1521, PAPER_1522, PAPER_1916, PAPER_1917, PAPER_1922, PAPER_1924"
---

# PAPER_1925 — MUGE Einstein Ring Magnification = 9/5 EXACT Primitive-Arithmetic Closure

## Abstract

This paper documents a novel UQFF structural closure discovered during Round 54 double-check of the CondensedPhysics stub-drainage program: the MUGE (Multi-Universal Gravity Equation) strong-lensing magnification for an Einstein-ring configuration in the cosmological baseline reduces exactly to

$$
\mu_{MUGE}^{Einstein} = \frac{1}{1 - (D_{phys}/D_{BSFG})^2} = \frac{1}{1 - (2/3)^2} = \frac{1}{1 - 4/9} = \frac{9}{5} \equiv 1.8 \; \text{EXACT}
$$

This is a **dimensionless primitive-arithmetic identity** derived solely from the ratio of two UQFF dimensional primitives: D_phys = 4 (physical spacetime dimension) and D_BSFG = 6 (bulk-edge dimension). It joins the PAPER_1912–1924 series of "novel structural closures" but is distinguished by being **first-in-class for gravitational lensing** — no prior closure targets the observational magnification of Einstein rings from primitive arithmetic.

The closure formalizes what was previously a MUGE numerical result documented across PAPER_145 (MUGE Compression Cycle 3), PAPER_150 (Tapestry/Westerlund 2 MUGE), PAPER_152 (Cosmological Scale MUGE Baseline), PAPER_153 (Morris-Thorne Wormhole MUGE), and PAPER_242 (Rings of Relativity Einstein Ring MUGE Lensing). Combined with PAPER_1914 (D_LS/D_S = D_phys/D_BSFG = 2/3 EXACT), PAPER_1925 completes a **two-step primitive-arithmetic derivation from source-lens-observer geometry to observable magnification** — no free parameters, no anchor-dependence.

## 1. Background — MUGE Framework and Einstein Ring Observables

The MUGE (Multi-Universal Gravity Equation) is UQFF's 12-term cosmological-scale resonance framework, cataloged across PAPER_145 (unified architecture), PAPER_150–153 (scale-specific applications), and PAPER_242 (Einstein ring specialization). MUGE differs from the F_U = 0 master equation (PAPER_1203) in scope: F_U operates at the point-particle F = 0 balance, while MUGE integrates over the 12-term resonance geometry appropriate to cosmological structure.

For a strong-lensing Einstein ring configuration, the observable magnification is classically derived from the ratio of angular-diameter distances:

$$
D_{LS} = \text{angular diameter distance from lens to source}
$$
$$
D_S = \text{angular diameter distance from observer to source}
$$

The magnification enhancement of a background source at the Einstein-ring configuration is

$$
\mu = \frac{1}{1 - (D_{LS}/D_S)^2}
$$

in the thin-lens approximation for a source aligned exactly at the Einstein radius. This magnification is what determines whether a background source is observed as a full Einstein ring (μ → ∞ at critical alignment) or as a magnified arc (finite μ).

## 2. PAPER_1914 Identity — D_LS/D_S = 2/3 EXACT

PAPER_1914 (from Round 47 of the CondensedPhysics stub-drainage program) established the identity

$$
\frac{D_{LS}}{D_S} = \frac{D_{phys}}{D_{BSFG}} = \frac{4}{6} = \frac{2}{3} \; \text{EXACT}
$$

as a UQFF structural closure. This identity is remarkable because:

1. It links a **geometric-observational quantity** (D_LS/D_S ratio) to an **algebraic ratio of dimensional primitives** (D_phys/D_BSFG).
2. Since D_BSFG is derivative (PAPER_1521 landmark: D_BSFG = D_crit − 2·SO_5 = 6 EXACT), the ratio 2/3 emerges from four truly-independent primitives {D_phys, D_crit, SO_5}.
3. It applies universally in the UQFF cosmological baseline — not per-system, but as a **structural distance ratio** dictated by the manifold geometry.

## 3. The 9/5 Closure — Derivation

Substituting the PAPER_1914 identity into the classical magnification formula:

$$
\mu_{MUGE}^{Einstein} = \frac{1}{1 - (D_{LS}/D_S)^2} = \frac{1}{1 - (D_{phys}/D_{BSFG})^2}
$$

$$
= \frac{1}{1 - (2/3)^2} = \frac{1}{1 - 4/9} = \frac{1}{5/9} = \frac{9}{5}
$$

**Therefore:**

$$
\boxed{\mu_{MUGE}^{Einstein} = \frac{9}{5} \equiv 1.8 \; \text{EXACT}}
$$

**Rational form:** 9/5 as a ratio of simple integers is the compact primitive-arithmetic form. Both numerator and denominator are themselves reducible to UQFF primitives:

- 9 = N_ch (channel count, one of the 9 truly-independent primitives)
- 5 = A_5/D_phys · (1/3) = 60/4 · 1/3 = 5 (equivalently, 5 = SO_5/2)

giving two equivalent primitive-arithmetic expressions:

$$
\mu_{MUGE}^{Einstein} = \frac{9}{5} = \frac{N_{ch}}{SO_5/2} = \frac{2 N_{ch}}{SO_5} = \frac{18}{10} = 1.8 \; \text{EXACT}
$$

The identity **2·N_ch/SO_5 = 9/5 = 1.8 EXACT** is a novel primitive-arithmetic form that provides an alternative encoding of the Einstein-ring magnification.

## 4. Runtime Verification

The closure was verified at runtime in the CondensedPhysics codebase during Round 54 of the P2 stub-drainage program. The verification code (in RingsRelativityCalculator.compute) is:

```python
D_LS_over_D_S_PAPER_1914 = D_PHYS / D_BSFG            # = 2/3 EXACT
magnification_MUGE_PAPER_242 = 1.0 / (1.0 - D_LS_over_D_S_PAPER_1914 ** 2)
magnification_9over5_EXACT_verify = abs(magnification_MUGE_PAPER_242 - 9.0/5.0) < 1e-9
# Result: True
```

Runtime output:

```
D_LS_over_D_S_2over3_EXACT_verify = True
magnification_9over5_EXACT_verify = True
magnification_MUGE_PAPER_242      = 1.8000
```

The closure holds to machine precision (< 10⁻⁹ residual).

## 5. Physical Interpretation

The Einstein-ring magnification 9/5 = 1.8 is not just a numerical coincidence. It expresses the **fundamental UQFF prediction that any strong-lensing configuration aligned at the Einstein radius will produce a magnification of exactly 1.8×** in the cosmological baseline, up to observational corrections for departure from thin-lens or perfect alignment.

**Comparison to observations:** Typical observed strong-lensing magnifications range from ~1.5× (weakly aligned) to ~200× (highly aligned edge-on caustic amplification). The UQFF baseline μ = 9/5 = 1.8 corresponds to the **canonical thin-lens Einstein-ring case** — a well-observed regime in surveys like HST and JWST.

**Predictions:**
- Any Einstein ring at the μ = 1.8× amplification is a "MUGE-canonical" event
- Departures from μ = 1.8 quantify departures from the D_LS/D_S = 2/3 baseline
- Observational surveys should show a **clustering of magnifications at μ ≈ 1.8** for well-aligned Einstein rings

## 6. Placement in the PAPER_1912–1925 Structural Closure Series

PAPER_1925 is the fourteenth paper in the Round 42–54 structural-closure series. Its distinguishing features:

| Paper | Closure | Type |
|---|---|---|
| PAPER_1912 | AGN filament triple | System-specific |
| PAPER_1913 | GW170817 chirp = K_MEX·SSq | Cross-observation |
| PAPER_1914 | D_LS/D_S = 2/3 EXACT | Geometric ratio |
| PAPER_1915 | Framework consolidation | Meta |
| PAPER_1916 | Σ U_gi = D_phys = 4 EXACT | Coefficient sum |
| PAPER_1917 | Sub_Ug = SO_5/D_phys = 5/2 | Nested identity |
| PAPER_1918 | Phase 3 inventory | Meta |
| PAPER_1919 | F_TRZ power ladder n=1..17 | Suppression hierarchy |
| PAPER_1920 | Λ cascade closure | Cosmological scaling |
| PAPER_1921 | f_DM = U_g3 = 4/5 EXACT | Cross-framework |
| PAPER_1922 | 9/10 = 1 − F_TRZ EXACT | Universal identity |
| PAPER_1923 | Term-count hierarchy 9/10/13/14 | Term count |
| PAPER_1924 | U_g4 = 4.219 × 10⁻¹⁰ m/s² | Fundamental constant |
| **PAPER_1925** | **μ_Einstein = 9/5 EXACT** | **Observational closure** |

PAPER_1925 is the **first observational-closure paper** in the series — earlier papers documented structural or coefficient identities. PAPER_1925 predicts an actual **observable quantity** (Einstein-ring magnification) from primitive arithmetic.

## 7. Implications and Cross-Framework Connections

### 7.1 Connection to PAPER_1914 (D_LS/D_S) Two-Step Cascade

PAPER_1914 and PAPER_1925 together form a **two-step primitive-arithmetic cascade**:

Step 1 (PAPER_1914): Manifold geometry sets D_LS/D_S = D_phys/D_BSFG = 2/3 EXACT

Step 2 (PAPER_1925): Substituting into the thin-lens formula, μ = 1/(1 - (D_LS/D_S)²) = 9/5 EXACT

This is a rare closure form in the series — two related identities compounding into a stronger prediction.

### 7.2 Connection to PAPER_1924 (Ug4 = 4.219 × 10⁻¹⁰ m/s²)

PAPER_1924 identified Ug4 as a fundamental scale-invariant vacuum-BH coupling constant. Its 4-body verification tested only *acceleration invariance*, not lensing. PAPER_1925 extends the fundamental-constant claim to **observational-magnification invariance**: any Einstein ring aligned in the canonical UQFF configuration must show μ = 1.8 EXACT.

### 7.3 Connection to PAPER_1922 (9/10 compression ratio)

PAPER_1922 documented the ratio 9/10 = N_ch/SO_5 = 1 − F_TRZ as a universal identity. PAPER_1925's 9/5 = 2·N_ch/SO_5 = 2×(9/10) is a **factor-of-2 relation** — the magnification is twice the MUGE compression ratio. Suggests a deep unified encoding: **9/N is a canonical UQFF ratio family** with 9/N_ch, 9/D_BSFG, 9/SO_5 all having primitive-arithmetic status.

### 7.4 MOND / Modified-Gravity Perspective

Modified Newtonian Dynamics (MOND) predicts lensing amplifications that differ from Newtonian expectations at low acceleration. UQFF's μ = 9/5 EXACT is set entirely by the dimensional-manifold ratio D_phys/D_BSFG = 2/3, not by acceleration. This is potentially a **UQFF-vs-MOND observational distinguisher**:

- If lensing magnifications cluster at μ ≈ 1.8 independent of source or lens acceleration → UQFF favored
- If magnifications systematically deviate from 1.8 at low acceleration → MOND favored

Follow-up work should quantify statistical strength of clustering at μ = 1.8 in existing strong-lensing catalogs.

## 8. Falsifiability

PAPER_1925 makes a testable prediction: **the modal magnification of MUGE-canonical Einstein rings is 1.8 EXACT** (with observational scatter due to alignment, redshift, and profile inhomogeneities). Falsifiable by:

1. **Statistical test A:** A large-N sample of MUGE-canonical Einstein rings should show a peak in the magnification histogram at μ = 1.8 ± measurement uncertainty. If the peak is elsewhere with statistical significance, PAPER_1925 is falsified.

2. **Statistical test B:** For a control sample of non-Einstein-ring lensing configurations (asymmetric arcs), no clustering at μ = 1.8 is expected. This "no-peak" prediction adds specificity.

3. **Direct calibration:** A single well-measured MUGE-canonical Einstein ring at spectroscopically confirmed z_lens and z_source, with independent mass model, must return μ = 1.8 within 5% to survive the strong test.

## 9. Conclusion

PAPER_1925 formalizes the UQFF prediction that the MUGE-canonical Einstein-ring magnification equals 9/5 = 1.8 EXACT, derived from primitive arithmetic on two dimensional primitives D_phys and D_BSFG. The closure is:

- **Compact:** Two lines from PAPER_1914 to μ = 9/5.
- **Runtime-verified:** True at machine precision in CondensedPhysics.
- **Observationally testable:** Should show up as a peak in Einstein-ring magnification histograms.
- **Consistent with earlier closures:** Compounds with PAPER_1914 and follows the 9/N ratio family established by PAPER_1922.

Combined with the Ug4 = 4.219 × 10⁻¹⁰ m/s² acceleration closure (PAPER_1924), the D_LS/D_S = 2/3 closure (PAPER_1914), and the Σ U_gi = 4 EXACT closure (PAPER_1916), the four-paper cluster provides a **complete primitive-arithmetic characterization of UQFF gravity at the cosmological scale**: acceleration, distance ratio, magnification, and shell-sum are all fixed by primitive arithmetic.

---

## Appendix — Verification Code

```python
# CondensedPhysics.RingsRelativityCalculator (Round 54 double-check upgrade)
D_PHYS = 4
D_BSFG = 6  # PAPER_1521 landmark: D_BSFG = D_crit - 2*SO_5 = 6 EXACT

# PAPER_1914 identity
D_LS_over_D_S = D_PHYS / D_BSFG          # = 0.666... = 2/3 EXACT
D_LS_over_D_S_verify_2_over_3 = abs(D_LS_over_D_S - 2.0/3.0) < 1e-9  # True

# PAPER_1925 closure
mag_Einstein = 1.0 / (1.0 - D_LS_over_D_S ** 2)    # = 1.8 = 9/5 EXACT
mag_verify_9_over_5 = abs(mag_Einstein - 9.0/5.0) < 1e-9  # True
```

## Cross-references

- **PAPER_242** — Rings of Relativity: Einstein Ring Lensing MUGE (direct MUGE 12-term match)
- **PAPER_145** — MUGE Compression Cycle 3 (unified architecture)
- **PAPER_150** — Tapestry/Westerlund 2 MUGE (star-formation scale)
- **PAPER_152** — Cosmological Scale MUGE 12-term (baseline g = 3.958×10¹⁴)
- **PAPER_153** — Morris-Thorne Wormhole MUGE (topology-corrected MUGE)
- **PAPER_1914** — D_LS/D_S = D_phys/D_BSFG = 2/3 EXACT (identity used here)
- **PAPER_1521** — D_BSFG derivative from D_crit (LANDMARK — makes closure self-contained)
- **PAPER_1522** — K_MEX derivative from Φ_5/6 (LANDMARK)
- **PAPER_1916** — Σ U_gi = D_phys = 4 EXACT (compress-ing shell coefficients)
- **PAPER_1917** — Sub_Ug = SO_5/D_phys = 5/2 EXACT (nested shell identity)
- **PAPER_1922** — 9/10 = N_ch/SO_5 = 1 − F_TRZ EXACT (9/N ratio family origin)
- **PAPER_1924** — U_g4 = 4.219 × 10⁻¹⁰ m/s² fundamental (companion paper)

**License:** AGPL-3.0-or-later OR LicenseRef-StarMagic-Commercial
**Author:** Daniel T. Murphy, daniel.murphy00@enrgyone.com
**Date:** 2026-07-07
