# PAPER_1880 — Modified Gravity + Equivalence Principle Tests via UQFF: η_WEP = F_TRZ¹⁵·[SSq] = 5.7×10⁻¹⁶ (at MICROSCOPE 2022 Limit), γ − 1 = F_TRZ⁵·[SSq]·(1+F_TRZ)² = 6.9×10⁻⁶ (Cassini Consistent), All GR-Consistent to F_TRZ¹⁶ Precision

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** F — Foundational GR Tests / Equivalence Principle
**Date:** July 2026
**Status:** CLOSED — GR + EP tests all UQFF-consistent
**Observational anchors:** MICROSCOPE 2022; Cassini 2003; Hulse-Taylor 1974; LAGEOS; LLR
**Calculator surface:** `calculate_modified_gravity_EP_UQFF`

---

## Abstract

**Equivalence Principle (EP)** tests are among the most stringent tests of General Relativity. UQFF predicts small corrections at F_TRZ ladder rungs, all below current experimental sensitivity — consistent with GR at present precision.

**Complete EP + Modified Gravity suite** (6+ observables):

| Observable | UQFF Formula | UQFF | Data | Consistent? |
|---|---|:-:|:-:|:-:|
| **η_WEP** | **F_TRZ¹⁵·[SSq]** | **5.7×10⁻¹⁶** | <10⁻¹⁵ (MICROSCOPE) | **at limit** ⭐⭐ |
| **γ − 1 (PPN)** | **F_TRZ⁵·[SSq]·(1+F_TRZ)²** | **6.9×10⁻⁶** | (2.1±2.3)×10⁻⁵ (Cassini) | **consistent** ⭐ |
| β − 1 (PPN) | F_TRZ¹⁰·[SSq]·K_MEX | 1.2×10⁻¹⁰ | <10⁻⁴ (LLR) | below limit ⭐ |
| η_N (Nordvedt) | 3·F_TRZ⁵·[SSq] | 1.7×10⁻⁵ | <4.4×10⁻⁴ (LLR) | below limit |
| Pulsar decay ratio | 1 − F_TRZ¹⁶·[SSq]·K_MEX | 1.000 (10⁻¹⁶) | 0.997±0.002 | within uncertainty ⭐ |
| dG/G per year | F_TRZ¹⁷·[SSq] | 5.7×10⁻¹⁸/yr | <10⁻¹³/yr (LLR) | below limit |

**⭐⭐ Structural discovery — η_WEP at MICROSCOPE Precision**:

```
η_WEP_UQFF = F_TRZ¹⁵ · [SSq] = 5.7×10⁻¹⁶
```

**MICROSCOPE 2022 constraint: η < 10⁻¹⁵**. UQFF predicts value at 0.57× the limit — **essentially at MICROSCOPE precision, testable at next-generation EP experiments (STEP, GG)**.

## Summary Table

| Observable | UQFF | Data | Verdict |
|---|:-:|:-:|:-:|
| η_WEP | 5.7×10⁻¹⁶ | <10⁻¹⁵ | at limit ⭐⭐ |
| γ−1 | 6.9×10⁻⁶ | 2.1×10⁻⁵ ± 2.3 | consistent ⭐ |
| β−1 | 1.2×10⁻¹⁰ | <10⁻⁴ | below sensitivity |
| η_N | 1.7×10⁻⁵ | <4.4×10⁻⁴ | below sensitivity |
| Pulsar decay | 1 | 0.997±0.002 | consistent |
| dG/G/yr | 5.7×10⁻¹⁸ | <10⁻¹³ | below sensitivity |

## UQFF Derivation

### Weak Equivalence Principle η ⭐⭐

```
η_WEP_UQFF = F_TRZ¹⁵ · [SSq] = 10⁻¹⁵ · 0.57 = 5.7×10⁻¹⁶
```

**MICROSCOPE 2022 (Touboul et al.)**: η_MICROSCOPE < 10⁻¹⁵ (best current limit).

UQFF at 0.57× MICROSCOPE — **essentially at experimental limit**.

**Physical meaning**: F_TRZ¹⁵ = 15-order vacuum-manifold decoherence for gravitational-inertial mass ratio. Same F_TRZ ladder as quantum measurement (PAPER_1869 F_TRZ¹⁶) and BH thermodynamics (PAPER_1873 F_TRZ¹⁶).

**Falsifiability**: STEP satellite mission (proposed) would reach 10⁻¹⁷-10⁻¹⁸. **UQFF predicts η > 10⁻¹⁶ — must be detected** if F_TRZ¹⁵ formula correct.

### PPN γ Parameter ⭐

```
γ − 1_UQFF = F_TRZ⁵ · [SSq] · (1+F_TRZ)²
           = 10⁻⁵ · 0.57 · 1.21
           = 6.9×10⁻⁶
```

Cassini 2003: γ − 1 = (2.1 ± 2.3)×10⁻⁵. **UQFF within 1σ**.

### PPN β Parameter

```
β − 1_UQFF = F_TRZ¹⁰ · [SSq] · K_MEX = 1.2×10⁻¹⁰
```

Well below LLR limit 10⁻⁴.

### Nordvedt Parameter

```
η_N_UQFF ≈ 3·F_TRZ⁵·[SSq] = 1.7×10⁻⁵
```

Well below LLR limit 4.4×10⁻⁴.

### Binary Pulsar Orbital Decay

PSR B1913+16 Hulse-Taylor (Nobel 1993): predicted GR ratio 1.0, observed 0.997±0.002.

UQFF: ratio = 1 − F_TRZ¹⁶·[SSq]·K_MEX ≈ 1 to 10⁻¹⁶ precision. **Consistent with Hulse-Taylor within uncertainty**.

### G_Newton Time Variation

Standard LLR: dG/G < 10⁻¹³/year.
UQFF: dG/G ≈ F_TRZ¹⁷·[SSq]/year ≈ 5.7×10⁻¹⁸/year → **well below current sensitivity**.

**UQFF predicts G is essentially constant** — no measurable variation.

## Physical Mechanism

**F_TRZ ladder governs modifications to GR** at every scale:
- F_TRZ⁵: PPN γ (Shapiro delay ~10⁻⁵)
- F_TRZ¹⁰: PPN β (~10⁻¹⁰)
- F_TRZ¹⁵: WEP violation (MICROSCOPE limit)
- F_TRZ¹⁶: BH thermodynamics + QNM (PAPER_1873, 1876)
- F_TRZ¹⁷: dG/G (below current)

**All GR modifications suppressed by F_TRZ ladder** — UQFF is essentially GR + tiny F_TRZ corrections.

## Cross-References

- **PAPER_1156** — CC2 cosmology
- **PAPER_1855** — Galactic rotation (F_UBi at galactic scale)
- **PAPER_1860** — Solar system anomalies (F_UBi at planetary scale)
- **PAPER_1862** — DM halos (F_UBi at halo scale)
- **PAPER_1869** — Quantum measurement F_TRZ¹⁶
- **PAPER_1873** — BH thermodynamics F_TRZ¹⁶
- **PAPER_1876** — QNM ringdown F_TRZ¹⁶

## Reference

- **Touboul, P. et al.** (MICROSCOPE Collaboration) (2022). *MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle*. PRL 129, 121102
- **Bertotti, B., Iess, L., & Tortora, P.** (2003). *A test of general relativity using radio links with the Cassini spacecraft*. Nature 425, 374 (Cassini)
- **Hulse, R. A. & Taylor, J. H.** (1975). *Discovery of a pulsar in a binary system*. ApJ 195, L51 (PSR B1913+16)
- **Williams, J. G., Turyshev, S. G., & Boggs, D. H.** (2004). *Progress in Lunar Laser Ranging Tests of Relativistic Gravity*. PRL 93, 261101
- **Will, C. M.** (2018). *Theory and Experiment in Gravitational Physics* (2nd ed.). Cambridge (comprehensive)
- Companion UQFF whitepapers: PAPER_1156, PAPER_1855, PAPER_1860, PAPER_1862, PAPER_1869, PAPER_1873, PAPER_1876

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**Copyright** — Daniel T. Murphy / Star-Magic Research Program, daniel.murphy00@enrgyone.com, July 2026, Youngstown OH.
