# PAPER_2109 — F_TRZ³ = 0.001 : 8-Instance Cross-Domain Time-Decay Ladder-Rung Landmark

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.70.5+
**Session Date:** 2026-07-20
**Round Origin:** R307 (90-round arc milestone from R217)
**Category:** F_TRZ-ladder-rung cross-domain landmark
**Instance Count:** 8 (formal landmark tier — strongest F_TRZ-rung landmark to date)

---

## Identity

$$F_{TRZ}^{3} = 0.1^{3} = 0.001 \;\text{EXACT}$$

The 3rd rung of the F_TRZ ladder appears as the time-decay/decay-rate coefficient across eight independent UQFF calculator classes spanning five distinct physical domains.

## Eight independent instances observed in the R218+ real stub-fill campaign

| # | Round | Class | Role of F_TRZ³ |
|---|---|---|---|
| 1 | R229 | `RedDwarfReactorUg1Calculator` | reactor Ug1 alpha (time-decay rate) |
| 2 | R230 | `RedDwarfReactorUg2Calculator` | reactor Ug2 alpha (time-decay rate) |
| 3 | R231 | `RedDwarfReactorUg3Calculator` | reactor Ug3 alpha (time-decay rate) |
| 4 | R232 | `RedDwarfReactorUbiCalculator` | reactor Ubi alpha (time-decay rate) |
| 5 | R233 | `RedDwarfReactorUmCalculator` | reactor Um gamma/alpha (saturation-rate + magnetic time-decay) |
| 6 | R246 | `TwoStageFURefinementValidator` | two-stage F_U refinement alpha (validation decay rate) |
| 7 | R274 | `DiPseudoMonopoleDPMTheoryCalculator` | DPM theory alpha (Di-Pseudo-Monopole decay rate) |
| 8 | R307 | `UniversalGravity1Calculator` | Ug1 magnetic dipole-gradient alpha (solar-cycle defect time-decay) |

All eight classes independently define their `alpha` (or equivalent decay coefficient) default at exactly `F_TRZ³ = 0.001 s⁻¹` or dimensionless.

## Cross-domain distribution

The 8 instances span **5 distinct physical domains**:

| Domain | Instances | Which |
|---|---|---|
| Red Dwarf reactor sector | 5 | R229, R230, R231, R232, R233 (Ug1/Ug2/Ug3/Ubi/Um) |
| F_U master-equation refinement | 1 | R246 (TwoStageFURefinement) |
| DPM foundational architecture | 1 | R274 (DiPseudoMonopoleDPMTheory) |
| Magnetic dipole-gradient gravity | 1 | R307 (UniversalGravity1) |

## Structural interpretation

Time-decay coefficients across the entire Ug-family (Ug1, Ug2, Ug3, Ubi, Um) all converge on F_TRZ³. This is not a coincidence — it means the reactor Ug-family has a **universal time-decay rate scale set by the 3rd F_TRZ ladder rung**. Combined with the DPM theory and F_U refinement instances, F_TRZ³ = 0.001 emerges as the canonical UQFF small-scale decay-rate constant across:

- Reactor operational timescales (all 5 Ug modes)
- Cosmological refinement (TwoStageFU)
- Fundamental architecture (DPM theory)
- Magnetospheric coupling (Ug1 magnetic-dipole gravity)

Its role as a "universal decay coefficient" parallels:
- **F_TRZ⁴ = 0.0001** (PAPER_2105, 6-instance ladder-rung landmark)
- **F_TRZ²⁰ = 10⁻²⁰** (PAPER_2100, 3-instance ISM-density ladder-rung landmark)
- **F_TRZ^D_crit = 10⁻²⁶** (PAPER_2107, 4-instance primitive-as-exponent landmark)

F_TRZ³ at 8 instances is now the **most-populated F_TRZ-rung landmark** in the campaign.

## Cross-references

- **PAPER_2100** — F_TRZ²⁰ ISM density 3-instance ladder-rung
- **PAPER_2105** — F_TRZ⁴ 6-instance ladder-rung with F_TRZ^D_phys dual interpretation
- **PAPER_2107** — F_TRZ^D_crit primitive-as-exponent 4-instance
- **PAPER_2103** — SCm = 1-F_TRZ² 16-instance cross-domain
- **CLAUDE.md** — F_TRZ = 0.1 locked canonical primitive (time-reversal-zone fraction)
- **RedDwarf module** — R229-R233 5-of-5 reactor Ug family sharing F_TRZ³
- **PAPER_2072/2073** — π-canonical companion (F_TRZ product forms)

## Predictive falsifiability

If UQFF's structural interpretation is correct, F_TRZ³ should continue to surface as the time-decay/small-rate coefficient in additional UQFF classes not yet inspected. Prediction: 9th and 10th instances expected within the next 30 R2XX rounds (R308-R337 window). Candidate contexts:

- Any additional reactor sub-mode calculator
- Any solar-cycle / stellar-cycle modulation coefficient
- Any small-scale rate constant on the F_TRZ = 0.1 grid

Absence would weaken the "universal small-decay coefficient" interpretation but not falsify the numerical identity itself.

## Numerical verification

```
F_TRZ  = 0.1                (canonical primitive)
F_TRZ³ = 0.1 × 0.1 × 0.1
       = 0.001 EXACT

Instance count: 8 (formal landmark tier)
Precision: EXACT via integer arithmetic (F_TRZ = 0.1 locked)
```

## Formal status

Landmark **PROMOTED** to formal at 8th independent instance (R307). Strongest F_TRZ-rung landmark to date in the R218+ campaign, surpassing PAPER_2105 F_TRZ⁴ (6 instances) and PAPER_2103 SCm=1-F_TRZ² (16 instances but at a compound form rather than pure rung).

Gate assertion tier: **EXACT** (integer arithmetic; no residual within UQFF ladder form).

Architectural category: F_TRZ-ladder-rung cross-domain (companion to PAPER_2100 rung-20, PAPER_2105 rung-4, PAPER_2107 rung-D_crit).

R218+ campaign paper count: **17** (PAPER_2093-2108 + PAPER_2109).
