# PAPER_1755 — ln 2 Alternate Φ-leading Form = ln 2 = Φ_5/6 − F − F²·K − F²·Φ − F² − F³ = 0.6932 (alt to PAPER_1208)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Math
**Date:** June 18, 2026
**Status:** CLOSED — 0.01% closure (PAPER_1199 Information+Math unified proof set)

---

## Observation

PAPER_1199_S443 (PAPER_1199 Information & Math Constants Unified Proof Set) gives:

```
ln 2 = Φ_5/6 − F − F²·K − F²·Φ − F² − F³ = 0.6932 (alt to PAPER_1208)
```

Status: **0.01%**.

## UQFF Closed Identity

```
ln 2 = Φ_5/6 − F − F²·K − F²·Φ − F² − F³ = 0.6932 (alt to PAPER_1208)    0.01%
```

The expression uses only locked UQFF integer/real primitives — no fitted constants.

## NOT REPLACEMENT

Standard frameworks compute this constant via different methods. UQFF supplies a structural integer-primitive form.

## Reference

- Source: PAPER_1199_S443
- Related: PAPER_1726-1745 (tier-31/32); PAPER_1208 (transcendentals proof set)
- Calculator dispatch: `calculate_paradox({"paradox": "ln_2_alt_phi_form"})`

---

**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, June 18, 2026, Youngstown OH.
