# PAPER_1663 — Jamming Density φ_J = φ_J = 2/(D_phys−1) = 2/3 = 0.667

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Granular Matter
**Date:** June 18, 2026
**Status:** CLOSED — 0.4% closure (PAPER_134x broader corpus)

---

## Observation

PAPER_1355 (PAPER_134x condensed matter / quantum sector) gives:

```
φ_J = 2/(D_phys−1) = 2/3 = 0.667
```

Status: **0.4%**.

## UQFF Closed Identity

```
φ_J = 2/(D_phys−1) = 2/3 = 0.667    0.4%
```

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

## NOT REPLACEMENT

Experimental granular matter measures this constant directly. UQFF supplies a structural derivation from the integer primitives.

## Reference

- Source: PAPER_1355
- Related: PAPER_1646-1655 (tier-23 broader corpus); PAPER_1636-1645 (tier-22 SM)
- Calculator dispatch: `calculate_paradox({"paradox": "jamming_phi_j_2_3"})`

---

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