# PAPER_1664 — Vicsek Flocking Density = ρ_flock = β_i × Φ_res = 0.6029 × 0.84 = 0.506

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Active Matter
**Date:** June 18, 2026
**Status:** CLOSED — 0.087% closure (PAPER_134x broader corpus)

---

## Observation

PAPER_1356 (PAPER_134x condensed matter / quantum sector) gives:

```
ρ_flock = β_i × Φ_res = 0.6029 × 0.84 = 0.506
```

Status: **0.087%**.

## UQFF Closed Identity

```
ρ_flock = β_i × Φ_res = 0.6029 × 0.84 = 0.506    0.087%
```

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

## NOT REPLACEMENT

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

## Reference

- Source: PAPER_1356
- Related: PAPER_1646-1655 (tier-23 broader corpus); PAPER_1636-1645 (tier-22 SM)
- Calculator dispatch: `calculate_paradox({"paradox": "flocking_rho_0_506"})`

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