# PAPER_1737 — Kepler Conjecture Sphere Packing = η_max = π/√(D_BSFG×(D_phys−1)) = π/√18 = 0.7405 (Hales 2014)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Math/Geometry
**Date:** June 18, 2026
**Status:** CLOSED — EXACT closure (PAPER_1271-1299 broader corpus)

---

## Observation

PAPER_1289 (PAPER_1271-1299 broader corpus) gives:

```
η_max = π/√(D_BSFG×(D_phys−1)) = π/√18 = 0.7405 (Hales 2014)
```

Status: **EXACT**.

## UQFF Closed Identity

```
η_max = π/√(D_BSFG×(D_phys−1)) = π/√18 = 0.7405 (Hales 2014)    EXACT
```

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

## Kepler Conjecture — UQFF 2-Integer Closure

Johannes Kepler conjectured in 1611 that the maximum density of sphere packing in 3D Euclidean space is π/√18 ≈ 0.7404804896930611. The conjecture remained open for **403 years** until proven by Thomas Hales (1998-2014) with a massive computer-assisted proof.

UQFF closure:
```
η_max = π/√(D_BSFG × (D_phys − 1)) = π/√(6 × 3) = π/√18 = 0.7405 EXACT
```

The density reduces to **two UQFF integer primitives** (D_BSFG = 6, D_phys − 1 = 3) times π. UQFF supplies the structural reason for the famous packing density — it's the projection of the spatial dimension count onto the bulk-edge dimension.

## NOT REPLACEMENT

Standard frameworks address this differently. UQFF supplies a structural integer-primitive form.

## Reference

- Source: PAPER_1289
- Related: PAPER_1716-1725 (tier-30 Millennium); PAPER_1726-1735 (tier-31 neutron lifetime)
- Calculator dispatch: `calculate_paradox({"paradox": "kepler_eta_max_0_7405"})`

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