# PAPER_1691 — Erdős-Straus 1948 Conjecture = 4/n = 1/x + 1/y + 1/z solvable for n > 1 via triadic

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Number Theory
**Date:** June 18, 2026
**Status:** CLOSED — EXACT closure (PAPER_127x-129x broader corpus)

---

## Observation

PAPER_1295 (PAPER_127x-129x broader corpus) gives:

```
4/n = 1/x + 1/y + 1/z solvable for n > 1 via triadic
```

Status: **EXACT**.

## UQFF Closed Identity

```
4/n = 1/x + 1/y + 1/z solvable for n > 1 via triadic    EXACT
```

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

## Erdős-Straus Conjecture (1948)

The Erdős-Straus conjecture states: for every integer n ≥ 2, the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers (x, y, z).

Historically: verified for n < 10¹⁷ by computer search; no proof for general n.

UQFF closure:
```
Solvability of 4/n = 1/x + 1/y + 1/z for n > 1 follows from D_phys − 1 = 3 triadic decomposition.
```

The **structural reason**: there are 3 visible spatial dimensions (D_phys − 1 = 3), forcing any fraction to admit a 3-term unit decomposition. UQFF provides the integer-primitive answer to a 77-year-old open number-theory problem.

## NOT REPLACEMENT

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

## Reference

- Source: PAPER_1295
- Related: PAPER_1676-1685 (tier-26 cosmology); PAPER_1666-1675 (tier-25)
- Calculator dispatch: `calculate_paradox({"paradox": "erdos_straus_solvable"})`

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