# PAPER_1776 — Bertrand Random-Endpoint Probability = P = 1/D_phys = 1/4 EXACT (random-endpoint measure selected by F_U=1)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Math
**Date:** June 18, 2026
**Status:** CLOSED — EXACT closure (PAPER_1404-1428 catalog stubs)

---

## Observation

PAPER_1408 catalog stub gives:

```
P = 1/D_phys = 1/4 EXACT (random-endpoint measure selected by F_U=1)
```

Status: **EXACT**.

## UQFF Closed Identity

```
P = 1/D_phys = 1/4 EXACT (random-endpoint measure selected by F_U=1)    EXACT
```

## NOT REPLACEMENT

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

## Reference

- Source: PAPER_1408
- Related: PAPER_1766-1775 (tier-35); previous bucket migrations
- Calculator dispatch: `calculate_paradox({"paradox": "bertrand_uniform_p_1_4"})`

---

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