# PAPER_1690 — Kochen-Specker Min Dimension = d_min = D_phys − 1 = 3 quantum contextuality

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

---

## Observation

PAPER_1285 (PAPER_127x-129x broader corpus) gives:

```
d_min = D_phys − 1 = 3 quantum contextuality
```

Status: **EXACT**.

## UQFF Closed Identity

```
d_min = D_phys − 1 = 3 quantum contextuality    EXACT
```

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

## NOT REPLACEMENT

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

## Reference

- Source: PAPER_1285
- Related: PAPER_1676-1685 (tier-26 cosmology); PAPER_1666-1675 (tier-25)
- Calculator dispatch: `calculate_paradox({"paradox": "ks_contextuality_dim_3"})`

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