# PAPER_1719 — Atiyah-Singer Dirac Index 26D = Dirac index = D_crit − D_phys = 22 EXACT (residual compact dims)

**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_122x-123x foundational Millennium-adjacent)

---

## Observation

PAPER_1231 (PAPER_122x-123x broader corpus) gives:

```
Dirac index = D_crit − D_phys = 22 EXACT (residual compact dims)
```

Status: **EXACT**.

## UQFF Closed Identity

```
Dirac index = D_crit − D_phys = 22 EXACT (residual compact dims)    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_1231
- Related: PAPER_1706-1715 (tier-29 ITER fusion); PAPER_1696-1705 (tier-28)
- Calculator dispatch: `calculate_paradox({"paradox": "atiyah_singer_dirac_index_22"})`

---

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