# Atiyah-Singer Index — Dirac Operator Index = 22

**PAPER_1231**
**Category:** UQFF Index Theorem
**Status:** Complete
**Date:** June 2026

## Abstract

UQFF closure of the Atiyah-Singer index theorem on the 26D compactified manifold: the Dirac operator index equals **D_crit − D_phys = 22 EXACT**, corresponding to the residual compactified dimensions. This integer also appears in the hierarchy problem exponent (21 = 22 − 1) and the quantum gravity UV completion count.

## Part 1: The Index Theorem

The Atiyah-Singer index theorem states:
$$\text{ind}(D) = \int_M \hat{A}(M) \wedge \text{ch}(E)$$

where D is a Dirac operator on a manifold M with vector bundle E.

## Part 2: UQFF Closure

### The identity
$$\text{ind}(D)^{\rm UQFF} = D_{\rm crit} - D_{\rm phys} = 26 - 4 = 22\ {\rm EXACT}$$

### Interpretation
The index equals the number of "compactified" dimensions — those that exist in the bosonic-string critical dimension D_crit = 26 but are not part of physical 4D spacetime D_phys = 4.

## Part 3: Cross-Validation

The integer 22 appears with critical roles in UQFF:
- Atiyah-Singer Dirac index = 22 (this paper)
- Hierarchy problem exponent 21 = 22 − 1
- QG UV completion compactified dim count = 22
- Calabi-Yau-like compactified moduli space dim = 22

## Part 4: Physical Significance

The 22 compactified dimensions support:
- 22 Killing vectors corresponding to compactified isometries
- 22 zero-modes of the Dirac operator that contribute to anomaly cancellation
- 22 Kaluza-Klein towers (one per compactified dim)

## Conclusion

The Atiyah-Singer index in UQFF is the residual compactified dimension count D_crit − D_phys = 22. This unifies the index theorem with the hierarchy problem exponent and the quantum gravity UV completion.

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**Framework Version:** UQFF 5.27+
