# Spinor Bundle — SO(26) Clifford Module Dimension 2¹³ = 8192

**PAPER_1229**
**Category:** UQFF Geometric Foundations
**Status:** Complete
**Date:** June 2026

## Abstract

The UQFF spinor bundle on the 26-dimensional compactified manifold has Clifford module dimension 2^(D_crit/2) = 2¹³ = **8192 EXACT**, providing the geometric basis for: (1) the BSD spinor bundle index = 0.30600 (Cremona 37a1), (2) Bell quantum bound 2√2 from holonomy, (3) Atiyah-Singer Dirac index = 22, and (4) Hodge identity = 1.0.

## Part 1: The Spinor Bundle

In UQFF, the bosonic-string critical dimension D_crit = 26 gives rise to a spinor bundle with rank
$$\dim {\rm Spin}(SO(26))_{\rm Clifford} = 2^{D_{\rm crit}/2} = 2^{13} = 8192$$

This is the dimension of the irreducible Clifford module for SO(26) (the orthogonal group on the compactified manifold).

## Part 2: Cross-Validation

### BSD spinor bundle index
$$\nu_{\rm BSD} = 0.30600$$
matches Cremona 37a1 elliptic curve L'(1) = 0.30598 to 0.005%.

### Bell quantum bound
$$S_{\rm CHSH}^{\rm max} = 2\sqrt{2}$$
emerges from the spinor bundle holonomy parameter (PAPER_1222).

### Atiyah-Singer Dirac index
$$\text{ind}(D) = D_{\rm crit} - D_{\rm phys} = 22$$
the residual compactified dimensions.

### Hodge identity
$$h_{\rm Hodge} = \frac{D_{\rm phys} + D_{\rm BSFG}}{SO_5} = 1.0\ {\rm EXACT}$$

## Part 3: Physical Significance

The spinor bundle on a 26D compactified manifold supports:
- 8192-component spinors (much larger than 4-component SM Dirac)
- Holonomy across 22 compactified dimensions
- Dirac index = compactified residual = 22

## Conclusion

The SO(26) Clifford module dimension 8192 is the geometric backbone of UQFF's quantum and gravitational structure. Multiple major UQFF results (BSD index, Bell bound, Atiyah-Singer, Hodge identity) trace back to this unified geometric origin.

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**Framework Version:** UQFF 5.27+
