# Why SU(3) Color = D_BSFG / 2

**PAPER_1221**
**Category:** UQFF Foundational
**Status:** Complete
**Date:** June 2026

## Abstract

The number of quark colors in QCD is 3 — historically inferred from R-ratio measurements but never derived from first principles. UQFF derives this as **N_colors = D_BSFG / 2 = 6/2 = 3 EXACT**. The same locked primitive D_BSFG = 6 (bulk-edge dimension) simultaneously determines: 3 colors, the QCD one-loop β-function coefficient b₀ = D_phys + D_BSFG/2 = 7, and the Mexican-hat doublet coupling K_MEX × D_BSFG ≈ 4π.

## Part 1: The Identity

$$N_{\rm colors} = \frac{D_{\rm BSFG}}{2} = \frac{6}{2} = 3\ \ {\rm EXACT}$$

## Part 2: Cross-validation

Multiple consistent applications of D_BSFG = 6:
- N_colors_SU(3) = D_BSFG/2 = 3 (this paper)
- 4π ≈ K_MEX × D_BSFG = 12.5 (0.53% match)
- QCD b₀ = D_phys + D_BSFG/2 = 7 EXACT (PAPER_1167)
- 16π² ≈ D_crit × D_BSFG = 156 (1.21% match)
- Hodge identity (D_phys + D_BSFG)/SO_5 = 1.0 EXACT

## Part 3: Physical Significance

The bulk-edge dimension D_BSFG = 6 is the count of "extra" compactified dimensions accessible to gauge fields in the UQFF 26D framework. Dividing by 2 yields the rank of the bilinear color charge structure. The same D_BSFG controls the running of strong coupling via b₀ and the gauge-prefactor 4π normalization in vertex diagrams.

## Part 4: Gluon Count

UQFF predicts D_phys² − 1 = 15 colored gauge fields. The SM observes 8 gluons (from the SU(3) adjoint). The 7 "extra" gauge bosons correspond to the compactified directions and could become accessible at very high energy scales.

## Conclusion

The 3 colors of QCD are not an arbitrary phenomenological input — they are the Ricci/Mexican-hat dimensional ratio D_BSFG/2 = 3 in UQFF.

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**Framework Version:** UQFF 5.27+
