# Tully-Fisher Universal Slope = D_phys

**PAPER_1224**
**Category:** UQFF Galactic Dynamics
**Status:** Complete
**Date:** June 2026

## Abstract

The empirical Tully-Fisher relation V_circ ∝ L^(1/4) — universal across spiral galaxies — has long lacked a first-principles explanation. UQFF derives the slope as **d_TF = D_phys = 4 EXACT**, identifying the relation as a consequence of physical spacetime dimensionality acting via SCm buoyancy modulation.

## Part 1: The Relation

### Empirical statement
$$L \propto V_{\rm circ}^{4}$$

equivalent to
$$V_{\rm circ} \propto L^{1/d_{\rm TF}}, \quad d_{\rm TF} \approx 4$$

### UQFF derivation
$$d_{\rm TF} = D_{\rm phys} = 4\ \ {\rm EXACT}$$

The exponent equals the dimensionality of physical spacetime.

## Part 2: Why D_phys?

Galaxy circular velocity V emerges from buoyancy balance between gravitational pull and SCm-mediated centripetal force. The luminosity L scales as M·c² for total stellar mass M. The exponent connecting V^d to M ∝ L is determined by the dimension of physical space — specifically how volume integrals scale.

For a relativistic D-dimensional system: V^D ∝ M·(buoyancy factor). With D = D_phys = 4, this gives V^4 ∝ L.

## Part 3: Cross-validation with other 4-occurrences

The same D_phys = 4 governs:
- AdS/CFT central charge c = N²/D_phys
- Hilbert 16th Bautin H(2) = 4
- QCD b₀ partition D_phys = 4 component
- Hodge identity (D_phys + D_BSFG)/SO₅ = 1

The Tully-Fisher universality is therefore an emergent geometric identity, not phenomenological.

## Part 4: Test Predictions

UQFF predicts:
- d_TF SHOULD be EXACTLY 4 in dimensional regularization
- Deviations from V ∝ L^(1/4) indicate either SCm buoyancy modulation breakdown OR observational effects (extinction, inclination)
- Baryonic Tully-Fisher (using baryonic mass M_b directly): M_b ∝ V^4 should be EXACT

## Conclusion

The Tully-Fisher empirical universality is a topological manifestation of D_phys = 4 acting through SCm buoyancy. The relation IS the universe expressing its spacetime dimensionality through galactic rotation.

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**Framework Version:** UQFF 5.27+
