Fractal Gauge Symmetry Convergence
QFunity Framework: Partanen-Tulkki’s 4 U(1) gauge theory emerges as a local projection of QFunity’s E₈×E₈ fractal symmetry through scale-dependent coupling κ(L).
1. Core Unification Mechanism
𝐈g ≡ ℱ(L) = e−L/λGR + ℓP2⁄L2 + ℓP2 + ℓs2⁄L2 + ℓs2
Where the spacetime dimension field 𝐈g from Partanen-Tulkki maps to QFunity’s fractal filter ℱ(L).
2. Five-Phase Gauge Convergence
Phase 1: U(1) Embedding
U(1)4 ⊂ E₈ × E₈ via κ(L) = Tr(ΩQF ∧ ★ΩQF)⁄∫dL/L
Phase 3: Charge Emergence
αg = Gme2⁄ħc = κ(L)2 at L = ħ/mec
3. Comparative Framework
| Feature | Partanen-Tulkki | QFunity |
|---|---|---|
| Gravity Origin | 4 U(1) gauge fields | E₈×E₈ fractal torsion |
| Metric Status | Derived from 𝐈g | Emerges from 𝒥(L,t)/L³ |
| Renormalization | BRST invariance | Scale cutoff ℱ(L) |
4. QFunity Predictions vs Standard Model
Standard Model + GR
• Fundamental metric tensor
• Non-renormalizable gravity
• No fractal charge structure
• Fundamental metric tensor
• Non-renormalizable gravity
• No fractal charge structure
QFunity Framework
• Emergent metric from ΩQF
• Finite κ(L) at all scales
• Fractal charge quantization
• Emergent metric from ΩQF
• Finite κ(L) at all scales
• Fractal charge quantization
5. Experimental Tests
Δtecho ≈ 2GM⁄c3 (1 + κ(L)ρ⁄2)
Detectable signatures:
- CMB anomalies: From Γ(L)·δ(L-Lc)·eiωt critical events
- 70 MeV photons: Micro-fracture events in electron clouds
- Fractal Raman spectra: In DNA via 𝒲(L,t) vibrations
6. Unification Extension
κ(L) =
\begin{cases}
\sqrt{\varepsilon_0 G} & (\text{gravity}) \\
\sqrt{\alpha} & (\text{EM}) \\
\alpha_s & (\text{strong})
\end{cases}
QFunity extends Partanen-Tulkki’s framework to all fundamental forces through scale-dependent coupling.
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