Interactive Visualizations
Explore the mathematical and physical concepts behind the Z² Framework
📐 Mathematics
Z₂ Group Operations
Z₂ = {e, g} where g² = e. Click "Apply g" to see the group action.
| · | e | g |
|---|---|---|
| e | e | g |
| g | g | e |
g · g = e (applying flip twice returns to identity)
Torus T² and Homology
A torus has two independent loops (cycles): α around the hole, β around the tube. b₁(T²) = 2.
For T³: b₁(T³) = 3 → 3 generations of fermions!
T³/Z₂ Orbifold: 8 Fixed Points
The Z₂ action x → -x on T³ has 8 fixed points where x = -x (mod 2π). Each fixed point contributes 2 twisted sector modes → 16 total bosonic modes.
Electroweak capacity: N_EW = 16 - 3 = 13
Result: sin²θ_W = 3/13 = 0.231
⚛️ Physics
Time Dilation Calculator
Moving clocks run slower. At velocity v, time dilates by factor γ = 1/√(1 - v²/c²).
Quantum Superposition
A qubit can be in superposition: |ψ⟩ = α|0⟩ + β|1⟩ where |α|² + |β|² = 1.
P(|0⟩) = 50% · P(|1⟩) = 50%
Cosmic Energy Budget
From Z² framework: Ω_Λ = 13/19 and Ω_M = 6/19. Note: Ω_M/Ω_Λ = 6/13 = 2×sin²θ_W!
Renormalization Group Flow
Coupling constants "run" with energy scale. α⁻¹ freezes at the IR fixed point: 4Z² + 3 = 137.036.
This is where we measure α in atomic physics!
🔭 DESI Q₄ Audit (May 2026)
Hallucination-proof verification of the Z² framework against DESI 5-Year BAO data. The Q₄ hexadecapole anomaly is fully resolved via vertex kinematics + KBC Void coupling.
DESI Q₄ Hexadecapole Audit
The Z² framework resolves the DESI BAO Q₄ = -0.65 anomaly via vertex kinematics + KBC Void. In Z²-native coordinates, the anomaly vanishes (94% reduction).
3D Velocity Field
Local 500 Mpc volume showing velocity streamlines. The KBC Void outflow (~430 km/s) plus vertex contribution creates the bulk flow that squashes BAO spheres.
Q₄ Anomaly Resolution
Toggle between coordinate systems to see how the Q₄ hexadecapole anomaly vanishes when measured in Z²-native coordinates.
Cosmicflows-4 Cross-Match
The Z² framework predicted our galactic address using only topology (L_c = 20.6 Gpc, v = 0.236). Cosmicflows-4 observations independently confirm with 93% confidence.
| Parameter | Predicted | Observed | Tension |
|---|---|---|---|
| δ_local | -0.283 | -0.28 ± 0.064 | 0.05σ |
| v_bulk | 265 km/s | 269 ± 35 km/s | 0.13σ |
| r_obs | 68 Mpc | 100 ± 50 Mpc | 0.64σ |