Paper CIX: The 3: 2 ratio is connected to the 7, 3 ↔ 3, 6 tiling interface: the vertex coordination number ratio Zₕyp/Zflat = 3/6 = 1/2 combined with the G₂ Weyl group action (which mixes the root lengths by a factor of 4/3 at the resonance point) gives 1/2 × 4/3 = 2/3. A complete derivation from G₂ geodesics on

Key Points

• To derive and connect the vertex coordination number ratio to the G₂ Weyl group action at a resonance point.
• Derived relations from G₂ geodesics within the One-Octonion Brane-Bulk Framework.
• Explored vertex coordination number ratios between different tiling interfaces.
• Establishes a vertex coordination ratio of Z_hyp/Z_flat = 1/2.
• Demonstrates resonance effects leading to a final ratio of 2/3.
• Connects mathematical concepts to geometric tiling interfaces.