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Status: exact coupled-kernel identity for a specified periodic graph; numerical gaps depend on the graph convention.

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First-principles formalism

Let

$$ \Lambda=\mathbb Z_5^3 $$

be a connected periodic spatial graph with Laplacian $L_{T^3}$. Define

$$ D_{T^3}=L_{T^3}\otimes I_{125}+I_{125}\otimes K^2. $$

Since both summands are positive semidefinite,

$$ \ker D_{T^3}=\ker L_{T^3}\otimes\ker K. $$

For a connected periodic graph,

$$ \ker L_{T^3}=\operatorname{span}\{\mathbf1\}. $$

Therefore

$$ \ker D_{T^3}=\operatorname{span}\{\mathbf1\}\otimes E_{47}, $$

and

$$ \dim\ker D_{T^3}=47. $$

Logical deductions

• The internal protected sector survives the spatial lift exactly.
• Periodic boundary conditions eliminate boundary-degree irregularity.
• Protected states are spatially uniform in the zero-Laplacian mode.
• The coupled kernel remains $47$-dimensional.