Minimal Robust Relational Closure and an Induced Rank-Three Relational Metric: A Restricted Graph-Theoretic Note for Registration Theory

This note establishes a restricted graph-theoretic result motivated by Registration Theory. It asks for the smallest finite simple undirected relational structure that is connected, vertex-transitive, cyclically closed, robust to deletion of any single member, and robust to deletion of any single relation while retaining both connectivity and nontrivial cyclic closure.

Under these conditions, the minimal admissible structure is the complete graph K4. After removal of the uniform common mode, the remaining relational subspace is three-dimensional:

H4={x∈R4:1Tx=0}.

The K4 graph Laplacian satisfies the exact identity

LK4=4PH4,

so the relational quadratic form is positive definite, isotropic, and rank three on the non-uniform sector. Projection of the four member states into this subspace yields a nondegenerate regular 3-simplex under the intrinsic relational metric.

The identity LKm=mPHm and the associated regular-simplex construction are standard results. The specific contribution of this note is the robust-closure characterization that selects K4 as the minimal admissible relational structure under the stated conditions, together with its interpretation within the Registration Theory program.

The result establishes a canonical rank-three relational metric cell without assuming three external spatial coordinates. It does not claim a derivation of physical three-dimensional space. Extending the local relational cell into persistent homogeneous large-scale geometry remains an open problem.