Rank-Three Invariant Relational Geometry from Robust Closure in Registration Theory: Age-Quotiented ACRF-4 Center Dynamics and the Cubic Departure from Flat Lattice Composition

This paper develops a restricted theorem-level bridge between robust relational closure and local three-dimensional effective dynamics in Registration Theory.

The first part establishes an exact graph-theoretic and geometric result. Under a specified set of robustness conditions, the minimal relational closure is the complete graph K4. Its nonuniform relational sector is the three-dimensional subspace H4=1⊥, on which the K4 Laplacian induces an isotropic positive-definite rank-three metric. The four projected face modes satisfy one linear closure relation, and the resulting additive consequence geometry generates a rank-three lattice with polynomial ball growth of degree three.

The second part connects this exact closure geometry to the canonical ACRF-4 registration dynamics. After quotienting the uniform transverse aging factor, the symmetric fixed point has a three-dimensional neutral relational sector and a contracting retained-geometry sector. Standard center-manifold structure therefore yields a local three-dimensional invariant effective manifold. The retained-geometry graph is even under relational reversal and begins quadratically in the relational variables.

The reduced dynamics on this manifold admit the explicit S4-equivariant normal form

R(x)=x+0.41847826087∥x∥2x−0.82402173913PH4(x∘3)+O(∥x∥5).

Thus the exact K4/Z3 closure law is the leading-order or tangent consequence geometry of the reduced ACRF-4 dynamics, while the first departure from flat additive composition occurs at cubic order. The cubic term analytically accounts for the nonzero pairwise commutator, permutation-spread, and closure-loop defects observed in the reduced dynamics.

The result does not claim a derivation of macroscopic physical three-space. It establishes, within the restricted ACRF-4 model, a precise local relation between minimal robust relational closure, a rank-three invariant dynamical manifold, exact tangent lattice geometry, and its first nonlinear correction.