From Isotropic Tangent Geometry to Discrete Nonlinear Channels in Registration Theory: An Exact Four-Channel Zero-Set Theorem for ACRF-4

This technical note extends the rank-three relational geometry developed in Registration Theory by examining the first nonlinear structure that appears beyond the isotropic tangent regime of canonical ACRF-4 dynamics.

Starting from the previously derived reduced cubic normal form on the rank-three relational sector H4,

C3(x)=A∥x∥2x+BPH4(x∘3),

the note defines the mixed cubic registration-transport defect

H(u,v)=C3(u+v)−C3(u)−C3(v).

This defect is shown to be generated by a symmetric trilinear, S4-equivariant tensor and to decompose exactly into an isotropic term and a tetrahedrally organized nonlinear term. For unit relational directions $u,v$,

H(u,v)=A(1+2u⋅v)(u+v)+3BPH4[(u∘v)∘(u+v)].

At relational angle 120∘, the isotropic contribution vanishes identically. The remaining tetrahedral term vanishes only for a discrete set of orientations. For the reference direction

u=21(−1,1,0,0),

the simultaneous conditions

v∈H4,∥v∥=1,u⋅v=−21,

and

PH4[(u∘v)∘(u+v)]=0

are proved symbolically to have exactly four solutions. These define four discrete nonlinear zero-transport channels, and the construction is preserved under the S4 symmetry of the underlying relational system.

The result shows that an isotropic rank-three tangent geometry can retain a discrete symmetry-selected nonlinear structure at cubic order. The note does not identify these channels with physical directions, spacetime curvature, or temporal structure; rather, it establishes an exact intermediate result concerning the nonlinear organization of emergent relational geometry.

This work follows directly from the rank-three invariant relational geometry developed in DOI 10.5281/zenodo.22023004.