An Algebraic Renormalization Group for Regular Tetrahedra in Projective 3-Space

This paper studies the orthic construction for tetrahedra in projective 3-space equipped with a nondegenerate symmetric bilinear form. We consider the orthic map sending a tetrahedron to the tetrahedron formed by the orthocenters of its faces. Within the class of regular centered tetrahedra—defined by equal self-quadrance, equal pairwise bilinear products, and centroid at the origin—we prove: The regular centered tetrahedron is a fixed point of the orthic map up to projective equivalence. The induced affine scaling at this configuration has exact multiplier of magnitude 1/31/31/3. The corresponding eigenvalue on quadrance variables is exactly 1/91/91/9. The derivation is purely algebraic and depends only on bilinearity and nondegeneracy of the form. No positivity or signature assumptions are required, so the result applies uniformly in Euclidean, hyperbolic, and more general quadratic geometries. This work provides an exact computation of the linearization of the orthic map at its symmetric fixed point and establishes a precise contraction factor for the induced dynamics on quadrance variables.