The Centroidal Morley Transform of a Regular Simplex: Rectification and an Exact Scale Law

On each triangular 2-face of a regular d-simplex, intersect the first n-sector rays issued from the two endpoints of a given edge; for n = 3 this is the vertex of the Morley triangle of that face attached to the edge. To each edge we attach the centroid of the d − 1 corresponding face points. For all d ≥ 2 and n ≥ 2, the resulting points are the vertices of a homothetic copy of the rectified simplex, concentric with the original one, whose edges are r_d(n) times those of the simplex, where r_d(n) = 1/2 − (d+1)/(2(d−1)) · tan(π/(3n))/√3. The combinatorial part is forced by symmetry: every rule attaching points to edges equivariantly under the symmetric group yields a homothetic rectification, so the content is the scale. We record its monotonicity, limits, positivity, the closed form at n = 2, an affine relation with the planar case, and a comparison at the regular tetrahedron with the face-normal construction of a companion paper, which gives a concentric octahedron with edge ratio r_2(3) instead of r_3(3). The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive (Python scripts, figures, and a Lean 4 file checked with the AXLE verifier for the vector identity, the affine law and the bisection value). Paper 3 of 3 in a series on three-dimensional Morley constructions.