Face-Normal Morley Geometry in 3D: Complete Classification and a Seven-Branch Singularity

On each face of a Euclidean tetrahedron, take the point where the two trisectors adjacent to a given edge meet, and draw through it the line normal to that face. The tetrahedron is admissible if, at all six edges, the two normals attached to the edge meet. We show that, up to relabeling, the admissible tetrahedra are exactly the regular tetrahedron, a one-parameter family of type 3+1 and a one-parameter family of type 2+2. The proof reduces each edge condition to an equality of ratios of tangents, factors the six conditions through vertex and face potentials, and excludes three distinct potential levels by a strict mixed-curvature inequality, closed by an explicit rational reduction and a Bernstein certificate with 39 positive coefficients. Modulo similarities, the admissible locus is the union of seven embedded real-analytic curves in the five-dimensional shape space, meeting pairwise only at the regular tetrahedron, where the tangent cone spans a three-dimensional subspace with angle classes π/2, arccos(1/√3) and arccos(1/3). In the regular tetrahedron the twelve normals meet in exactly ten points. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive (Python scripts, figures, and Lean 4 files checked with the AXLE verifier for the algebraic steps P ≥ 8019 and the identity for 1 − F(x)F(π/2 − x)). Paper 1 of 3 in a series on three-dimensional Morley constructions.