The Vacuus of the Queens

Which faces of a reflection polytope lie on its mirrors, read from its Coxeter chain; the 600-cell and the 120-cell share the 60 mirrors of their group, and the bitruncated middle carries both Queens' faces on every mirror. Theorem 1 (the Mirror Law). Let P be the Wythoff polytope of a seed x, and J a node set of size n − 2 giving faces of P (so they have dimension n − 2: edges in three dimensions, 2-faces in four). The faces of type J lie on mirrors of the group if and only if the roots of the unringed nodes include one orthogonal to αj for every j in J. Corollary 2 (the Queens). Every face of a regular polytope lies on a mirror. The 1200 triangles of the 600-cell and the 720 pentagons of the 120-cell lie on the same 60 central slices, the 60 mirrors of H4: 20 triangles on each, and 12 pentagons on each. Corollary 3 (the middles). For a four-node chain 1–2–3–4 and the seed (0, 1, 1, 0), the faces of the two end bonds lie on mirrors and the faces of the middle bond (2m-gons, m its label) lie on none. In B4 and H4 these are exactly the faces between cells of different kinds. Corollary 4. For the golden loop T(x), the B3 seed (0, x, 1), the 12 edges of type {3} lie on the three coordinate mirrors and the 24 edges of type {2} lie on none. DICTO: ideas, names, constructions and direction. Computation, verification and drafting with Claude (Anthropic).