DICTO 1:2 Truncated Octahedron: The 4D Vacuus of the Permutohedron

Version 2 adds the hero shots from the page The 4D Permutohedron (cells pulled apart), a figure of the orderings on every corner, cutaways of the heart and of the DICTO 1:2 Truncated Octahedron's own 3D Vacuus, and names the cells as permutohedra. No result changes. Write 1 to 5 in every order and link two orders that differ by exchanging two neighbouring values: the web of exchange, 120 knots and 240 exchanges, is the skeleton of the 4D permutohedron. Its 4D Vacuus is the set of central slices that carry its faces. Theorem 1. The 150 two-dimensional faces lie in exactly 65 central slices, in four congruence classes (30, 20, 10, 5), each described exactly. Theorem 2. The five slices that hold the middle value fixed are the DICTO 1:2 Truncated Octahedron (squares of side 1, hexagons with sides 1 and 2 in turn); any two cross in a regular hexagon through the centre. Theorem 3. The number of central hyperplanes through the ridges of the permutohedron of 1..n: 3, 15, 65, 225, 770, 2443, 7761 for dimensions 2 to 8, with a closed formula. Theorem 4. The semi-uniform truncated octahedra T(x) of DICTONIC's golden bond (Koca et al. 2015; #48) are the permutohedra of four values placed symmetrically about zero; at x = 1/2 the slider passes through the DICTO 1:2 Truncated Octahedron, whose space-filling partner is the DICTO 2:1 Truncated Octahedron. Classical parts credited (the weak order, reduced words, majorization, translation tilings). Checked before release by an in-house panel of four automated referees and one independent attack. DICTO: ideas, names, constructions and direction. Computation, verification and drafting with Claude (Anthropic).