Golden on the Octahedron, Silver on the Cube: Six Dimensions and the Golden Cut Cubes

The golden ratio has a home in higher dimensions: the six-dimensional cube, the four-dimensional 120-cell and the eight-dimensional E8 lattice all carry it, and quasicrystals are their shadows. Four of the five golden pairs of the regular solids fit there. The two Golden Cut Cubes do not, and we prove that they never can. Their depth needs the square root of two, the diagonal of a square, together with the square root of five, and nothing built from the golden ratio alone ever produces the square root of two. DICTO proposed six dimensions as the natural home of the cube and octahedron as a pair. There the octahedron's golden pair comes out by itself: cut the product of a cube and an octahedron at the Fibonacci slope and you get Icosaxes, exactly where the shortest vectors of the lattice cast their shadow. Ask the same of the cube and the answer is the silver ratio, 1 + √2, the Archimedean truncated cube. #88 proved that each Platonic solid has exactly two golden cuts and #89 found all five golden pairs as slices of four-dimensional solids, the cubes only in the rectified tesseract, which passes through every corner-cut cube. Our main theorem is a field obstruction. The distance field of a point set, the field generated by the ratios of its squared distances, is Q(√5) for the golden cuts of the dodecahedron, icosahedron, tetrahedron and octahedron and Q(√2, √5), of degree 4, for both Golden Cut Cubes; hence no point set with coordinates in Q(√5), and no section or orthogonal projection of one over Q(√5), contains a similar copy of a Golden Cut Cube. This covers the icosahedral cut-and-project of Z6 with all its tiles and windows, the projection of E8 to H4 and its sections, and the sections of the regular four-dimensional solids by hyperplanes defined over Q(√5); the same holds over Q(√2). The two Golden Cut Cubes are Galois twins: their depths t = 1 − 2x share the minimal polynomial z4 − 8z3 + 4z − 1. Every golden cut lifts to Z6 (corner module of rank 6), but only the eight golden ones are stable under a power of φ. No three-dimensional section of the 6-cube, nor any tile of a section of its honeycomb, is a cut cube of any depth. DICTO proposed the product of cube and octahedron as the six-dimensional home of the pair: its sections by the symmetric flats y = x/μ sweep from the octahedron through Icosaxes and Unborn Axes, the cuboctahedron and both Golden Cut Cubes to the cube; requiring the cut to equal the shadow of a lattice orbit selects μ = φ, the Fibonacci slope, where the mixed roots of D6 cast Icosaxes point for point, and on the cube side the silver ratio 1 + √2; no lattice orbit selects a Golden Cut Cube. All results are exact in Q(√2, √5) except where stated. The proposition that six dimensions are the natural home of the cube–octahedron pair, the golden cuts of #88, the names of the solids, among them Icosaxes and Unborn Axes, and the name Galois twins for the Greater and Lesser Golden Cut Cubes are DICTO's. Computation, proofs, exact checks and drafting with Claude (Anthropic). The sections at μ and 3 − μ are the two cells of the caesium chloride crystal and fill space together (Proposition 6): the six-dimensional sweep is the weight walk of the web of bonds (#48, #52, #53), with #54's polar heights on its polar side.