The Geometry of Primes: The 16-cell as the seat of the eight coprime classes modulo 30

DEFINITION: The Prime Hyperoctahedron — Definition and Properties. The Prime Hyperoctahedron is the four-dimensional regular cross-polytope β₄, also known as the 16-cell or hexadecachoron, whose eight vertices carry the eight residue classes coprime to 30, namely 1, 7, 11, 13, 17, 19, 23, and 29. It is the unique convex regular 4-polytope with Schläfli symbol 3, 3, 4, dual to the tesseract 4, 3, 3, and a member of the infinite family of cross-polytopes (βₙ) generalising the line segment, the square, and the octahedron to higher dimensions. Its f-vector is (8, 24, 32, 16): eight vertices, twenty-four edges, thirty-two equilateral triangular faces, and sixteen regular tetrahedral cells. Its symmetry group is the hyperoctahedral group B₄ (also denoted BC₄ or 3, 3, 4) of order 384, which contains four orthogonal reflection axes through opposite vertex pairs and acts transitively on vertices, edges, faces, and cells. Realised in standard coordinates, the eight vertices sit at (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), and (0, 0, 0, ±1), giving a unit circumradius, edge length √2, and a regular tetrahedral vertex figure. In the present work the eight vertices are identified with the eight coprime residue classes modulo 30 in such a way that antipodal vertex pairs realise the four mirror pairs 1 ↔ 29, 7 ↔ 23, 11 ↔ 19, and 13 ↔ 17 around the reflection axis at the value 15. Under this identification the body acquires four structural pillars: the eight vertices carry the eight coprime classes, the twenty-four edges carry the fourteen even residues from 2 to 28 (omitting 30), the thirty-two faces carry the fourteen odd residues from 1 to 29 omitting 15, and the sixteen cells carry the fifteen even residues from 2 to 30. The exclusion of 15 from the face level and of 30 from the edge level reflects the same reflection axis appearing twice across dimensions. Three of the thirty-two faces are equilateral triangles with side length √2 and residue sums congruent to 1 modulo 30; one of these, the triangle with vertices 7, 11, 13, is taken as the Ur-Dreieck or canonical reference face onto which the full body projects as a hexagram (Star of David) carrying 7, 11, 13 on one triangle, 17, 19, 23 on the antipodal triangle, and 1 and 29 at the central point. SHORT DESCRIPTION We identify the eight residue classes coprime to 30, namely 1, 7, 11, 13, 17, 19, 23, 29, with the eight vertices of the four-dimensional regular cross-polytope, also known as the 16-cell or hyperoctahedron beta₄. Schlaefli's classification of regular convex 4-polytopes singles out the 16-cell as the unique such polytope with eight vertices, so the identification is forced once one accepts that the four mirror pairs r, 30 - r act as four mutually orthogonal antipodal axes. The 24 edges, 32 triangular facets, and 16 tetrahedral cells of the 16-cell then distribute over the residues modulo 30 in a structured way that interacts with the omitted classes 15 and 30. LONG DESCRIPTION This paper continues the series Geometrie der Realitaet and refines the elementary matchstick framework introduced one day earlier in Prime Geometry: Triangles, Mirror Axes, and the Construction of Primes from Geometric Necessity. The earlier paper used n = 3 triangles and a mirror axis at the even number 2, and observed that the primes emerge bottom-up as a consequence of these elementary moves rather than being imposed top-down. The present paper answers the question left open at the end of that note: which geometric object naturally hosts the fourfold mirror symmetry of the eight coprime residue classes modulo 30? Main statements: 1. The four mirror pairs of V = 1, 7, 11, 13, 17, 19, 23, 29, namely 1, 29, 7, 23, 11, 19, 13, 17, are the four antipodal axes of the four-dimensional cross-polytope. 2. Schlaefli's classification of the six regular convex 4-polytopes singles out the 16-cell as the unique such polytope with exactly 8 vertices. The identification is therefore forced if one demands regularity. 3. The 24 edges of the 16-cell, labelled by vertex sums modulo 30, cover the fourteen even residues from 2 to 28. The four multiples of 6, namely 6, 12, 18, 24, are each covered three times. The residue 30 is omitted from the edge support. 4. The 32 triangular facets, labelled by vertex sums modulo 30, cover the fourteen odd residues in 1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29. The residues 5 and 25 are each covered four times. The residue 15 is omitted from the facet support. 5. The 16 tetrahedral cells, labelled by vertex sums modulo 30, cover the fifteen even residues from 2 to 30 nearly bijectively. The residue 30 is covered exactly twice, by two cells that are antipodal in the symmetry group of the 16-cell. 6. The two omissions, 15 in the facets and 30 in the edges, are exactly the products 3 * 5 and 2 * 3 * 5. They record, inside the 16-cell, the small prime factors that were excluded from V by the coprime condition. 7. The Ur-triangle 7, 11, 13 has class sum 31, congruent to 1 modulo 30, and is one equilateral triangular facet of the 16-cell. It anchors the class 1 of multiplicative identity inside the geometric structure. Negative findings: We honestly report what was tested and falsified. - Orbits of the matchstick rule on residues coprime to 30, 210, and 2310 show no Petrie-projection signature of H₄, F₄, or E₈. - The Fourier spectrum of the orbit class trace is statistically flat. - The mean-squared class displacement scales sub-diffusively with exponent 0. 47 over 10⁷ primes, with no obvious 16-cell interpretation. - The numerical coincidences phi (210) = 48 and phi (2310) = 480 with vertex counts of dual 24-cells and root counts of E₈ are not matched by any geometric structure beyond the cardinality. The B17 self-protection rule of the series forbids us from converting these coincidences into structural claims. Mathematical implications discussed in the paper: - Goldbach decomposition modulo 30 inside the cell support. - Hardy-Littlewood prime-constellation correction for multiples of six in the edge support. - Dirichlet characters of (Z/30Z) ᵗimes pair with the four mirror axes. - Chebyshev's bias and primes in arithmetic progressions, as potential repackaging. For non-specialists the paper closes with a one-paragraph simple description of the construction.