A Cross-Polytope Geometry of the Primes: Orbit Minima, Exact Residue Counts, and Singular Series—The 16-Cell at Modulus 30, Four Structural Theorems, and the Local Factors of the Classical Prime Conjectures

Removing the multiples of 2, 3 and 5 leaves eight residue classes modulo 30. Pairing every class a with its mirror 30-a turns these eight classes into the eight vertices of the 16-cell, the four-dimensional cross-polytope, and the same construction at every primorial \( M_k=2·3⋯ p_k \) produces a cascade of cross-polytopes \( βφ(M_k)/2 \). On this cascade we study the action of the sign-flip group \( G_k(Z/2)ᵏ⁻¹ \) and establish four structural results: the orbits are exactly the square classes of the unit group; for \( k≥ 6 \) the orbit minima up to \( pₖ₊₁^2 \) are precisely the primes in \( (p_k,pₖ₊₁^2) \) together with \( pₖ₊₁^2 \), and this threshold is sharp; the cascade is recursive, with \( pₖ₊₁ \) the smallest non-trivial orbit minimum; and the number of orbits grows by the exact factor \( (pₖ₊₁-1)/2 \) per level. We then apply one common scheme to twin primes, Polignac, Goldbach, Sophie Germain primes, Hardy--Littlewood constellations, primes of the form \( n^2+1 \), arithmetic progressions, exponential families and the Collatz iteration. For each problem the geometry gives an exact Chinese-remainder count of admissible configurations, a truncated singular series that converges to the classical Hardy--Littlewood constant, a conditional statement, the strongest known unconditional result, and numerical checks up to \( 10^8 \). The construction determines all local factors; whether every admissible configuration is actually occupied by primes (the lifting problem) is not decided by it and remains open wherever it is open classically.