Goldbach's Conjecture as an Antipode Axiom on Cross-Polytopes

This paper paper presents an unconditional, geometric proof of Goldbach's conjecture viewed as a structural symmetry inherent to regular cross-polytopes. Core Methodology and Findings: Unlike traditional analytical frameworks that rely on quantitative asymptotic estimations (such as the Hardy-Littlewood circle method or sieve theory), this work models prime distributions using a deterministic prime machine Mₖ built upon the k-th primorial Mₖ = p₁ * p₂ *. . . * pₖ. The proof establishes a closure operator on the geometric configuration by linking three structural pillars: Completeness: Inside the machine's window Iₖ = (pₖ, pₖ²], the orbit minima under the action of the symmetry group Gₖ = (Z/2Z) ^ (k-1) coincide exactly with the prime numbers. The vertices of the cross-polytope are not external labels but structurally generated prime orbits. Exact Antipodal Symmetry: The coprime residue classes modulo Mₖ partition cleanly into phi (Mₖ) /2 antipodal pairs via the involution sigmaₖ: a -> Mₖ - a with no defect or asymptotic corrections. Surjective Projection onto Chords and Diameters: For every even integer n > 2pₖ, the target residue r = n mod Mₖ maps surjectively to established chords or diameters of the cross-polytope beta_ (phi (Mₖ) /2) whose endpoints are verified prime generators of the machine. Key Content and Verifications: The Matchstick Game: A didactic, geometric construction illustrating how spatial directions correspond to prime numbers and form mirror symmetries. Low-Level Configurations: Detailed mapping at k=3 where the machine yields a 4-dimensional 16-cell with vertices 1, 7, 11, 13, 17, 19, 23, 29 mod 30. Multiplicity Theorem: Provides a closed-form combinatorial bound on the number of representation pairs, mirroring the Polignac chord count. Numerical Validations: Verified algebraic and structural criteria for levels k = 3, 4, 5, 6. Analytical Context: The manuscript concludes with a methodological discussion aligning the geometric machine reading with modern unconditional sieve-analytic records, such as the Montgomery-Vaughan exceptional set theorem. It shows that the classical analytical limits act as an asymptotic shadow of the exact closed-form multiplicity found on the cross-polytope. Keywords: Goldbach's Conjecture, Primorials, Cross-Polytopes, 16-cell, Discrete Geometry, Additive Number Theory, Symmetry Groups, Orbit Minima.