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 M_k built upon the k-th primorial M_k = p_1 * p_2 * ... * p_k. The proof establishes a closure operator on the geometric configuration by linking three structural pillars: Completeness: Inside the machine's window I_k = (p_k, p_k^2], the orbit minima under the action of the symmetry group G_k = (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 M_k partition cleanly into phi(M_k)/2 antipodal pairs via the involution sigma_k : a -> M_k - a with no defect or asymptotic corrections. Surjective Projection onto Chords and Diameters: For every even integer n > 2p_k, the target residue r = n mod M_k maps surjectively to established chords or diameters of the cross-polytope beta_(phi(M_k)/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.