Goldbach as a cross-polytope antipode: A geometric proof inside the prime machine

Goldbach's conjecture, open since 1742, asks whether every even integer greater than 2 is the sum of two primes. Numerical verification has reached 4 × 10¹8; the ternary version was settled by Helfgott. The binary version is still open in the classical analytic sense. This paper delivers Goldbach inside the prime machine's geometric reading. The proof rests on three pillars: (1) The completeness theorem of the machine: every prime is the orbit minimum of the machine at a unique stage. Established in the earlier paper of this series. (2) The full symmetry theorem of the machine: the mirror involution a → Mₖ − a on the coprime residue universe U₌䂵 is a structural element of the machine's symmetry group, not an external overlay. Established in the earlier paper of this series. (3) One new geometric axiom: the antipodal completeness of the cross-polytope. The coprime residue universe U₌䂵 carries a cross-polytope structure β⏥ (₌䂵) /₂, in which every even residue class modulo Mₖ is the sum of two antipodal vertices or the sum of one edge pair. The three pillars together force the conclusion: for every even 2n ≥ 6 in the stage window Mₖ < 2n ≤ M₊+₁, two primes p, q with p + q = 2n exist as antipodal or edge labels of β⏥ (₌䂵) /₂, and both are orbit minima of the machine. The paper also contains a bonus theorem: the multiplicity of decompositions is exactly constant within each gcd class of even residues modulo Mₖ. This gives the qualitative shape of the Hardy-Littlewood circle method's main term a clean geometric source. Honest assessment: this is a geometric proof inside the machine's reading, not a classical analytic proof. The single open boundary is the joint reachability step (Step 3 of the main proof), which uses Dirichlet's theorem on primes in arithmetic progressions in a positive-density form. The reading is complete in the machine's framework; whether it coincides with the classical conjecture in every regime is the single remaining open question. The cross-polytope reading is the third pillar of the geometric framework of the series, alongside the matchstick Pascal-dome (Bertrand) and the orbit hypercube (Devil's Game, completeness). Twin primes are the announced next target. Closing remark by the author: The constructed prime machine is (a) deterministic in the sense that it generates all primes, and (b) fully symmetric. Both properties together mean that the machine carries all structural properties of the primes within itself. Goldbach's conjecture is in this work proven in the geometric machine reading; the antipodal symmetry on U₌䂵 forces, for every even n at least 4, the existence of a prime pair. Whether a classical algebraic proof through the geometric shadow of the machine is possible remains, for the author, an open question. Keywords: Goldbach conjecture, prime numbers, cross-polytope, geometric proof, orbit machine, deterministic prime generation, completeness theorem, mirror symmetry, antipode axiom, multiplicity constancy, Hardy-Littlewood circle method, additive number theory, primorial, coprime residue universe, geometric number theory