The Prime Machine and Polignac's Conjecture - A Versatz Axiom on the Cross-Polytope

This paper treats Polignac's conjecture inside a deterministic geometric apparatus called the prime machine M_k. The machine is constructed once and is the same for every problem in the series: a finite, symmetric, complete structure on the residue classes modulo the primorial M_k = p_1 p_2 ... p_k, realized as a cross-polytope betan_k in Rn_k with n_k = phi(M_k)/2. Polignac's conjecture states that for every even positive integer 2d, infinitely many pairs of primes differ by 2d. The case d=1 is the twin-prime conjecture. We frame the conjecture as a Versatz axiom on chords inside the machine (German "Versatz" = offset; we retain the original term as a series-specific designation, parallel to Ansatz or Eigenwert in the wider mathematical lexicon), parallel to the Goldbach reading as an antipode axiom on diagonals. Main result (geometric machine theorem): The chord count PolRes_d(M_k) is strictly positive and strictly increasing in k above an explicit threshold k_0(d), with the closed-form expression PolRes_d(M_k) = prodp | M_k, p > 2, p does not divide d (p - 2) x prodp | M_k, p > 2, p divides d (p - 1). Three thresholds are distinguished and ordered: - k_0(d): the smallest k with p_k > 2d, above which non-degeneracy holds, - k_1*(d): the smallest k from which every odd prime divisor of d is captured by M_k, - k_1(d): the smallest k >= k_1*(d) with 2d < M_k, above which the closed form is exact. Lemma 5.5 establishes k_1(d) <= k_0(d) in general, verified numerically for d = 1, ..., 99. The paper proves explicit symmetry and completeness theorems for the machine (mandatory in this series since the Fermat paper), gives a direct numerical verification over 26 cases for k = 2 through k = 8, and derives the Hardy-Littlewood constellation constant 2 C_2 e⁻ᵍᵃᵐᵐᵃ prodp | d, p > 2 (p-1)/(p-2) as a corollary of the geometric closed form combined with the prime number theorem. New in version 10: Subsection 8.5 places the geometric reading inside the modern classical landscape. Maynard's small-gaps theorem and the Polymath8b refinement establish unconditionally that liminf (pₙ₊₁ - p_n) <= 246, so the Versatz axiom is unconditional for at least one specific even 2d* <= 246. Granville's BAMS survey (Corollary 1.4) and the Banks-Freiberg-Maynard limit-points work establish that a positive proportion of admissible 2d (at least 1/5460, in fact at least 1/8 - o(1)) are Polignac numbers unconditionally. The residual question — Lesart B for each individual d — coincides with the global state of the field, not with a gap specific to this paper. What this paper claims: A geometric reading of Polignac's conjecture inside the machine, with explicit symmetry and completeness proofs, a precise differentiation from Holt's sieve-cycle work, a derivation of the Hardy-Littlewood constellation constant as a geometric corollary, and an honest classical bridge via Maynard-Tao. What this paper does not claim: A classical resolution of Polignac's conjecture for each individual d. That gap is the current global state of the field, not a defect of the geometric reading. The Versatz axiom is an axiomatic geometric reading inside the machine. Methodological stance: The prime machine is the original object; algebraic appearances such as the Euler totient, the polynomial resultant, and the Hardy-Littlewood constants are its shadows. Maynard-Tao is read here as the strongest classical complement currently available to the geometric machine reading, not as a replacement for it. MSC2020:11A41 (Primes), 11N05 (Distribution of primes), 52B11 (n-dimensional polytopes), 11N36 (Applications of sieve methods).