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 Mₖ. 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 Mₖ = p₁ p₂. . . pₖ, realized as a cross-polytope beta₍䂵 in R^nₖ with nₖ = phi (Mₖ) /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 PolResd (Mₖ) is strictly positive and strictly increasing in k above an explicit threshold k₀ (d), with the closed-form expression PolResd (Mₖ) = prod₏ | ₌䂵, ₏ > ₂, ₏ ₃₎₄ₒ ₍₎ₓ ₃₈ₕ₈₃₄ ₃ (p - 2) x prod₏ | ₌䂵, ₏ > ₂, ₏ ₃₈ₕ₈₃₄ₒ ₃ (p - 1). Three thresholds are distinguished and ordered: - k₀ (d): the smallest k with pₖ > 2d, above which non-degeneracy holds, - k₁* (d): the smallest k from which every odd prime divisor of d is captured by Mₖ, - k₁ (d): the smallest k >= k₁* (d) with 2d 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 (p₍+₁ - pₙ) <= 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). Keywords: Polignac's conjecture, prime gaps, twin primes, cross-polytope, residue classes, closure operator, geometric number theory, prime machine, Hardy-Littlewood constellations, sieve cycles, Maynard-Tao, bounded gaps.