The Prime Machine and Polignac's Conjecture - A Versatz Axiom on the Cross-Polytope
Year: 2026
Type: preprint
Abstract: The Prime Machine and Polignac's Conjecture: A Versatz Axiom on the Cross-Polytope Please Note: This paper does not claim a traditional analytic resolution of the classical Polignac conjecture on \(\mathbb{Z}\). Instead, it introduces an unconditional geometric and operator-theoretic framework showing that additive prime gaps are structural invariants of higher-dimensional polytopes. In this paper, part of the Geometry of Reality series, the author treats Alphonse de Polignac's conjecture (that every positive even integer 2d occurs infinitely often as a prime gap) as a Versatz Axiom on Chords (from the German Versatz for offset) within the deterministic apparatus of the Prime Machine \(\mathcal{M}_{k}\). By collapsing the prime coordinate lattice modulo a primorial \(M_{k}\), the state space is mapped onto a regular cross-polytope \(\beta _{n_{k}}\) of dimension \(n_k = \varphi(M_k)/2\). Within this fixed architecture, classical prime-number phenomena emerge as local, rigid consequences of a single geometric object. 🏛️ The Core Geometric Theorems Proved Inside: 1. The Chord-Count Closed Form (Theorem 5.4): Proves that the exact number of admissible offset chords, \(PolRes_d(M_k)\), is strictly positive and governed by a rigid product formula over the prime factors of \(M_{k}\), requiring no probabilistic heuristics. 2. Strict Monotonicity (Theorem 5.4, ii): Proves that above an explicit existence threshold k₀(d), the number of chords increases strictly monotonically with k, driven by an exact geometric growth factor of precisely \(p_{k+1} - 2\) per stage. 3. Exact Topological Invariance (Theorem 5.4, iii): Proves that the ratio \(PolRes_d(M_k) / PolRes_1(M_k)\) crystallizes exactly into the product \(\prod_{q\vert{}d, q>2} (q-1)/(q-2)\) as soon as a non-degeneracy threshold k₁(d) is crossed. This invariant is reached exactly and finitely, not asymptotically. 🔍 The Lift Question: Locating the Analytic Boundary The paper establishes a strict methodological partition to ensure absolute transparency: Lesart A (The Machine Statement - Proved Here): For every d ≥ 1 and every stage above the threshold, the Versatz-chord set is non-empty, exact, and strictly increasing. The machine is a complete apparatus that omits no residue-class pair required by the Polignac question. Lesart B (The Classical Statement - Open Boundary): The translation of these stable geometric chords into concrete, infinite prime pairs on \(\mathbb{Z}\). Rather than competing with modern analytic records (Zhang, Maynard, Polymath), this framework serves as their structural companion. The machine provides the deterministic source of admissibility, while classical sieve theory (such as Brun's or Maynard-Tao density) handles the realization in density, successfully narrowing the residual open problem to an explicitly located boundary. File Content: Full peer-review ready PDF containing complete foundational proofs of the Polignac chord theorem, exact 26-case verification matrices (k=2 to 8), strict monotonicity logs, and a precise differentiation from dynamic sieve-cycle models (Holt). Keywords: Polignac's conjecture, prime gaps, twin primes, cross-polytope, residue classes, closure operator, geometric number theory, prime machine, Hardy-Littlewood constants, Maynard-Tao bounded gaps.
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166555456
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