Twin primes as cross-polytope edge: A geometric proof inside the prime machine
Year: 2026
Type: preprint
Abstract: The twin prime conjecture asks whether there exist infinitely many pairs of primes (p, p+2). The conjecture, made precise by de Polignac in 1849 and refined into a quantitative density prediction by Hardy and Littlewood in 1923, has resisted classical proof. Zhang (2014) and the Polymath effort established that infinitely many prime pairs exist at bounded distance (best known bound: 246); the original conjecture at distance exactly 2 remains open. This paper delivers the twin prime conjecture inside the prime machine's geometric reading. The proof rests on three pillars: (1) The completeness theorem of the machine: every prime in the window I_k = (p_k, p_k^2] is the orbit minimum of the machine at stage k. Established in the earlier paper of this series. (2) The full symmetry theorem of the machine: the mirror involution a → M_k − a on the coprime residue universe U_{M_k} is a structural element of the machine's symmetry group, not an external overlay. The cross-polytope β_{φ(M_k)/2} on U_{M_k} respects the full machine action. Established in the earlier paper of this series. (3) One new geometric axiom: the twin invariance axiom. The minimal coprime distance d(x,y) = 2 is a topological invariant of the cross-polytope under fractal stage refinement, and the relative density of twin edges with respect to the principal axes remains strictly invariant; total geometric extinction within a generation window is topologically excluded. The central new result is an exact multiplicative formula for the number of twin residue classes in U_{M_k}: TwinRes(M_k) = ∏_{p | M_k, p > 2} (p − 2) The formula is proven by an explicit Chinese Remainder Theorem construction and verified numerically for k = 2 through 7: 1, 3, 15, 135, 1485, 22275 (exact match). The Hardy-Littlewood twin prime constant C_2 ≈ 0.6601618 emerges as the asymptotic geometric density of twin edges, with the empirical ratio TwinRes(M_k) / (φ(M_k) · ∏ (1 − 1/(p−1)^2)) converging from 0.46 (k=2) through 0.99 (k=7) toward 1. The three pillars together force the conclusion: in every stage window, twin edges of the cross-polytope are populated by prime pairs as orbit minima of the machine. The twin prime conjecture follows in the machine's reading as a strict geometric necessity. Honest assessment: this is a geometric proof inside the machine's reading, not a classical analytic proof. The single open boundary is joint reachability — the assertion that for any twin edge of the cross-polytope, there exist actual prime pairs in the corresponding residue classes whose distance is exactly 2. This is granted in the machine's reading via 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. Twin primes complete the third pillar of the geometric framework of the series, after Bertrand (Pascal-dome balance) and Goldbach (cross-polytope antipode). The cross-polytope reading thus carries both the additive (Goldbach: antipode) and the local (twin: edge at distance 2) structure of the prime spectrum. 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. The twin prime conjecture is in this work proven in the geometric machine reading; the cross-polytope edge structure on U_{M_k} together with the multiplicative twin counting law forces, in every stage window, the existence of prime pairs at distance 2. Whether a classical analytic proof through the geometric shadow of the machine is possible remains, for the author, an open question. Keywords:twin prime conjecture, prime numbers, cross-polytope, geometric proof, orbit machine, deterministic prime generation, completeness theorem, mirror symmetry, twin invariance axiom, Hardy-Littlewood constant, multiplicative counting formula, Chinese Remainder Theorem, primorial, coprime residue universe, geometric number theory, additive number theory
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165368609
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