The TCAP Transformation (Ξ) VII – Ξ⁺ and Twin Primes: Diamond Confluence, Fragmentation, and Direct Preimage Structure

Abstract

This work studies the arithmetic dynamics induced by the TCAP variant Ξ⁺ and its unexpected connection with twin primes. For an integer n = p₁p₂⋯pₖ with ordered prime factors, Ξ⁺ is defined by: Ξ⁺(n) = ∑ᵢ₌₁ᵏ pᵢ + ∑ᵢ₌₁ᵏ⁻¹ pᵢpᵢ₊₁. A central identity is: Ξ⁺(p²) = p(p + 2), which produces the gap-2 structure directly from the square of a prime. For every odd prime p, the paper proves that the two Ξ⁺ branches p² → p(p + 2) → (Ξ⁺)²(p²) and 2p² → Ξ⁺(2p²) converge exactly when p + 2 is prime: (Ξ⁺)²(p²) = Ξ⁺(2p²) ⇔ p + 2 is prime. This configuration is called the TCAP diamond. When the common summit Aₚ = (p + 2)² − 2 = p² + 4p + 2 is itself prime, it becomes a prime fixed point of Ξ⁺. These terminal prime diamonds provide a dynamical interpretation of the previously known OEIS sequences A065017, introduced by Stephan Wagler, and A128550, introduced by Zak Seidov. The paper also studies open diamonds, where p + 2 is composite, through an exact opening defect D(p), a congruential residue R(p), and a stratification governed by the factorization of p + 2. Direct preimages of terminal summits are analyzed as well. In particular, semiprime preimages satisfy: (a + 1)(b + 1) = A + 1, and each terminal summit has a unique associated twin-prime semiprime predecessor. Computational experiments include all odd primes p ≤ 10⁶ and the first 10 000 terminal prime diamonds. Reproducibility data and scripts are included with the deposit. The results provide an exact dynamical characterization of twin-prime pairs within Ξ⁺, but do not constitute a proof of the infinitude of twin primes. The infinitude of closed TCAP diamonds is equivalent to the twin-prime conjecture, while the infinitude of terminal prime diamonds is a strictly stronger requirement.