TCAP — Ξ⁺: Atomized Factorial Dynamics and Deterministic Prime Certification

This article presents the fifth part of the TCAP — Transformations, Collisions, Attractors and Primes program and investigates a new construction derived from the Ξ⁺ arithmetic dynamics, based on controlled factorial atomization using the fixed prime atom 2. The construction associates with an initial integer n a deterministic dynamics in which an explicitly controlled part of the factorial structure is separated from an opaque arithmetic remainder. At each rank, this remainder undergoes saturated reduction by primes belonging to a reservoir fixed by n and is subsequently reinjected into the Ξ⁺ dynamics. This procedure yields a particularly simple affine identity along reduction-free branches: T(R) = 3R + 2 and, after r iterations, Tʳ(R) = 3ʳ(R + 1) − 1. It also makes it possible to associate certain orbits with an elementary deterministic primality certificate, based on the exhaustive elimination of prime factors up to a sufficient bound. An exhaustive computational campaign over 3 ≤ n ≤ 100 000 comprising 99,998 cases produced 93,109 certifications, corresponding to 93.11086222%, with a maximum observed certification depth of 12 ranks. 6,886 cases terminate through complete exhaustion of the remainder, while only three orbits — n = 3, 4, and 5 — survive after 50 ranks. These three cases exhibit structured expansive branches and constitute specific obstructions requiring further investigation. The paper systematically distinguishes between proved identities, experimental results, and conjectures. In particular, it investigates the modular structure of reduction-free branches. For a prime p belonging to the reservoir, the condition p ∣ Tʳ(R) is equivalent to 3ʳ(R + 1) ≡ 1 mod p, revealing a natural connection between the Ξ⁺ₐₜ{2} dynamics, congruences, and multiplicative orders of 3. The study also addresses certification depth, the experimentally observed decay of the surviving population, algorithmic complexity, and the requirements for computational reproducibility. The method is not presented as a replacement for standard primality-proving algorithms. Rather, it defines a new object of deterministic arithmetic dynamics, intrinsic to Ξ⁺, in which generation, reduction, and certification interact within a single construction. The numerical results lead to several open problems, foremost among them the theoretical understanding of orbit termination and the characterization of branches capable of persistently avoiding the fixed reservoir of prime factors.