TCAP Transformation Ξ V: Controlled Atomization and Deterministic Prime Certification

Abstract

TCAP Transformation Ξ V: Controlled Atomization and Deterministic Prime Certification This fifth paper develops a deterministic prime-selection and certification dynamics from the TCAP transformations Ξ and Ξ⁺. The starting difficulty is structural: Ξ and Ξ⁺ are defined from ordered prime factors, so literal iteration normally requires factorizing every successive image. The paper introduces controlled atomization: selected integers are treated as opaque entries without asserting that they are prime and without factorizing them. The canonical construction uses H₂(n)=(2,3,5,7,9,…,m), where m is the largest odd integer ≤n. This state is derived from the classical odd double factorial; its formal product is 2m!!, but that product is never computed. For m=2r−1, the first image has the closed form X₁=(4r³−3r²−r+15)/3. Each image is then reduced progressively by exact GCD operations for B=2,3,…,n. After saturation through B, if the remainder R satisfies 1 n², R is kept as one opaque atom and the next rank is exactly Ξ⁺(2,R)=3R+2. If R=1, complete reduction occurs and the base branch stops. The exhaustive canonical campaign over 3≤n≤500,000, comprising 499,998 inputs, produced 499,951 certifications (99.99059996%), 46 full reductions R=1, and one survivor after 50 ranks. That survivor is n=3, whose non-termination under the base rule is proved exactly. The mean certification rank is 1.8492932307 and the maximum observed rank is 11. These are finite computational observations and imply neither an asymptotic density nor a universal depth bound.