The TCAP (Ξ) Transformation: Definition, Dynamics and Affine Structures

This record presents a study of the Transformation by Combination of Prime Adjacencies (TCAP), denoted by Ξ, an arithmetic transformation defined from the ordered prime factorisation of integers greater than one. Let n = p₁p₂⋯pₖ, with 2 ≤ p₁ ≤ p₂ ≤ ⋯ ≤ pₖ, be the ordered prime factorisation of n. The transformation is defined by Ξ(n) = p₁ + Σᵢ₌₁ᵏ⁻¹ pᵢpᵢ₊₁. Unlike many classical arithmetic functions, which treat prime factors individually through their values or multiplicities, the TCAP transformation depends on the interactions between consecutive prime factors in the ordered factorisation. Despite its elementary definition, iteration of the transformation produces a non-trivial discrete dynamical system. The numerical investigation reported in the accompanying article reveals prime fixed points, the observed composite fixed point 12, the observed period-two orbit 14 ↔ 16, highly non-uniform attraction basins, and exact affine structures associated with fixed factorisation patterns. A central theoretical result establishes that, for primes a ≤ b ≤ p, Ξ(abp) = bp + a(b + 1). More generally, fixing an ordered factorisation prefix and allowing the largest prime factor to vary produces an exact affine function. The article also investigates the empirical fertility of 1,081 three-prime affine families. A greedy maximum-coverage procedure selects nineteen complementary families that cover every prime parameter in the calibration interval 200 ≤ p ≤ 100,000, in the sense that at least one associated affine value is prime. When applied without modification to independent intervals extending to p = 10⁶, the corresponding prime-parameter coverage remains above 99.25%. These results are finite-range computational observations and are not presented as asymptotic theorems. The purpose of this work is to introduce the TCAP transformation, establish its elementary mathematical properties, investigate its discrete dynamics and affine organisation, and formulate open questions concerning fixed points, periodic orbits, preimages, affine fertility, and asymptotic behaviour. This work is presented as an independent exploratory study and has not undergone formal peer review. Every effort has been made to ensure the accuracy of the definitions, computations, and interpretations. The author welcomes comments, corrections, and reports of any remaining errors or imprecisions. This record contains the complete manuscript of the article and its associated reproducibility materials. This work is presented as an independent exploratory study and has not undergone formal peer review. Every effort has been made to ensure the accuracy of the definitions, computations, and interpretations. The author welcomes comments, corrections, and reports of any remaining errors or imprecisions.