The TCAP Transformation (Ξ) III

This record presents the third paper in the series devoted to the Transformation by Combination of Prime Adjacencies, abbreviated TCAP and denoted by Ξ. The first article introduced the transformation, its arithmetic dynamics, and empirically fertile affine families. The second article established exact structural identities and proved that no fixed finite collection of affine TCAP families can produce a prime image for every prime parameter. The present study develops the computational and algorithmic significance of obtaining a prime image at the very first iterate. For primes a ≤ b ≤ p, the composite integer n = abp has a factorisation known by construction, and its first TCAP image is given by the exact affine identity: Ξ(abp) = bp + a(b + 1). If this value is prime, it is already a fixed point of Ξ. The process therefore stops at Ξ¹, without requiring the factorisation of an intermediate composite image in order to compute later iterates. Fertile affine TCAP families can consequently be interpreted as a collective mechanism for first-step prime-candidate generation. The study analyses 1,081 three-prime affine families over the 9,546 prime parameters satisfying 200 ≤ p ≤ 100,000. This calibration represents 10,319,226 exact family-parameter evaluations. Among the 1,035 primitive non-diagonal families, a truncated singular-series factor predicts the observed fertility hierarchy with a Pearson correlation of 0.9883 and a Spearman rank correlation of 0.9849. A greedy maximum-coverage procedure selects nineteen complementary families and achieves complete coverage of the calibration interval. By comparison, the nineteen families with the highest individual fertility cover 99.8638% of the same prime parameters. Both fixed systems are then evaluated without modification on exactly 1,000,000 prime parameters, from p = 100,003 to p = 15,643,421. This massive validation represents 19,000,000 primality tests per system, or 38,000,000 tests in total. The greedy system covers 987,159 prime parameters, corresponding to 98.7159%. The system formed by the nineteen most fertile families covers 989,704 parameters, corresponding to 98.9704%. The latter system produces an average of 3.992256 prime outputs among nineteen candidates. When the process stops as soon as the first prime output is found, only 4.548982 primality tests are required per parameter on average. This early-stopping strategy avoids approximately 76.0580% of the complete nineteen-test workload. These results clarify the algorithmic interest of first-step TCAP generation. The factorisation of the initial composite is known in advance, each candidate is obtained through a direct affine evaluation, and a successful prime output at Ξ¹ avoids the additional factorisation that would be necessary to continue the orbit from a composite image. All percentages reported in this paper are finite-range computational results. They do not define a universal deterministic prime generator. In accordance with the theoretical obstruction proved in the second paper, every fixed finite affine TCAP system must fail simultaneously for infinitely many prime parameters.