The TCAP Transformation (Ξ) IV

This fourth paper in the TCAP series introduces the TCAP Translation map, denoted Ξ_T, a derived arithmetic transformation built from the ordered prime factorisation of an integer. For a composite number n = p₁p₂…pₖ, the map is defined by Ξ_T(n) = pₖ + p₁ + p₁p₂, while every prime remains a fixed point. For a strictly ordered prime triplet a < b < p, the construction yields the exact identity Ξ_T(abp) = p + a(b + 1), thereby converting the TCAP image of the two-factor prefix ab into a pure additive translation p + c, where c = a(b + 1). This unit-slope structure provides a direct framework for generating structured prime candidates from a known prime parameter. The study develops a primorial design strategy based on constants divisible by the kernel 210 = 7#, which automatically eliminate several small-prime divisibility obstructions. An exact enumeration up to 23# = 223,092,870 produces 763,619 admissible translations. Hierarchical multi-scale screening, exact bitset coverage analysis, maximin optimization, restricted exchanges, residual set-cover refinement, and early-stopping reordering are then used to construct nested systems containing 21, 22, 23, and 24 complementary translations. The final systems are evaluated without modification on 953,711 prime parameters distributed across three independent windows centred near 2 × 10¹³, 5 × 10¹³, and 10¹⁴. The 23-family system reaches 99.014062% aggregate coverage. The 24-family system covers 946,055 parameters, corresponding to 99.197241% aggregate coverage, and remains above 99% in every validation window. Under sequential early stopping, only 5.328701 primality tests are required per parameter on average. The construction is presented as a high-coverage generator of structured prime candidates, not as a universal prime generator or a replacement for primality certification. A theorem proves that every fixed finite translation system necessarily fails for infinitely many prime inputs, and all reported coverage percentages are explicitly restricted to the tested finite ranges. The accompanying reproducibility materials include the complete Python/Google Colab implementation, optimized translation systems, coverage data, comparison tables, examples of uncovered parameters, and CSV files used to generate the reported results.