The TCAP Transformation (Ξ) II

This preprint is the second article in the series devoted to the Transformation by Combination of Prime Adjacencies (TCAP), denoted by Ξ. Unlike the first paper, which combined theoretical results with computational investigations, the present work is exclusively theoretical: every stated result is accompanied by a complete proof, and no numerical experiment is used as evidence for a general claim. Starting from the ordered prime factorisation of an integer, we establish an exact concatenation formula describing how two ordered blocks of prime factors interact under Ξ. This identity yields the affine extension theorem and the uniqueness of its coefficients, as well as closed formulae for repeated-prime extensions, prime powers and semiprimes. We then analyse integers with exactly three prime factors, obtaining a coefficient dichotomy for the associated affine families, a parity classification, and a proof that 12 is the unique composite fixed point of Ξ having exactly three prime factors counted with multiplicity. Finally, we characterize the local congruence obstructions of every affine TCAP family. Using the Chinese Remainder Theorem and Dirichlet’s theorem on primes in arithmetic progressions, we prove that no fixed finite collection of affine TCAP families can produce a prime image for every prime parameter: every such finite system fails simultaneously for infinitely many prime inputs. The paper provides a rigorous foundation for further investigations into TCAP fixed points, periodic orbits, preimages and affine prime-generation systems.