The Bilayer Worlds of the Twin Primes: Privileged Prime Systems, the Factorization of Zeta, and the Self-Similar Arithmetic of Prime Pairs

I construct and analyze a family of arithmetic worlds generated by the twin primes. A world here is an elementary object: a chosen set of primes together with all the integers those primes generate by multiplication, and nothing else. Viewing the integers through a bilayer—two parallel copies of the number line separated by the twin gap 2, classifies every odd position into four types, written 11, 10, 01, 00 according to the primality of the position and of its upper neighbor, that obey an exact transition grammar with one forbidden set of transitions and one forced local word, and a balance law: the two singly-prime populations agree to within 2 at every height. Removing the upper twin primes from the alphabet of all primes splits the multiplicative structure into two complementary worlds whose zeta functions factor Riemann’s: ${\zeta }_L\left(s\right){\zeta }_R\left(s\right)=\zeta \left(s\right)$ for $\mathrm{Re}s>1$ . World L inherits the pole behavior, the boundary growth, and every classical analytic difficulty; world R, the integers built exclusively from upper twin primes, has a convergent zeta at $s=1$ (by Brun’s theorem), no parity problem (its Liouville sum is an absolutely convergent product), and exactly one hard statement: whether its alphabet is infinite, which is the twin prime conjecture. Statements about abscissae of convergence, boundary behavior at $s=1$ , and analytic continuation of the two factors are separate distributional questions, kept strictly apart from that equivalence (Section 7.6). I prove a general Privileged World Theorem: for any set P of primes, the world $W\left(P\right)$ satisfies a division law ($m$ -multiples divided by $m$ reproduce