TITLE:
The Bilayer Worlds of the Twin Primes: Privileged Prime Systems, the Factorization of Zeta, and the Self-Similar Arithmetic of Prime Pairs
AUTHORS:
Michael M. Anthony
KEYWORDS:
Twin Primes, Bilayer Classification, Beurling Generalized Numbers, Euler Products, Self-Similarity, Singular Series, Goldbach Decomposition, Parity, Multiplicative Monoids
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.10,
October
8,
2026
ABSTRACT: 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:
ζ
L
(
s
)
ζ
R
(
s
)=ζ(
s
)
for
Res>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(
P
)
satisfies a division law (
m
-multiples divided by
m
reproduce the world, for every member
m
), gap covariance, a head law stamping the alphabet’s initial spacings at every scale, an exact tiling by smallest-factor branches, and recovery of the alphabet from the member list alone by Möbius-von Mangoldt inversion, verified on twin, congruence, and Bertrand alphabets, the last two provably infinite. The worlds organize into a singularity spectrum in which the real-axis boundary behavior of
ζ
W
at
s=1
measures the density of the prime set, and this spectrum exposes an honest two-layer decomposition of the twin prime conjecture into existence (the Euler product of
ζ
R
has infinitely many factors) and density (the Hardy-Littlewood layer). Within world R the twin phenomenon regenerates: 1874 second-generation pairs below 107 obey a stabilized density law, every pair anchored on the prime 5 by a proven congruence theorem, while the pure third generation is impossible by a congruence argument and the mixed third generation is empty below 109. Cross-world structure includes quadruple and hexad patterns binding the two worlds and a decomposition of Dubner’s twin-Goldbach conjecture into three channels modulo 6 whose synchronized exceptions reproduce the classical list exactly, together with a proven divisibility law for Goldbach representations at the integer level of world R. Throughout, the prime 5—the unique prime that is both a lower and an upper twin—appears as the two worlds’ only shared letter, the generator of world R’s grading modulo 3, and the mandatory ancestor of every future generation of twins. The author’s published work “The Mathematical Principles of Causal Conspiracy” (SCIRP) describes a dual-world representation of nature with a paired structure that corresponds to the bilayer of the twin primes.