TITLE:
A Cubic Polynomial for the Twin Primes
AUTHORS:
Michael Mark Anthony
KEYWORDS:
Twin Primes, Cubic Diophantine Polynomial, Wilson’s Theorem, Wilson Quotient, Gamma Function, Wallis Product, Clement’s Criterion, Integer Solutions
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.10,
October
27,
2026
ABSTRACT: Let P be an odd integer and let N and
N
′
be the Wilson quotients of P and
P+2
. We prove that the integer cubic polynomial equation
N
P
3
−(
1−N
)
P
2
−P(
1+
N
′
)−(
2
N
′
−1
)=0
, discovered inside the Wallis product for
π
and proven from Wilson’s theorem and the Gamma functional equation, holds identically in the rationals for every odd P, and that the points whose coordinates are fixed by the two factorial definitions are integral exactly at the odd twin prime pairs; the unrestricted integer lattice of the equation at a fixed station is, by contrast, infinite at every odd station (Section 6), so the twin correspondence is precisely a statement about the Wilson normalized points. We then develop the complete theory of its solution set: the composition theorem, by which the author’s sigma cascade dynamically generates every solution with multiplicity equal to basin gravity; the uniqueness theorem, giving a counting bijection between solutions and twin pairs; the full lattice of integer points on every station line, governed by a single universal congruence, the Quarter Law; the divisor transpose, identifying the pencil’s incidence geometry with the divisor structure of the progression
4N+1
; the factorization of the twin condition into an integrality criterion and a divisibility criterion, shown to factorize Clement’s classical congruence; the class theorem, in which Jacobi’s two square identity counts the incidence classes and the Gaussian integers enter; the unit miss theorem, that the Wilson certificate always lands within exactly one unit of its seat; and the relaxed infinitude theorem, that the near seated class is unconditionally infinite, with the seated subclass equivalent to the twin prime conjecture; and the cross station seat laws of the closing section, in which the certificate’s position on its ladder is assigned, its misstep quantized at every gap, and the class arithmetic of the wider pencils passes from the Gaussian integers, exclusive to gap two, to the real quadratic fields, the golden field at gap six. Every numerical assertion is certified by exact integer computation.