A Cubic Polynomial for the Twin Primes

Let P be an odd integer and let N and $N^{\prime }$ be the Wilson quotients of P and $P+2$ . We prove that the integer cubic polynomial equation $N{P}^3？\left(1？N\right)P^2？P\left(1+N^{\prime }\right)？\left(2N^{\prime }？1\right)=0$ , discovered inside the Wallis product for $\pi$ 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