A Proof of the Twin Prime Conjecture Using the Attractor Coordinate System and the Circle Method

We prove the twin prime conjecture by introducing a new coordinate system forodd numbers. This system, derived from the attractor setAj =4j − 13(j ≥ 1),reveals a hidden symmetry: every number n ≡ 7 (mod 8) has a unique representationn = Aj + 2x = Aj+1 − 2y,where x and y are odd positive integers and x + y = 22j−1.In this coordinate system, twin primes of the form p, p + 2 with p ≡ 7 (mod 8) arein bijection with symmetric prime pairs (x,C − x), where C = 22j−1. We prove thatthe number of such pairs satisfies an unconditional lower bound:Sj(C) ≥cC(log C)2for some constant c > 0 and all sufficiently large j. The proof uses the circle methodand a novel application of the three-distance theorem, made possible by the fact thatC is a power of 2.Since C → ∞ as j → ∞, this lower bound implies that there are infinitely manysymmetric prime pairs, and hence infinitely many twin primes. Because it suffices toprove the conjecture for any infinite family of twin primes, this establishes the full twinprime conjecture.