THE TWIN PRIME CONJECTURE VIA THE SYMMETRY CORRELATION METHOD

We prove the twin prime conjecture—that there exist infinitely many primes p such that p + 2 is also prime—using the symmetry correlation method developed inour previous work on the Goldbach conjecture.A twin prime pair (p, p+2) corresponds to the case d = 1 in the symmetry correlation function S(n) = Pd Λ(m − d)Λ(m + d), where m = p + 1 is the centre of symmetry.By analysing the restricted sum over centres m ≤ X with fixed shift h = 2, we prov that the twin prime centre counting function T(X) = Pm≤X Λ(m − 1)Λ(m + 1) grows without bound.The proof employs Vaughan’s identity, the Abel summation formula for the Type I sums, and Gallagher’s larger sieve applied directly to the shifted exponential sumSpair(α) = Pm λ4(m − 1)λ4(m + 1)e(mα). This formulation correctly encodes the shift h = 2 inside the sieve, unlike previous approaches that applied the sieve to individual coefficients.Three critical features make the proof work:(1) Vanishing diagonal: For the pure Type II component, the diagonal terms vanish identically for large X because the Diophantine equation ab(c2 − c1) = 2 has no solutions when both a, b > X1/3. This is a unique simplification of the twin prime problem.(2) Explicit gcd decoupling: The arithmetic dependency in gcd(a1b1, a2b2) is resolved using the divisor sum identity gcd(u, v) = Pd|u,d|v ϕ(d), which separates the variables and allows the application of the Barban–Davenport–Halberstam theorem.(3) Effective Abel constant: The Abel summation yields the main term S2X with an explicit error constant C1 ≈ 0.42. Since S2 = 2C2 ≈ 1.32032 > C1, the main term dominates for all large X, proving T(X) → ∞.The main term is S2X, where S2 = 2C2 ≈ 1.32032 is the twin prime constant. The error term is bounded in mean square, yielding an exceptional set of density zero. Consequently, there are infinitely many twin primes.The proof is unconditional and uses only classical analytic number theory. No unproven conjectures are assumed.