The Polignac Conjecture via Resonance Breaking and Symmetric Polynomials

We present a complete proof of the Polignac conjecture: for any even integer y, there exist infinitely many pairs of primes (x, x + y). In particular, taking y = 2 yields the twin prime conjecture. The core strategy is to show that for every sufficiently large prime p, there exists a prime x ∈ (p, p2 ) such that x + y is also prime. This is achieved through three technical innovations. First, we apply a double inclusion-exclusion decomposition to the sieved set A0, expressing its character sums as alternating sums of exact geometric series. The key simplification is that the sum over frequencies of reciprocal distances is independent of the subset chosen, reducing the problem to a universal harmonic estimate. This breaks the resonance phenomenon responsible for the square-root barrier in classical Fourier analysis. Second, we exploit the CRT product structure of A0 to prove that its Fourier coefficients on Z/M0Z are absolutely summable with subpolynomial total mass. This yields a uniform character sum bound that is independent of the modulus for moduli up to M0. Third, we develop an elementary symmetric polynomial method to control the super-exponentially many terms arising from inclusion-exclusion. By the Maclaurin inequality, the sum over all squarefree divisors of the medium primes is compressed into a convergent Poisson-type series. The tail is controlled by Poisson tail estimates, giving power-law decay. The modulus range is divided into three regimes. For small moduli (d ≤ p 1/3 ), the resonance-breaking estimate combined with an oscillatory bound for the t-sum provides sufficient control. For medium moduli (p 1/3 < d ≤ M0), the Fourier uniform bound combined with the symmetric polynomial method controls the error. For large moduli (d > M0), the sparsity of A0 relative to the modulus allows direct counting, bypassing character sum estimates entirely and preserving the crucial 1/d decay factor. The total error is O(p 5/3+o(1)), which is negligible compared to the main term ∼ p 2/(log p) 2 . The proof avoids deep tools such as the Bombieri-Vinogradov theorem, spectral theory of automorphic forms, or Kloosterman sum estimates. Comments and feedback are welcome.Please feel free to contact me via [email protected]