A Complete and Detailed Proof of Polignac's 2 Conjecture 3 by the Method of Double Arithmetic 4 Progressions 5 and Physical-Space Period Cancellation

We present a complete, rigorous, and unconditional proof of Polignac’s conjec10 ture, which asserts that for every even integer y ≥ 2, there exist infinitely many 11 pairs of primes (q, q +y). The twin prime conjecture corresponds to the special case 12 y = 2. 13 The proof introduces a fundamentally new framework that combines five key in14 novations. First, a geometric primality criterion establishes that in the interval (p, p2 15 ), an integer is prime if and only if it is coprime to all primes ≤ p. This de16 terministic equivalence completely circumvents the parity problem of classical sieve 17 theory. Second, a double arithmetic progression structure with CRT pre-sieving 18 processes small primes r ≤ log p via the Chinese Remainder Theorem, constructing an admissible set A0 modulo M0 = Q r≤log p 19 r. This ensures the crucial condi20 tion gcd(M0, d) = 1 for all subsequent moduli d. Third, physical-space period 21 cancellation exploits the fact that the arithmetic progression a + tM0 (mod d) 22 traverses a complete residue system, causing error contributions from complete pe23 riods to vanish identically. Fourth, resonance breaking decomposes exponential sums over A0 into 3 π(log p) = p o(1) 24 explicit terms, eliminating the large factor |A0| 25 from estimates. Fifth, a four-zone modulus partition applies distinct estimation 26 techniques to different ranges of moduli d in the Möbius expansion. 27 All error terms are rigorously shown to be of strictly lower order than the main term C0p 2/(log p) 2 28 , where C0 > 0 equals the Hardy-Littlewood constant for the 29 pair (0, y). The proof uses only classical Fourier analysis on finite abelian groups, 30 elementary number theory, and standard estimates from analytic number theory. 31 No unproven hypotheses are assumed. Comments and feedback are welcome.Please feel free to contact me via wuhaizhu0512@163.com