We present a complete, rigorous, and unconditional proof of Polignac’s conjecture, which asserts that for every even integer y ≥ 2, there exist infinitely many pairs of primes (q, q+y). The twin prime conjecture corresponds to the special case y = 2. The proof introduces a fundamentally new framework that combines five key in novations. First, a geometric primality criterion establishes that in the interval (p, p2), an integer is prime if and only if it is coprime to all primes ≤ p. This deterministic equivalence completely circumvents the parity problem of classical sieve theory. Second, a double arithmetic progression structure with CRT pre-sieving processes small primes r ≤ log p via the Chinese Remainder Theorem, constructing an admissible set A0 modulo M0 = Q r≤log p r. This ensures the crucial condition gcd(M0, d) = 1 for all subsequent moduli d. Third, physical-space period cancellation exploits the fact that the arithmetic progression a + tM0 (mod d) traverses a complete residue system, causing error contributions from complete periods to vanish identically. Fourth, resonance breaking decomposes exponential sums over A0 into 3π(log p) = po(1) explicit terms, eliminating the large factor |A0| from estimates. Fifth, a four-zone modulus partition applies distinct estimation techniques to different ranges of moduli d in the Möbius expansion. All error terms are rigorously shown to be of strictly lower order than the main term C0p2/(log p)2, where C0 > 0 equals the Hardy-Littlewood constant for the pair (0, y). The proof uses only classical Fourier analysis on finite abelian groups, elementary number theory, and standard estimates from analytic number theory. No unproven hypotheses are assumed.