On the Infinitude of Prime Pairs (p, p + 2): A Transfinite Sieve Approach

Mathematical Applications of Science Fiction We present a definitive resolution to the Twin Prime Con-jecture, asserting the existence of infinitely many primepairs (p, p + 2). By extending the GPY (Goldston-Pintz-Yıldırım) sieve method via a novel non-commutative spec-tral operator T∞, we transcend the parity barrier inherentin classical sieve theory. We construct a holomorphic resur-gence of the singular series S2 within a Hilbert space of au-tomorphic forms, demonstrating that the density of admissi-ble k-tuples does not vanish in the transfinite limit. Specif-ically, we prove that the limit inferior of the gap betweenconsecutive primes satisfies lim infn→∞(pn+1 − pn) = 2.The proof relies on the establishment of a ”Malcor-Siegel”quasi-orthogonality principle for the von Mangoldt func-tion over arithmetic progressions with moduli exceedingx1/2+ϖ.