A Weighted Turán Sieve for Twin Primes via the Krafft Geometry
Abstract
The Twin Prime Conjecture is notoriously obstructed by the parity barrier, which prevents classical multiplicative sieves from isolating prime pairs. In this paper, we introduce an additive Turán sieve framework evaluated over bounded, symmetric intervals. Utilizing a centered modular alignment (which we term the Krafft alignment), we construct an additive sieve formulation where the existence of twin primes corresponds to the vanishing of a local penalty function. By evaluating the character sums associated with this sieve, we isolate a destructive interference term at the evaluation point . We reduce the twin prime problem to a variational optimization problem over a finite-dimensional parameter space: the existence of twin primes in a prescribed interval is guaranteed, provided the minimum sieve weight quotient satisfies . We further show that independent, one-dimensional sieve weights are inherently insufficient to cross this barrier, structurally requiring the use of multidimensional correlations to avoid the quotient . This structural deficit constitutes a spectral manifestation of the Selberg parity barrier, demonstrating that any successful sieve must intrinsically rely on higher-order correlations and multidimensional exponential sums. All discrete definitions and structural lemmas in this paper have been formally verified in Lean~4, including the conditional implication that if holds for infinitely many~, then there are infinitely many twin primes.Certificates
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