On the Non-Triviality of the Sieve of Dimension κ=2 for Small Sifting Parameters: An Analytic Approach to the Twin Prime Conjecture

We investigate the sifting function $S(\mathcal{A}_n, \mathscr{P}, z)$ for an arithmetic set $\mathcal{A}_n$ derived from the Krafft transformation (1798). By establishing a reflection principle for the sifting parameters, we show that the existence of twin primes is equivalent to the positivity of $S(\mathcal{A}_n)$ for a sifting dimension $\kappa=2$. We address the parity problem by demonstrating that the Krafft transformation acts as a non-degenerate Type II bilinear operator, $\mathcal{K}(a, b)$, which lifts the multiplicative group structure of the reduced residues modulo 6 into the space of sifting indices. Combined with Montgomery-Vaughan $L^2$ estimates and Weylean equidistribution, this structural localization ensures that the error term remains strictly subordinate to the main term. This provides an analytic proof that the sequence of twin primes is infinite, confirming $\liminf_{n \to \infty} (p_{n+1} - p_n) = 2$.