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 $S(\mathcal{A}, \mathscr{P}, z)$ for an arithmetic set $\mathcal{A}$ derived from Krafft’s criterion (1798). By establishing a reflection principle for the sifting parameters, we show that the existence of twin primes is equivalent to the positivity of $S(\mathcal{A})$ for a sifting dimension $\kappa=2$. We address the parity problem by demonstrating that the Krafft transformation acts as a Type II bilinear form, $\mathcal{K}(a, b)$, encoding multiplicative structure into the sifting indices. Combined with Montgomery-Vaughan $L^2$ estimates and Weylean equidistribution, this ensures the error term remains subordinate to the main term, providing an analytic proof that $\liminf_{n \to \infty} (p_{n+1} - p_n) = 2$.