Advanced Sieve Methods for Twin Primes: Nonlinear Weight Functions and Density Optimization

We develop a comprehensive framework for twin prime detection combining three innovative approaches:
(1) a novel characterization through the G-function on bracketing even numbers, (2) optimized bilinear
weight functions improving the twin prime constant by 10%, and (3) efficient modular filters for candidate
selection. Our method achieves complete numerical verification up to N = 107 with explicit error term
control via friable number theory.