Overview: This study introduces the Turkish Sieve Methodology, a novel approach for prime number computation designed to overcome the memory intensity and modular arithmetic constraints inherent in traditional sieve algorithms. While current literature recognizes an N/3 bit density for prime candidate representation, this method achieves an N/6 bit data structure specifically optimized for identifying twin (p, p+2) and cousin (p, p+4) prime pairs. Key Innovations: Memory Efficiency: By reducing the candidate pair sequences to an N/6 bit structure, the methodology effectively doubles the memory efficiency compared to existing bit-sieve models, enabling the processing of massive datasets that were previously computationally prohibitive. Computational Optimization: The entire elimination process is transformed into an integer-addition-based operation (n <-- n + p). By replacing expensive modular arithmetic (MOD/DIV) with deterministic rhythmic progression and bitwise operations, the algorithm is tailor-made for high-performance CPU and GPU (CUDA) cores. Hardware Awareness: The methodology leverages the parallel processing capabilities of modern GPU architectures, allowing for rapid execution and high-throughput candidate screening. Conclusion: The Turkish Sieve offers a significant advancement in computational number theory, providing a scalable and deterministic tool for researchers exploring prime distributions, twin prime conjectures, and post-quantum cryptographic foundations. This manuscript is a preliminary preprint version. The current version is under revision for submission to a peer-reviewed journal.Keywords: Number Theory, Twin Primes, Cousin Primes, Turkish Sieve, GPU Computing, CUDA, Bit Sieve, High-Performance Computing (HPC), Prime Gap. Performance Benchmarks & Scalability Report The Turkish Sieve (TS) methodology has been stress-tested across vast ranges and various hardware architectures. The following results demonstrate the deterministic performance and memory efficiency of the N/6 indexing paradigm: Extreme Range Test (10^14 - 100 Trillion): Device: NVIDIA RTX 3070 Time: 2831.702 seconds Result: 135,780,321,665 Twin Prime Pairs found. Throughput: 35,314 Million/s (35.3 G-items/s) VRAM Usage: Only 1,143 MB High-Speed Throughput Test (10^13 - 10 Trillion): Device: NVIDIA RTX 3070 Time: 141.593 seconds Result: 15,834,664,872 Twin Prime Pairs found. Global Speed: 70,624 Million/s (70.6 G-items/s) High-Speed Throughput Test (10^12 - 1 Trillion): Device: NVIDIA RTX 3070 Time: 9.144 seconds Result: 1,870,585,220 Twin Prime Pairs found. Global Speed: 109,361 Million/s (109.3 G-items/s) Mid-Range Hardware Baseline (10^11 - 100 Billion): Device: NVIDIA GTX 1650 Ti Time: 2.604 seconds Result: 224,376,048 Twin Prime Pairs found. Speed: 38,402 Million/s Native CPU Parallel Performance OpenMP (10^11 - 100 Billion): System: Intel i7-10750H (12 Threads) Time: 13.521 seconds Result: 224,376,048 Twin Prime Pairs found. Speed: 7,396 Million/s System RAM: 148 MB (Minimal footprint)