A Study of Modulo-10 Channels and the TGS-MAR Model for Large-Prime Generation and Micro Security Chips

This study addresses the computational bottleneck caused by passive blind traversal in large-prime generation within computational number theory. This paper introduces an optimization framework leveraging modulo-10 residue-class subspace partitioning and tiered step-size jumping. By constructing geometric relative partition channels across the four characteristic tracks (1, 3, 7, and 9), the model eliminates up to 60% of non-prime structural density over an infinitely extended numerical set.Continuous stress tests within a single residue channel validate the theoretical acceleration of large-stride jumping during random-prime generation. Crucially, the experiments expose a software-level micro-impedance bottleneck, where large-number object memory allocation severely dampens performance in high-level computing environments.To achieve storage balance in embedded micro security chips, the TGS-MAR (Modulo-10 Channel Return) model is introduced. To address the vulnerability of a small 30-digit key space against GPU brute-force attacks, the model proposes a "Sparse Random Subsampling" strategy. Fixing 2,000 selected primes into read-only memory (ROM) compresses the map volume within 31.3 KB. This architecture is strictly positioned as a lightweight protection mechanism rather than an industrial-grade cryptographic defense. A ROM dump reduces the effective key space to the pairwise combinations of the stored primes, making it ideal for low-security embedded tokens.