Prime Distribution in Binary Space via Recursive Masking

This paper proposes a deterministic framework for extracting prime distributions directly within binary computational space through layered recursive IF-THEN masking algorithms. Classical analytic number theory has traditionally described only the macroscopic density behavior of primes using decimal-oriented continuous analysis, such as the Prime Number Theorem. However, these approaches do not directly describe the microscopic structural placement of primes inside binary arithmetic space. To address this limitation, this study reconstructs prime configurations using binary-layer semantics. First, all even-number noise is completely removed through recursive right-shift decomposition, isolating odd-number skeletal structures whose least significant bit is fixed at 1. Next, finite bit-length spaces are treated as constrained physical cages in which the total odd-number capacity and the number of interfering periodic waves become finite and controllable. Under these constraints, a recursive IF-THEN masking algorithm is constructed by dynamically synchronizing bit-length growth with low-order odd-multiple interference waves. By recursively extending the bit-length parameter toward infinity, the model attempts to isolate prime configurations through deterministic bit-pattern masking without direct division operations. Furthermore, the study identifies Mersenne prime configurations as structural saturation limits within binary arithmetic space, where carry propagation energy reaches maximal occupancy states. The paper presents this framework as a structural interpretation model and requests independent mathematical verification regarding the complete rigor of its recursive masking dynamics.