Dyadic Shell Structures in Semiprime Arithmetic: A Structural Framework for Structured-Prime Diagnostics

Abstract We develop a structural framework for analyzing odd semiprimes n = pq through the constrained dyadic representation in the form of difference of power of 2 and square equal to multiple of n. The framework combines dyadic shell geometry, odd-integer accumulation, parity and residue constraints, 2-adic valuation rigidity, and finite arithmetic lattice analysis via the Chinese Remainder Theorem. Rather than proposing a replacement for asymptotically superior factorization algorithms, the structures studied here are positioned as a theoretical diagnostic layer for examining arithmetic regularity in semiprime families and structured prime-generation settings. The strongest rigorous component is the valuation-coupling identity ν (α) = 2ν (s), valid under the constraint 0 <s<2ʳ. We also formulate finite CRT admissibility densities for quadratic-difference residue lattices and discuss their relevance to structured-prime diagnostics. Possible applications include weak-key analysis, entropy-collapse interpretation, structured-prime auditing, and cryptographic assessment of prime-generation procedures. This work does not claim polynomial-time factorization, improvement of GNFS, or practical replacement of existing algorithms. No empirical validation is provided; the paper should be read as a theoretical framework and research program rather than a validated cryptanalytic methodology.