Dyadic Shell Correspondences and Equivalence Relations in Semiprime Arithmetic

This preprint develops a structural correspondence framework for dyadic shell identities in semiprime arithmetic. The paper studies how exact dyadic shell configurations can be embedded into the broader algebraic setting of congruence-of-squares relations and how these relations project onto hidden CRT sign classes that determine the corresponding greatest-common-divisor outcomes. The work is not presented as a new factorization algorithm. Instead, it provides a classification and diagnostic framework for understanding why local congruence filters, two-adic profiles, recombination procedures, and dyadic-center constructions may have local capacity while still lacking public selectivity toward factor-revealing branches. The manuscript formalizes a monoid-to-group correspondence pipeline, separates local filter capacity from branch selectivity, analyzes finite-window endpoint effects, gives a cautious treatment of ultrametric local-unit residue profiles, and compares the framework with classical factorization paradigms such as Fermat-type identities, continued fractions, quadratic sieve, number field sieve, Coppersmith-type methods, ROCA-style structured-prime attacks, and elliptic-curve factorization. The preprint also includes small complete examples, computational-audit pseudocode, and a list of open problems for future work, including distributional models for public dyadic representatives, geometry-of-numbers questions, multi-base selectivity models, structured-prime visibility experiments, and possible quantum-period-finding analogues. This version is prepared as a clean preprint release for archival publication. It removes internal changelog material, keeps the mathematical scope focused, and states the framework as a structural theory of congruence-of-squares correspondences rather than as a practical factorization method.