Two-Adic Obstruction Theory and the Algebraic Limits of Dyadic Shell Screening

This preprint develops an obstruction-theoretic interpretation of the dyadic shell framework for semiprime arithmetic. The work reframes dyadic shell screening not as a general-purpose factorization method, but as a structured diagnostic framework for understanding why visible, bounded, and locally guided searches fail to reveal the hidden CRT split of a generic semiprime. The manuscript establishes a conservative algebraic core based on CRT branch decomposition, the distinction between homogeneous and mixed branches, the equivalence between useful mixed-root discovery and factor extraction, density bounds for blind visible-representative search, two-adic valuation constraints, and finite-depth two-adic orthogonality. It further analyzes the limitations of local perturbation, square-divisor visibility, standard small-root and Coppersmith-type approaches, auxiliary quadratic-residue filters, and periodic affine shell structure. The central conclusion is that the useful mixed branches are structurally hidden from public two-adic filters, bounded visible screening, and generic local search procedures unless additional non-public or non-generic structure is available. The framework therefore positions dyadic shell arithmetic as a tool for structural diagnostics, weak-key analysis, and obstruction theory rather than as a replacement for established integer factorization algorithms. The paper also includes a worked example, a notation index, a comparison with Fermat-type search, a discussion of classical versus quantum scope, and an extension of the branch-decomposition principle to squarefree multiprime moduli.