Classification of Binary Residual Structures in Quotient Word Categories Derived from Minimal Structural Constraints

This work presents a purely mathematical classification result. Starting from word categories generated over finite alphabets with Q_n ≅ (ℤ₂)^n relations (n ≥ 2), and assuming only minimal structural well-definedness constraints on quotient constructions, we establish: • Any admissible quotient layer avoiding cyclic collapse must satisfy D ≤ 1. • In the NC2 branch (non-commutative residue absorbed under thin/poset constraints without refinement), the induced residual label is necessarily binary. • The induced binary algebra splits into exactly two algebraic types: (M) Boolean monoid (irreversible accumulation) (G) Abelian group (ℤ₂, ⊕) No geometric, metric, physical, or continuity assumptions are used. This paper establishes a complete candidate classification under minimal structural constraints.