What question did this study set out to answer?

The aim is to classify binary residual structures associated with quotient word categories under minimal constraints.

February 17, 2026Open Access

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

Key Points

The aim is to classify binary residual structures associated with quotient word categories under minimal constraints.
Used mathematical classification techniques starting from finite alphabets.
Analyzed quotient constructions while enforcing minimal structural constraints.
Identified properties of quotient layers and their induced binary algebras.
Established that admissible quotient layers must have D ≤ 1 to avoid cyclic collapse.
Found that binary algebra splits into two types: Boolean monoid and Abelian group.
Demonstrated the induced residual label is binary in the NC2 branch under specific constraints.

Abstract

This work presents a purely mathematical classification result. Starting from word categories generated over finite alphabets with Qₙ ≅ (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.

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

Key Points

Abstract

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