Noncommutative Infinite Hierarchy

This paper develops a unified algebraic framework for hierarchical complex numbers, a discrete multi-layered structure in which each digit carries an independent sign bit. The central operation is the digit-wise sign inversion, whose behavior is fully determined by two XOR-based mechanisms: local inversion (Target1) and inter-digitpropagation (Target2). We derive a closed general formula (i, j) =₊=₀^n (iₖ jₖ) \;\;₊=₁^n ( (iₖ jₖ) j₊-₁), expresses the inversion parity purely in terms of bit agreement and disagreement. Using this formula, we show that the associated matrices possess a highly constrained and self-contained eigenvalue structure: all eigenvalues are 1, eigenvectorscorrespond directly to bit patterns, and the value component is governed by the XOR shift a₈ ₉. These properties yield a fully closed discrete eigenvalue algebra with intrinsic self-similarity across digit layers. The paper also examines the impossibility of collapsing the two-series structure (Target1 and Target2) into a single homogeneous series, demonstrating that digit k=0 forms an essential singular layer that prevents such unification. Finally, we outline preliminary considerations for defining multiplication on hierarchical complex numbers, highlighting the inherent asymmetry introduced by Target2 and the resulting possibility of non-commutative behavior.