Noncommutative Infinite Kaiso

Abstract

This paper develops a unified algebraic framework for kaiso complex numbers, a discrete multi-layered structure (kaiso) in which each layer (kai) contains a single digit carrying an independent sign bit. The central operation is the digit-wise sign inversion, governed by two XOR-based mechanisms: local inversion (Target1) and propagation toward lower digits across kai (Target2). We derive the closed formula\[t(i,j)=\bigoplus_{k=0}^{n} (\bar{i_k} j_k)\;\oplus\;\bigoplus_{k=1}^{n} ((i_k \oplus j_k) j_{k-1}),\]which expresses the inversion parity entirely in terms of digit agreement and disagreement across kai. Using this formula, we show that the associated matrices form a tightly constrained discrete eigenvalue system: all eigenvalues are $\pm 1$, eigenvectors correspond directly to digit patterns across kai, and the value component is determined by the XOR shift $a_{i\oplus j}$. These properties yield a closed,self-similar eigenvalue algebra that persists uniformly across kaiso depth. We further demonstrate that the two-series structure (Target1 and Target2) cannot be collapsed into a single homogeneous rule. The 0-digit (the least significant digit, LSD) acts as an intrinsic singular layer that prevents such unification and enforces the directional asymmetry of the inversion mechanism.Finally, we outline preliminary considerations for defining multiplication on kaiso complex numbers, emphasizing the structural asymmetry introduced by Target2 and the resulting possibility of non-commutative behavior.