Mathematical and Structural Analysis of Attention Mechanisms and Dimensional Invariance in the Kajino Multiplicative LLM

Abstract The Transformer architecture, which forms the foundation of contemporary natural language processing (NLP) and large language models (LLMs), processes conceptual representations additively within vector spaces as linear combinations (the additive model: \boldsymbol{v}_A + \boldsymbol{v}_B). However, as context lengths increase, this additive approach merely accumulates constituents in parallel, introducing fundamental limitations such as an inability to capture organic synergies among elements and a tendency toward thematic deviation (context collapse). In this paper, we apply the "Kajino Multiplicative Model (1^n = 1)"—originally formulated to describe the structural existence of life and family—to the computational paradigm of next-generation artificial intelligence, proposing the theoretical framework of the "Kajino Multiplicative LLM (KM-LLM)." Within this model, interactions among elements are represented through multiplicative tensor operations (\boldsymbol{v}_A \otimes \boldsymbol{v}_B) as a unified "Field." This yields a non-dispersive attention structure that maintains the overall semantic space's harmony at a value of 1, regardless of how many dimensions (exponents n) are added. Furthermore, by incorporating the "Law of Zero," wherein the omission of a prerequisite condition collapses the entire product to 0, we mathematically prove the architecture's effectiveness in establishing rigorous logical reasoning and ethical verification capabilities.