Extended Imaginary Number Theory: A Conceptual Framework for the Dual Structure of Existence

This paper presents Extended Imaginary Number Theory, a conceptual framework that describes the structure of existence as a superposition of a real-dimensional aspect (D) and an imaginary-dimensional aspect (iD). The real dimension comprises all structural aspects that currently bear meaning for the subject — whether observed, articulated, conceptualized, imagined, or projected as hypothesis. The imaginary dimension refers to the pre-meaningful, non-manifest, non-defined structural surplus that accompanies an object. Drawing on a structural analogy with complex numbers, the totality of these aspects is denoted Z = D + iD. The paper provides (1) conceptual definitions and explicit boundary clarification, (2) five axioms governing their relationship, (3) structural implications, and (4) connections to phenomenology, cognitive science, complex-systems theory, and latent-variable models in machine learning. Japanese title: 拡張虚数理論：存在の二重構造としての Z = D + iD 本稿は、存在の構造を「実次元」（real dimension, D）と「虚次元」（imaginary dimension, iD）の重ね合わせとして記述する概念的枠組み「拡張虚数理論」（Extended Imaginary Number Theory）を提示する。両側面の総体を、複素数との構造的類比に基づき Z = D + iD と表記する。本稿では、(1) 実次元と虚次元の概念的定義および両者の境界の明示、(2) 両次元の関係性を規定する五つの公理、(3) この枠組みから導かれる存在記述の構造的特徴、(4) 現象学・認知科学・複雑系理論・機械学習との接続点を論じる。