『Extended Imaginary Number Theory』 Officially Published on International Academic Repository Zenodo ── A New Ontological Framework Describing Existence as Z = D + iD

The first visual of the announced Extended Imaginary Number Theory.

il Co., Ltd. (Headquarters: Tokyo, Founder: Yuma Suguri) announced that Founder Yuma Suguri's uniquely systematized ontological framework, "Extended Imaginary Number Theory," was officially published on May 9, 2026, in Zenodo, an international academic repository operated by the European Organization for Nuclear Research (CERN). This paper has been assigned a permanent identifier (DOI: 10.5281/zenodo.20095255), connecting it to the international academic citation network.

◆ "Real Dimension" and "Imaginary Dimension" ── Describing Existence as a Dual Structure

This paper presents a conceptual framework that describes the structure of an object's existence as a superposition of a "real dimension (D)" and an "imaginary dimension (iD)." The real dimension refers to the totality of structural aspects of an object that currently hold meaning (taking the form of observation, verbalization, conceptualization, or imagination), while the imaginary dimension refers to the non-meaningful, non-manifest, and undefined structural surplus accompanying the object. The totality of both aspects is expressed as follows, based on a structural analogy with complex numbers:

Z = D + iD

Here, Z represents the conceptual totality for describing the existence of an object, and D and iD are positioned as the articulated component (corresponding to the real part in complex numbers) and the non-articulated component (corresponding to the imaginary part) of Z, respectively.

◆ Systematized as a Fourteen-Page Treatise Centered on Five Axioms

This paper is a 14-page (A4) academic paper, with both Japanese and English versions included in the same deposit. At the core of the paper are the following five axioms:

● Axiom 1: Non-closure of D ── The real dimension is always accompanied by a structural surplus that cannot be fully captured by its description.

● Axiom 2: Immanence ── The imaginary dimension component is not external to the object but belongs to its internal structure.

● Axiom 3: Orthogonality ── The real dimension and imaginary dimension coexist as mutually orthogonal components within Z.

● Axiom 4: Non-finality of Description ── A complete description of Z(X) is impossible at any subject or time.

● Axiom 5: Dynamism of Boundary ── The boundary between the real dimension and the imaginary dimension is not fixed and fluctuates with the progress of cognition.

This framework is positioned as an attempt at an integrated description, mediated by the structural analogy of imaginary numbers, for the common problem of "structures that cannot be fully captured by articulated/manifest descriptions," which existing academic traditions have individually addressed, such as phenomenology (Husserl's horizon structure, Merleau-Ponty's bodily potentiality), cognitive science (Polanyi's tacit knowledge), emergence in complex systems theory, and latent variable models in machine learning.

◆ Not a Claim of Mathematical Rigor, but an Ontological Framework

What is emphasized in this paper is that this framework does not claim algebraic rigor in pure mathematics. Z = D + iD does not hold as an object equipped with algebraic operations of complex numbers or the inner product structure of Hilbert space; the relationship between the two is a structural analogy, not an algebraic isomorphism.

Furthermore, this paper does not directly describe the imaginary dimension iD itself, but rather describes the non-closability of D, and iD is positioned as the structural surplus required by that non-closability. Moreover, to position this non-closability as a structurally independent direction within the description of existence, this paper introduces iD as an orthogonal axis. That is, iD is not negatively defined as the incompleteness of D, but is actively positioned within Z as a structural direction irreducibly different from D. This treatise attempts a unique theoretical positioning by connecting with multiple intellectual traditions, such as the argumentative structure that explicitly states this self-limitation from the outset, its connection to phenomenology, cognitive science, complex systems theory, and machine learning, and its explicit relationship with East Asian metaphysical traditions (Daoist "nothingness," Buddhist "emptiness").

◆ As a Foundation for Subsequent Papers

This paper is positioned as the ontological foundation for a series of theories systematized by Yuma Suguri to date. The following subsequent research is planned to be developed based on this paper:

● General theory of articulation ── Theorization of the transition mechanism from iD to D

● Mother vowel operator theory ── Concrete implementation of articulation using five operators ä → ï → ü → ë → ö

● Peer-reviewed journal submission paper mediated by Izutsu / Kūkai ── Preparation for submission to Philosophy East and West as a candidate journal.