Yuma Muranushi

This paper defines as R-ealism the tacit ontology on which the world is exhaustively describable, and argues for the structural limit this position meets when it carries its own inner logic to the end. R-ealism equates being real with being formally describable—with R-describability as its representative form—and has run through both the natural science and the metaphysics of the modern period. But this dominance comes at a price. However precisely we describe an object, the description always leaves a surplus that has not yet risen into meaning; a description can record what has already taken shape, but can never contain the very arising by which that shape comes to be. R-ealism has banished this surplus outside the real as "something subjective" or as "an appearance to be explained away in due course." Pursued to its limit, R-ealism runs up against the non-closure of description: in three regions—experience, meaning, and event—description necessarily fails to capture the arising that precedes its result. As a framework that positions this surplus without negating R-ealism, the paper sketches Extended Imaginary Number Theory (Z = D + iD; Muranushi 2026a) and proposes Extended Realism, which restores that surplus as the imaginary dimension iD on the side of the real. The conclusion: R-ealism is not false, but narrow.

This paper, the third in the Extended Imaginary Number Theory series, takes as its subject the higher-order concept of arising organized in the second paper, "The Structure of Arising" (Muranushi, 2026b) — namely Tōh (transparency, 透). Whereas the arising of the second paper was a discrete event premised on the separation between the capturing subject and the captured event, Tōh designates the limit at which that separation dissolves and the sub ject becomes the medium of the ceaseless self-arising of arising — not a sequel to arising but its inverted completion. The central claim of this paper lies in this inversion of the subject — the turn from one who captures to a medium. This paper formalizes the activation condition of Tōh as becoming-prayer. Becoming-prayer is the point at which the subject–object separation dissolves, and is expressed as the product Φ = P × K of purity P (the degree of connection to the first impulse) and resolve K (the opening of identification energy). Φ is not a quantity measuring the magnitude of intensity, but an order parameter for the phase called Tōh; the passage of its critical threshold activates Tōh as a phase transition. The mode that rises when Tōh reaches its critical point is named becoming-realm — a mode that sheds its determination as an individual subject and stands open onto a structural domain (the realm), appearing as bareness, compelling force, and structural opening-up, and producing, as its consequence, a resonance of phase (ontological induction). Its intensity is stipulated by Φ = P × K, its scale by the subject's perspectival scope. Formally, Tōh is defined as the ideal limit VTōh(Z) = Z of the observer function V(Z) = αD + βiD. The central thesis is that becoming realm comes to hold not under the structure of self-expansion but under the structure of self purification. The description proceeds throughout from the real-dimensional side and claims no direct access to the imaginary dimension. The precise empirical verifiability of Φ and V(Z), and the systematic unfolding of the series as a whole, are deferred to subsequent papers.

This paper, the second in the Extended Imaginary Number Theory series, takes as its subject arising (起こり)—the event in which fine perturbations of imaginary-dimensional origin are captured within cognitive space. Whereas the first paper, "Extended Imaginary Number Theory: Existence as the Dual Structure Z = D + iD" (Muranushi, 2026), described the dual structure of existence itself, the present paper asks under what structure arising comes to hold. This is the core of the inquiry. Within this framework, the imaginary-dimensional side is described as continuously exerting influence on the real-dimensional side. Within this continuous influence, what stands at the point of origin of upsurge on the real-dimensional side is arising. The accurate capture of arising substantially regulates the precision of the subsequent concretization (具現化). The present paper extracts the structure under which arising comes to hold, thereby providing a structural description of its relation to the precision of concretization. Intentional grasp—the work by which the subject actively cognizes an object—and arising are not mutually exclusive existential modes; rather, they form a gradation of qualitative texture in relation to the imaginary dimension. The present paper takes this continuum as its premise, phenomenologically describes the situation in which the arising-leaning pole becomes foregrounded, and further organizes its precision along two axes: perspectival scope (視座) and purity (純度). This paper proceeds throughout as a description from the real-dimensional side. It does not directly describe what occurs in the imaginary dimension but rather aims to extract the structure under which arising comes to hold on the real-dimensional side. The treatment of precision remains a preliminary structural description; its existential and practical development is deferred to a separate work.

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。