Yuma Muranushi

This paper formalises perspective in terms of the coordinate structure of thought itself. We propose three constitutive elements — range, orthogonality, and origin — and clarify the structural invisibility of the effective origin from within the coordinate system it defines, distinguishing two modes of that invisibility: the non-objectifiability of the origin arising from its indeterminacy, and the backgrounding of the presuppositional layer arising from its operating as the standard of measurement. We then distinguish the origin proper (a structural position of zero value) from the effective origin (the origin together with the presuppositional layer that overlays it), and show that what ordinary language calls "movement of the origin" is not translation but a three-phase process: detachment of that layer (zeroing), exposure of zero (the transitional phase), and re-formation under a newly constituted layer. This distinction dissolves the apparent tension between defining the origin as zero and asserting that it moves. The driving conditions of the process are organised as the product of three variables — core-tremor, duration, and plasticity — and the natural frequency that governs core-tremor is assigned neither to the origin itself nor to the presuppositional layer, but to a substratum (body and history) that persists across the three phases. Because the origin has zero value across all axes, we derive an epistemological reversal, articulated as a difference in temporality between derivation and generation: the origin is not a past source but the present locus that continuously sustains the possibility of axes. This generativity is specified strictly as an enabling condition rather than an efficient cause. Finally, given the structural correspondence between the indeterminate origin and emptiness (空, śūnyatā), everything else constituted within the model is relationally constituted in a manner corresponding to dependent origination (縁起, pratītya-samutpāda), and we re-address the long-standing question of self (ātman) versus no-self (anātman) — including the Mahāyāna reintroduction of a "true self" and the Buddhist traditions that reject such affirmative formulations — through the unified concept of generative emptiness.

Why does extension in mathematics terminate, while extension in ontological description does not? The field of real numbers, confronted with its failure to be algebraically closed, was closed by the addition of a single axis: the imaginary uniti. The description of the real dimension, by contrast, does not close even once the imaginary dimension iD has been acknowledged. This paper locates the ground of that difference in the recursivity of extension — the structure whereby a description that acknowledges a non-describable position generates, with each acknowledgement, a further residue. As the sixth paper of the Extended Imaginary Number Theory series, it undertakes two tasks. First, it shows that the self-limitation running through the preceding five papers — the mode of description that never describes the imaginary dimension directly but requires its existence structurally — accumulated across five stages, and that this accumulation functioned not as a negative brake but as the condition enabling entry into deeper territory. Second, it formalises the operation at the core of that mode as a theory of residue. Minimal extension, established as a principle in Paper V — a description that fails to close requires an independent axis, and only the minimum required — divides in two by the criterion of whether the introduced axis is itself describable: into instrumental extension and residual acknowledgement. Only the former can bring a system to closure (Proposition P1a); the latter necessarily recurs (Proposition P1b). The three operations the series has performed — the addition of the imaginary unit i, the requirement of imaginary time it, and the acknowledgement of the imaginary dimension iD — are unified under this single classification, and structural simultaneity, left by Paper V as a task for subsequent work, is likewise given as a corollary of the theory; its application is restricted to systems whose antecedent has been argued for (the real time axis t). The conclusion of this paper contracts into a single proposition: a descriptive system can be extended beyond itself only in virtue of failing to close, and whether that extension terminates or recurs is settled by whether the required residue itself becomes describable.

This paper sets out from a single circumstance: the description of what we ordinarily call "time"—the one-dimensional flow taken to run in one direction from past to future, the time axis t—does not close on its own. A description that does not close demands independent descriptive axes that complement it, and demands them only minimally. This paper names this general policy the Principle of Minimal Extension and applies it to the structure of time. As its consequence, time is described as complex time T = t + it, given as the composition of the real time axis t and the imaginary time axis it orthogonal to it. This unfolds into the structure of time the ontological formula presented in the first paper, "Extended Imaginary Number Theory: A Conceptual Framework for the Dual Structure of Existence" (Muranushi, 2026a), and corresponds structurally to the historical extension of the number system from the one-dimensional number line to the two-dimensional complex plane. The central claim of this paper condenses into one point. The present is a field in which different phases coexist. Onto the phase now occurring, the phase just passing away and the phase about to arrive are superimposed—and this coexistence of phases is not reducible to the before-and-after relations of t: it is not recovered however finely the real time axist is divided, nor by replacing it with any interval. That phases separated on t coexist within a single present—this structural simultaneity—can be placed only on an independent descriptive axis orthogonal to t. This irreducibility is the paper's central ground for demanding it as an independent descriptive axis. Unlike imaginary time in physics (the Wick rotation; the Hartle–Hawking no-boundary proposal), which is introduced technically as an analytic continuation of real time, the imaginary time it of this paper is demanded as a descriptive axis independent of t. The two do not compete; they are orthogonal at different descriptive strata. This paper does not assert the reality of imaginary time it; it addresses the descriptive circumstance that the description of real time t does not close on its own.

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。