虚次元を文学し、数学し、哲学する

存在の構造を、現時点で意味を持つ構造的側面の総体である実次元（D）と、それに伴う非意味・非顕在・非定義のままの構造的余剰である虚次元（iD）の重ね合わせ Z = D + iD として記述する概念的枠組みを提示する。五公理を中核として、実次元の非閉鎖性・内在性・直交性・記述の非最終性・境界の動性を規定し、本シリーズの存在論的基盤を据える。

Presents a conceptual framework that describes the structure of existence as a superposition Z = D + iD: the real dimension (D), the totality of structural aspects that currently bear meaning, and the imaginary dimension (iD), the structural surplus that accompanies it while remaining non-meaningful, non-manifest, and undefined. Centred on five axioms — the non-closure, immanence, orthogonality, non-finality of description, and mobility of boundaries of the real dimension — it lays the ontological foundation of this series.

@article{muranushi2026eint,
  author    = {Muranushi, Yuma},
  title     = {Extended Imaginary Number Theory: A Conceptual Framework for the Dual Structure of Existence},
  journal   = {ï.Fragmenta},
  volume    = {1},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.20095255},
  url       = {https://doi.org/10.5281/zenodo.20095255}
}

第一論文が存在の二重構造そのものを記述したのに対し、本稿は「起こり」——虚次元由来の微細振動が認知空間上で気づきとして主体に捕捉される事象——を主題とする。意図的把握と「起こり」を排他的様式ではなく虚次元との関係における質感のグラデーションとして位置づけ、「起こり」寄りの極が前景化する事態を現象学的に記述する。「成立を許す」構え、気づきの事後性、「起こり」を阻む実次元側の三つの構造を析出し、その精度を視座・純度の二軸として整理する。

Whereas the first paper described the dual structure of existence itself, this paper takes as its subject “arising” — the event in which subtle vibrations originating from the imaginary dimension are captured by the subject as awareness within cognitive space. It positions intentional grasping and arising not as exclusive modes but as a gradient of texture in their relation to the imaginary dimension, and phenomenologically describes situations in which the pole closer to arising comes to the foreground. It draws out three structures — the posture of “allowing to hold,” the aposteriority of awareness, and the real-dimensional aspect that obstructs arising — and organises their precision along the two axes of perspective and purity.

@article{muranushi2026arising,
  author    = {Muranushi, Yuma},
  title     = {The Structure of Arising: Passive Aposteriority and the Posture of Allowing to Hold},
  journal   = {ï.Fragmenta},
  volume    = {2},
  pages     = {1--14},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.20254107},
  url       = {https://doi.org/10.5281/zenodo.20254107}
}

本稿は拡張虚数理論の第三論文として、第二論文「『起こり』の構造論」が整理した「起こり」の上位概念 ──「透」── を主題とする。第二論文の「起こり」が、捕捉する主体と捕捉される出来事の分離を前提とした離散的事象であったのに対し、透はその分離が解消し、主体が起こりの絶え間ない自己生起の媒体となる極限を指す ── 起こりの続編ではなく、その反転的完成である。本稿の中心的主張はこの主体の反転 ── 捕捉する者から媒体への転回 ── にある。

As the third paper of the Extended Imaginary Number Theory, this work takes as its subject Tōh — the higher-order concept of arising organised in the second paper, “The Structure of Arising.” 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 subject becomes the medium of arising’s continuous self-emergence — not a sequel to arising but its inverted completion. The central claim of this paper rests on this inversion of the subject — the turn from one who captures to a medium.

@article{muranushi2026toh,
  author    = {Muranushi, Yuma},
  title     = {T{\=o}h: Undistorted Mediation, the Culmination of Prayer},
  journal   = {ï.Fragmenta},
  volume    = {3},
  pages     = {1--22},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.20509278},
  url       = {https://doi.org/10.5281/zenodo.20509278}
}

本稿は、「世界は記述され尽くせる」という暗黙の存在論を実数主義（R‑ealism）と定義し、その立場が自らの内的論理を貫徹したときに突き当たる構造的限界を論証する。実数主義とは、実在的であることを形式的記述可能性──その代表形としての R‑記述可能性──に等値する立場であり、近代以降の自然科学と形而上学を貫いてきた。どれほど精緻に記述しても、対象には必ず、まだ意味として立ち上がっていない余剰が残る。本稿が示すのは、実数主義はその論理を極限まで貫いたとき、経験・意味・出来事という三つの局面で、記述が立ち上がりの手前を必ず取り残すこと──記述の非閉鎖性──に突き当たる、という事態である。この取り残される余剰を、実数主義を否定せず包含する形で位置づける枠組みとして、本稿は拡張虚数理論 Z = D + iD を素描し、その余剰を虚次元 iD として実在の側に回復する拡張実在論を提示する。結論は一文に集約される──実数主義は誤っているのではなく、狭いのである。

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. However precisely we describe an object, the description always leaves a surplus that has not yet risen into meaning. 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) 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.

@article{muranushi2026realism,
  author    = {Muranushi, Yuma},
  title     = {R-ealism at Its Limit: Beyond Reductive Physicalism},
  journal   = {ï.Fragmenta},
  volume    = {4},
  pages     = {1--21},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.20676271},
  url       = {https://doi.org/10.5281/zenodo.20676271}
}

本稿は、我々が日常的に「時間」と呼ぶ一次元の流れの記述が単独では閉じない、という事態から出発する。閉じない記述は、それを補完する独立した記述軸を最小限だけ要請する──本稿はこの方針を最小拡張原理と呼び、記述の閉鎖の定義と軸の負荷根拠という規範からの帰結として定式化したうえで、時間構造へ適用する。中心的主張は一点に集約される。現在とは、異なる位相が共存する場である。いま起こりつつある位相に、過ぎ去りつつある位相と来たるべき位相とが重なり合う──この位相の共存は、実時間軸tをいかに細かく分割しても、いかなる区間に置き換えても回収されない。本稿はこの還元不可能性を論証し、位相の共存を位置づける非実在の記述軸itを導入して、時間を複素的時間 T = t + it として記述する。物理学における虚時間（ウィック回転、Hartle–Hawking 無境界仮説）とは競合せず、異なる記述層において直交する。

This paper sets out from a single circumstance: the description of what we ordinarily call “time”—the one-dimensional flow from past to future—does not close on its own. A description that does not close demands independent descriptive axes that complement it, and demands them only minimally. Formulating this policy as the Principle of Minimal Extension—derived from a definition of descriptive closure and a norm on descriptive axes—the paper applies it to the structure of time. The central claim condenses into one point: the present is a field in which different phases coexist, and this coexistence is recovered neither by dividing the real time axis t however finely nor by replacing it with any interval. Arguing for this irreducibility, the paper introduces the non-real descriptive axis it on which the coexistence of phases is placed, and describes time as complex time T = t + it. It does not compete with imaginary time in physics (the Wick rotation; the Hartle–Hawking no-boundary proposal); the two are orthogonal at different descriptive strata.

@article{muranushi2026imaginarytime,
  author    = {Muranushi, Yuma},
  title     = {Imaginary Time: Descriptive Non-Closure and the Principle of Minimal Extension},
  journal   = {ï.Fragmenta},
  volume    = {5},
  pages     = {1--15},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.21185463},
  url       = {https://doi.org/10.5281/zenodo.21185463}
}

なぜ数学の拡張は完結し、存在論的記述の拡張は完結しないのか。実数体は、代数的に閉じないという事態に一本の軸——虚数単位 i を加えることで閉じた。だが実次元の記述は、虚次元 iD を承認してもなお閉じない。本稿はこの差の根拠を、拡張の再帰性——不可記述な位置を承認する記述が、そのつど新たな残余を産む構造——に求める。本稿は、拡張虚数理論シリーズの第六論文として、二つの作業を行う。第一に、先行五論文を貫く記述様式——虚次元を直接記述せず、その存在を構造的に要請する様式——の自己制限が五段階を経て累積し、この累積が消極的なブレーキではなく、より深い領域への踏み込みを可能にする条件であったことを示す。第二に、この様式の核にある操作を残余の理論として定式化する。第五論文が原理化した最小拡張——閉じない記述は独立軸を最小限だけ要請する——は、導入される軸それ自体の記述可能性を基準に、装置的拡張と余剰的承認とに二分される。前者のみが体系を閉じさせうる（命題 P1a）のに対し、後者は必然的に再帰する（命題 P1b）。虚数単位 i の増設・虚時間 it の要請・虚次元 iD の承認というシリーズの三操作は、この単一の分類のもとに統合され、第五論文が後続の課題とした構造的同時性も、この理論の系として与えられる——その適用は、前件が論証された体系（実時間軸 t）に限定される。本稿の帰結は一つの命題に集約される——記述体系は、自らが閉じないことにおいてのみ、自らを超えて拡張されうる。そして、その拡張が完結するか再帰するかは、要請された残余それ自体が記述可能になるか否かで決まる。

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 unit i. 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.

@article{muranushi2026residue,
  author    = {Muranushi, Yuma},
  title     = {The Ontology of Residue: A Philosophy of Structural Non-Describability and a Theory of Minimal Descriptive Extension},
  journal   = {ï.Fragmenta},
  volume    = {6},
  pages     = {1--18},
  year      = {2026},
  publisher = {i.Institute},
  doi       = {10.5281/zenodo.21974402},
  url       = {https://doi.org/10.5281/zenodo.21974402}
}