Tōh : Undistorted Mediation, the Culmination of Prayer

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.