Imaginary Time: Descriptive Non-Closure and the Principle of Minimal Extension

This paper sets out from a single circumstance: the description of what we ordinarily call "time"—the one-dimensionalflow taken to run in one direction from past to future, the time axis t—does not close on its own. A description that doesnot close demands independent descriptive axes that complement it, and demands them only minimally. This papernames 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 axisit orthogonal to it. This unfolds into the structure of time the ontological formula presented in the first paper, "ExtendedImaginary Number Theory: A Conceptual Framework for the Dual Structure of Existence" (Muranushi, 2026a), andcorresponds structurally to the historical extension of the number system from the one-dimensional number line to thetwo-dimensional complex plane.The central claim of this paper condenses into one point. The present is a field in which different phases coexist. Onto thephase now occurring, the phase just passing away and the phase about to arrive are superimposed—and this coexistenceof phases is not reducible to the before-and-after relations of t: it is not recovered however finely the real time axis t isdivided, nor by replacing it with any interval. That phases separated on t coexist within a single present—this structuralsimultaneity—can be placed only on an independent descriptive axis orthogonal to t. This irreducibility is the paper'scentral 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 introducedtechnically as an analytic continuation of real time, the imaginary time it of this paper is demanded as a descriptive axisindependent of t. The two do not compete; they are orthogonal at different descriptive strata. This paper does not assertthe reality of imaginary time it; it addresses the descriptive circumstance that the description of real time t does not closeon its own.