Demonstrates the relationship between observed outputs and complex structures, implying novel applications in various fields.
This paper examines the physical structure represented by the imaginary unit when measured outputs are real. It distinguishes a zero-valued observation from an absent state using observation fibers, signed balance, and interaction-dependent readout. A real scalar oscillation is shown to require a two-dimensional minimal first-order realization whose positive-metric-preserving oriented exchange is generated by an operator satisfying J2=−IJ2=−I. Finite-time observability recovers the transverse quadrature hidden from an instantaneous scalar readout. Applications to mechanics, circuits, electromagnetism, transport, pattern formation, and quantum relative phase are discussed. The conclusions are conditional and do not identify the imaginary unit with a universal hidden substance or literal +1/−1+1/−1 constituents.
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Yoshida (2026) studied this question.
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