What question did this study set out to answer?

To extend Euler's identity to relate rotations in the context of different dynamical universality classes.

January 17, 2026Open Access

The Generalized Euler Identity: A Unified Rotation Between the Four Dynamical Universality Classes

Key Points

To extend Euler's identity to relate rotations in the context of different dynamical universality classes.
Replaced the imaginary unit i with an operator J satisfying J^2 = -I.
Encoded four fundamental PDE universality classes into a doubled dynamical state.
Developed a formalism of the Generalized Euler Identity e^{θJ} = cos θ I + sin θ J.
Established a unified rotation framework among elliptic, hyperbolic, parabolic, and unitary classes.
Demonstrated that this framework reveals a natural cyclicity among canonical dynamical regimes.

Abstract

Euler’s identity e^iθ = cos θ + i sin θ expresses rotation in the complex plane. We extend this identity by replacing the imaginary unit i with an operator J satisfying J² = −I, where J acts on a doubled dynamical state encoding the four fundamental PDE universality classes: elliptic, hyperbolic, parabolic (diffusive), and unitary (Schrodinger-type). This gives the Generalized Euler Identity e^θJ = cos θ I + sin θ J, which performs Wick rotation through physical law itself. This formalism reveals a natural holofractal cyclicity among the canonical dynamical regimes.

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