Unification of Information Curvature and Geometry

This work establishes a unified framework in which information curvature—defined as the trace of the Fisher information matrix on the information layer Ψ—serves as the fundamental source of spacetime geometry. The universe is formulated as a five‑layer hierarchical structure consisting of the information layer Ψ, the eigenvalue layer (black phase), the wave/coordinate layer (white phase), the geometric layer (gray phase), and the phenomenological layer Φ. Within this hierarchy, all physical quantities arise as successive projections of information curvature.

The central result of this paper is the unified equation

which shows that the spacetime curvature tensor is directly generated from information curvature. This formulation reinterprets the energy–momentum tensor in general relativity as the geometric image of information, thereby positioning Einstein’s equation as a special case of a more fundamental information‑geometric structure.

The five‑layer hierarchy naturally yields the seven fundamental equations of physics—mass, energy, wave dynamics, geometry, and observable phenomena—as a systematic decomposition of information curvature. In particular, the white phase is identified as the layer where time and space emerge through the relations x=ct and Et∼ℏ, while the gray phase elevates these coordinates into full spacetime geometry.

Finally, the phenomenological layer Φ is defined as the observer‑dependent projection of geometry, completing the mapping from information to observable reality. This unified framework provides a new conceptual foundation for connecting information geometry, spacetime structure, and physical phenomena, offering a potential pathway toward a deeper understanding of quantum gravity.