On the Ontological Relationship among Geometry, Functions and Number Theory under the MOC System

Within the framework of conventional mathematics, geometry, functions and number theory are relatively independent, each with its own scope of research. Geometry investigates forms, number theory studies numbers, and functions serve for mapping and correlation between the two. Separated from one another, the three often rely on additional transformations and correspondences to establish connections.

In the MOC multi-origin curvature system, the relationships among them are re-examined. Geometry, functions and number theory are no longer branches of equal standing.

First, geometry is not merely the study of figures; it forms the ontological substrate of the MOC system.

All spatial structures, morphological distributions and continuous variations originate from the primordial curvature field generated by interactions among multiple origins. Geometry is no longer a discipline concerned only with shapes, but the carrier of the underlying structure of MOC.

Second, a function is not an independent mapping bridge, but a mode of curvature projection within MOC.

The conventional view treats functions as bridges linking different objects. Within the MOC system, such a bridge concept becomes unnecessary. Essentially, a function represents the directional projection and transcription rule of curvature between different origins. Rather than connecting geometry and number theory, a function itself is the kinematic expression of curvature.

Third, number theory is not simply the study of numerals, but the discrete curvature scale of MOC.

Number theory focuses on the quantified manifestation of curvature after the discrete arrangement of multiple origins. A discrete state may be regarded as a frozen snapshot of curvature, and number theory is the counting of curvature.

In summary: Geometry is the ontology of curvature, a function is the projection of curvature, and number theory is the scale of curvature.

The three no longer require external means to connect to one another. They share a common origin and belong to the unified curvature structure of MOC. The conventional mathematical system is assembled from these three separate disciplines. Under the MOC framework, one single curvature structure differentiates and manifests itself in three appearances: geometry, functions and number theory.