Theoretical Division and Complementary Relationship between MOC and MIE

Author: Suhang Zhang, Luoyang

Within the present theoretical framework, Multi-Origin Curvature (MOC) and Maximum Information Efficiency (MIE) constitute two axiomatic systems with distinct hierarchical levels and mutual complementarity.

MOC governs the geometric structural layer, addressing the question of how the world is constructed. It extends traditional single-origin geometry to a structure of competing curvatures from multiple origins, turning geometry itself into an active generator of physics—forces, potentials, fields, and flows are all reduced to curvature distributions and competing origins. MOC provides geometric necessity for morphogenesis (e.g., brachistochrone, optical path length, biological networks, fractal structures), and unifies the inverse-square law, wave equations, geodesics, the principle of least action, and topological invariants within a consistent framework of curvature flow and extremal geometry. MOC serves as the mathematically rigorous backbone of the theory, formalizable via differential geometry, calculus of variations, curvature flow, and Dirichlet energy.

MIE governs the functional optimization layer, addressing the question of why systems are stable. It asserts that long-term stable dynamical systems necessarily extremize the information processing efficiency per unit energy consumption. In this way, MIE offers a unified meta-explanation for all historically isolated extremal principles (Fermat’s principle, brachistochrone, principle of least action, minimum entropy production, Murray’s law, Euler’s formula, etc.): they are all special cases of maximizing information efficiency. MIE further provides an evolutionary selection mechanism—only structures with the highest information efficiency can be preserved under long-term competition, allowing the theory to span physics, biology, networks, topology, and computation.

The two principles are strictly complementary: MOC generates the full space of possible geometric states (feasible curvature distributions, paths, branching patterns, and topological morphologies), while MIE selects from them the stable structures satisfying extremal conditions. MOC defines the possible modes of existence of a system; MIE determines its ultimately stable mode of existence.

In summary, MOC provides the geometric ontology and rules of structure generation, forming the mathematical foundation of the theory. MIE provides the criteria for functional optimization and evolutionary selection, serving as the unifying logical thread of the theory. Together, they form a complete, self-consistent framework ranging from geometric construction to functional optimization.