Multi-Origin Geometry (MOC) and the Dual Convergence Theorem: A Unified Framework for Curvature-Driven Fluid Interfaces

Author: Zhang Suhang, Luoyang
Affiliation: Independent Researcher

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Abstract

This paper proposes a unified analytical framework combining multi‑origin higher‑dimensional geometry (MOS) with the law of large numbers, tailored for incompressible Stokes flow problems with curved interfaces. The MOS framework partitions the spatial domain into several subdomains, each constructing its own curvature field with its own center of mass as an independent origin. Curvatures from different origins are coupled via transformation mappings and satisfy chain‑complex constraint conditions. The stress jump at the interface obeys a generalized Young–Laplace proportionality relation with curvature.

Two independent convergence paths are rigorously established:

· Geometric contraction convergence: Under the overlapping covering condition L < 1, the curvature coupling map K is a contraction mapping. Uniqueness of the solution is proved via the Banach fixed‑point theorem.
· Statistical convergence (law of large numbers): Independent of geometric conditions, as the number of independent samples N \to \infty, the statistical average curvature converges with probability 1 to a unique steady‑state configuration, with convergence order O(N^{-1/2}).

This dual convergence system simultaneously incorporates the advantages of deterministic geometric solution and stochastic statistical approximation. This paper extends the concepts of nested origin configurations, geometric invariant characterization, and statistical error estimation, explicitly formulating the Stokes equations and curvature–stress coupled interface conditions. This work establishes a self‑consistent mathematical foundation for multiscale, curvature‑dominated interfacial flows (e.g., biomembranes, multiphase flow).

Keywords: Multi‑origin geometry; Stokes flow; curvature coupling; law of large numbers; Banach fixed‑point theorem; dual convergence; interfacial flow

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Note at the end:

What can be directly used for numerical computation and programming solution is:

-\nabla p + \mu\Delta \boldsymbol{u} + \boldsymbol{f}_K = 0

This document is an abstract of the theoretical framework. A complete version containing full derivations, detailed equations, and numerical validation will be published separately.