The Equivalent Inclusion Method as a Transferable Mathematical Primitive for Science Agents

Keywords: equivalent inclusion method, Eshelby tensor, reaction--diffusion--advection, Green's function, autonomous science

Abstract: We formalize the Equivalent Inclusion Method (EIM) as an operator--theoretic primitive that an autonomous science agent can apply uniformly across disciplines. Many systems admit (i) a linear constant--coefficient operator on a homogeneous background, (ii) compact inhomogeneities representable as eigen--sources on bounded sets, and (iii) Green's function representations. Under these conditions, the heterogeneous problem is replaced by a homogeneous one with an unknown eigen--field supported on the inclusion and closed via an Eshelby map that depends only on the operator and the inclusion shape, not on the far--field forcing. We creatively extended this machinery from physics to reaction--diffusion--advection (RDA) dynamics for population dynamics and epidemiology, obtain a generalized Eshelby map and screened--Laplace limits, provide the two--inclusion interaction law, and develop analytical effective--medium closures (dilute/Maxwell--Garnett, Mori--Tanaka, and self--consistent) for the composite growth rate. Because only the operator and its Green's kernel are domain--specific, EIM serves as a reusable mathematical skill for agents transferring methods between scientific domains.

Submission Number: 182