This paper develops a formal framework for understanding why some research domains admit a plurality of stable theories while others rapidly collapse toward a small number of highly constrained formalisms. The starting intuition is that a theory is not merely a set of propositions but a way of selecting and organizing a high-dimensional explanatory domain into local regions, each with its own salience, conceptual granularity, and inferential constraints. We formalize a theory as a constrained explanatory atlas on a topological space of phenomena, then study the resulting refinement dynamics, the category of admissible theory-objects, and the local-to-global problem of theory unification. The central mathematical claim is that theoretical multiplicity is governed not only by cover refinement but by the geometry of the possibility space of admissible atlases, whose structure is naturally captured by sheaves, torsors, and cohomological obstructions. In this setting, theory pluralism arises when many inequivalent atlases are locally admissible yet globally non-gluable, while rigidity corresponds to strong descent conditions and vanishing obstructions. The paper gives rigorous definitions, proves several structural propositions, and outlines a program for further formal development.
Keywords: theory pluralism, cover refinement, sheaf theory, descent, torsors, cohomology, scientific explanation, theoretical rigidity.
Publication Date: 2026-06-18