Positive Topology and Feasible Refinement: Forcing Matrices, Positivity, and Information

Abstract

We develop a conceptual and operational account of Positive Topology starting from a basic relation between points or models and observable properties. From this relation, two complementary structures emerge. The first captures universal refinement and cover: what must hold across all relevant cases and how information can be systematically refined. The second captures positivity and witnessed existence: what can be positively realized and sustained without relying on classical complements. A central result shows that the underlying relation between points and observables can be reconstructed from either of these induced structures. We also clarify the distinction between point-based and pointfree formulations: when points are available, positivity can be derived from the underlying forcing relation, while in the formal pointfree setting positivity is taken as primitive and its compatibility with cover is imposed axiomatically. We develop two complementary interpretations of the framework. The first is information-theoretic, viewing cover as refinement of partial information and positivity as witnessed feasibility. The second is game-theoretic, viewing positivity as the ability of a witness or hypothesis to survive successive legitimate refinements. The final part of the paper is deliberately programmatic. We outline how resource constraints, verification costs, and finite budgets can be incorporated into the framework. This leads to resource-sensitive notions of forcing, cover, positivity, and refinement, and raises new questions about how these structures behave as available resources change. Examples from medical diagnosis, legal reasoning, and AI systems illustrate the potential relevance of the approach to grounded, explainable, and resource-aware inference.