正拓扑与可行精化：力迫矩阵、正性与信息
Positive Topology and Feasible Refinement: Forcing Matrices, Positivity, and Information
AI总结：
本文从点与可观察性质的关系出发构建正拓扑框架，区分基于点与无点表述，提出信息论与博弈论两种解释，并纲领性地引入资源约束，以支持可解释、资源感知的推理。
AI中文摘要：
我们从点或模型与可观察性质之间的基本关系出发，对正拓扑（Positive Topology）进行了概念性和操作性的阐述。由这一关系，涌现出两种互补的结构。第一种结构捕捉了普遍精化（universal refinement）与覆盖（cover）：在所有相关情形中必须成立的内容，以及信息如何被系统地精化。第二种结构捕捉了正性（positivity）与见证存在（witnessed existence）：无需依赖经典补集即可被正面实现和维持的内容。一个核心结果表明，点与可观察性质之间的底层关系可以从这两种导出结构中的任一种重构出来。我们还阐明了基于点的（point-based）与无点的（pointfree）表述之间的区别：当点可用时，正性可以从底层力迫关系（forcing relation）中推导出来，而在形式化的无点设定中，正性被视为原始概念，其与覆盖的相容性被公理化地施加。我们为这一框架发展了两种互补的解释。第一种是信息论的解释，将覆盖视为部分信息的精化，将正性视为见证的可行性（witnessed feasibility）。第二种是博弈论的解释，将正性视为见证者或假设在连续的合法精化中存续的能力。论文的最后部分刻意带有纲领性。我们概述了如何将资源约束、验证成本和有限预算纳入该框架。这引出了对力迫、覆盖、正性和精化的资源敏感（resource-sensitive）概念，并提出了关于这些结构在可用资源变化时如何行为的新问题。来自医学诊断、法律推理和人工智能系统的例子说明了该方法对基于证据的、可解释的和资源感知的推理的潜在相关性。
英文摘要：
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.
