发表机构
Conjecture Labs; The University of Wisconsin-Madison（康杰奇实验室; 威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出解释的数学理论（MTI），将解释视为与观察者相关的光谱测量，定义理性熵等概念，推导其性质，建立构建带显式假设与保证的解释方法的理论基础。
AI 中文摘要
本文呈现了《解释的数学理论》（MTI）的精简核心内容，该理论将解释视为访问结构下与观察者相关的光谱测量。MTI 将解释转化为方法设计问题：访问、查询、效用和媒介决定了观察者可选择、识别、交流或弃权（不执行）的内容。在学习不变的希尔伯特实现中，理性熵衡量知识、效用和媒介之间的残余不确定性。在有限有效 regime（ regime 译为“ regime”，此处保留）下，我们对其零集进行分类。成对可混淆性等价于均匀原子坍缩，而即使其他零成本状态仍为非原子时，唯一的效用最大值也可选择一个原子。这反转了可混淆性通常的零误差作用：至少一个观察者方向上的一致排除了未解决的多原子读数，而联合标签保留了识别结果。相应的自由设计容量是除最小方向预算外所有项的乘积。一个四条件证书刻画了有限交换码扇区上的清晰、可解码、媒介保真且与顺序无关的读出，并在这些保证失效时返回类型化阻碍。这些结果共同确立了 MTI 作为构建具有显式访问假设、保证和失败模式的解释方法的理论基础。
英文摘要
This article presents the abridged core of \emph{A Mathematical Theory of Interpretation} (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes.
CommentsAbridged core theory of the 2026 dissertation A Mathematical Theory of Interpretation
