In direct response to requests for a logico-mathematical test of the conjecture, we formally disprove a recently conjectured artificial intelligence trade-off between epistemic certainty and scope in its published universal hyperbolic product form, as introduced in Philosophy and Technology. Certainty is defined as the worst-case correctness probability over the input space, and scope as the sum of the Kolmogorov complexities of the input and output sets. Using standard facts from coding theory and algorithmic information theory, we show, first, that when the conjecture is instantiated with prefix (self-delimiting, prefix-free) Kolmogorov complexity, it leads to an internal inconsistency, and second, that when it is instantiated with plain Kolmogorov complexity, it is refuted by a constructive counterexample. These results establish a main theorem: contrary to the conjecture's claim, no universal "certainty-scope" hyperbola holds as a general bound under the published definitions. We further show that a subsequent "entropy-based" revision, replacing the Kolmogorov scope with Shannon joint entropy and redefining the epistemic certainty level accordingly, cannot restore universality either.

翻译：为直接回应针对该猜想进行逻辑-数学检验的请求，我们形式化地证伪了近期在《哲学与技术》期刊上提出的、以通用双曲乘积形式呈现的人工智能中认知确定性与范围之间的权衡假说。其中，确定性被定义为输入空间上的最坏情况正确概率，范围则定义为输入集与输出集柯尔莫哥洛夫复杂度之和。利用编码理论和算法信息论中的标准结论，我们首先证明：当该猜想以前缀（自定界、无前缀）柯尔莫哥洛夫复杂度进行实例化时，会导致内部矛盾；其次，当以普通柯尔莫哥洛夫复杂度进行实例化时，则可通过构造性反例予以反驳。这些结果确立了一个主要定理：与猜想声称相反，在已发表的定义下，不存在作为普遍界成立的通用"确定性-范围"双曲线。我们进一步证明，后续基于"熵的"修正版本——即用香农联合熵替代柯尔莫哥洛夫范围并相应重新定义认知确定性水平——也无法恢复其普遍性。