Monroe (2026) shows that the nonexistence of an optimal proof system can be read as an information constraint regarding canonical hard instances: no sound arithmetic theory simulates the extensions adjoining sufficiently large, unprovable Busy Beaver values. Furthermore, if the best-known route to simulation is also necessary -- that is, if simulation requires a relative-consistency explanation over a weak base theory -- then the same constraint holds for inaccessible Kolmogorov-randomness facts. Call this Kolmogorov Hardness (KH). We argue that open questions in computational complexity can likewise be reformulated as information constraints involving Kolmogorov-random strings. Variants of KH yield, as conditional consequences, dense families of small hard tautologies, no-mutual-help phenomena for independent random axioms, PH noncollapse with explicit dense separators at each level, $SAT\notin P/poly$, and canonical disjoint NP pairs arising from random-axiom constructions. Time-bounded and sparse-support variants extend the same template to one-way functions via Liu--Pass, derandomization, natural-proofs-style limitations, and Feige-style random refutation. This framework gives a unified working model of complexity theorists' beliefs, organized around canonical hard instances. It seems self-evident that efficient proofs in a theory should not leverage true randomness facts the theory cannot verify. Yet the structural features of KH and its variants suggest they may be formally independent of standard metatheories. They behave like reflection principles; their internal readings fail in nonstandard models even when the corresponding external readings are true; and the same information constraints may apply to the metatheories themselves. We propose a research program: extend the model, resolve questions of formal independence, and identify which principles are potential new axioms.

翻译：Monroe（2026）表明，最优证明系统的不存在性可被解读为关于规范硬实例的一种信息约束：不存在一个可靠的算术理论能够模拟其扩充了足够大且不可证明的忙碌海狸值之后的理论扩展。进一步地，如果已知最佳的模拟路径也是必要的——即，模拟需要基于一个弱基础理论的相对一致性解释——那么相同的约束也适用于不可达的柯尔莫哥洛夫随机性事实。我们将此称为柯尔莫哥洛夫硬性（KH）。我们论证，计算复杂性中的开放问题同样可以重新表述为涉及柯尔莫哥洛夫随机字符串的信息约束。KH的变体作为条件性推论，可推导出：稠密的小型硬重言式族、独立随机公理的无互帮助现象、具有显式稠密分离子的PH非坍缩、$SAT\notin P/poly$，以及由随机公理构造产生的规范不相交NP对。时间有界和稀疏支撑变体通过Liu--Pass、去随机化、自然证明类限制以及Feige式随机反驳，将同一模板扩展到单向函数。该框架给出一个围绕规范硬实例组织的复杂性理论家信念的统一工作模型。一个自明的观点是：理论中的有效证明不应利用该理论无法验证的真实随机性事实。然而，KH及其变体的结构特征表明，它们可能形式上独立于标准元理论。它们的行为类似于反射原理；其内部解读在非标准模型中失效，即便相应的外部解读为真；相同的信息约束可能也适用于元理论本身。我们提出一个研究计划：扩展该模型，解决形式独立性问题，并确定哪些原则是潜在的新公理。