This paper develops a model-theoretic framework for verifying context-conditioned language-model behavior by replacing benchmark labels with finite semantic certificates. The first problem is finite determinacy: when do examples in a context force the answer to a query without changing model parameters? In finite-field linear task families, we prove an exact row-space criterion, compute the residual hypothesis count, derive full and query-local identification curves, and show that extracting a smallest forcing subcontext is NP-complete even for binary outputs. The second problem is threshold emergence: when does an apparent benchmark jump reflect a semantic transition rather than a discontinuity of the scoring map? We prove an anti-mirage theorem separating thresholded metrics from semantic confidence and give a rate-sensitive crossing bound for latent commitments becoming visible above threshold. The common semantic object is a confidence functional on definable events. We show that it is a Boolean probability measure, equivalently a Keisler measure on the relevant type space, whose measure-one formulas form a proper filter and whose Stone-space representation is invariant under definitional expansion. The resulting calculus provides finite context certificates, pair-separator hitting sets, query teaching dimension, prompt-preservation criteria, and scale-limit witnesses. Exact-arithmetic ancillary scripts reproduce the finite-field and threshold calculations and generate the data used by the figures.

翻译：本文发展了一个模型论框架，通过用有限语义证书替代基准标签来验证上下文条件下的语言模型行为。首先研究有限确定性问题：在无需改变模型参数的情况下，上下文中的示例何时能强制确定查询答案？对于有限域线性任务族，我们证明了精确的行空间准则，计算了残差假设数量，推导出完整识别曲线和查询局部识别曲线，并证明即使对于二元输出，提取最小强制子上下文也是NP完全的。第二个问题是阈值涌现现象：基准测试中的明显跳跃何时反映语义转变而非评分映射的不连续性？我们证明了一个反幻象定理，将阈值化度量与语义置信度区分开来，并给出了隐性承诺在阈值之上变得可见的速率敏感交叉界。共同的语义对象是关于可定义事件的置信度泛函。我们证明它是一个布尔概率测度，等价于相关类型空间上的Keisler测度，其测度为一的公式构成一个真滤子，并且其Stone空间表示在定义扩张下不变。由此产生的演算提供了有限上下文证书、对分离子击中集、查询教学维度、提示保留准则和尺度极限见证。精确算术辅助脚本可复现有限域和阈值计算，并生成了图表所使用的数据。