Identification and Learning of Semantic Observation Kernels: Partial Observation, Uniform Recovery, & Minimax Limits

Probabilistic text generators supply conditional distributions over tokens and complete verbal continuations, whereas scientific use often requires a posterior over a finite state. Large language models are the leading example: phrase probabilities depend on prompt wording, and model-printed percentages are generated text rather than state posteriors. More generally, we ask when an observable language law can support a reproducible posterior over declared states. A semantic map groups meaning-equivalent continuations; held-out cases with reference posteriors identify a semiparametric inverse from grouped language probabilities to state probabilities. The language law remains nonparametric and no hidden model quantity is used. This is principally a theory and methods paper and makes several contributions. On the theoretical side, we derive conditions for existence, identification, stable recovery, and sequential updating; concentration, asymptotic, and nonparametric rates; identified sets under truncated probabilities; and a minimax boundary for uniform stability. On the empirical side, theorem- directed simulations verify recovery rates, compatible-set coverage, and stability gates, while two frozen language-model studies illustrate held-out recovery and conformal coverage. The results specify when observable language probabilities can provide an auditable state measurement without being interpreted as internal belief.