Latent Coherence: Do Frontier Language Models' Probability Judgments Obey the Law of Total Probability?

Typed-decision models such as TypeSafe's Jev answer a declared yes/no or multiple-choice question about a state with a probability instead of text, and their evaluations report accuracy and calibration. Neither requires that the probabilities a model gives to logically related questions fit together item by item, which is what a system that acts on those probabilities needs. We test this property, coherence, with a battery of logically linked questions that needs no labels. For 160 items from ChaosNLI and PubMedQA, each with three mutually exclusive labels, we ask whether the label is X, whether it is not X, whether it is one of the other two labels, and which label applies. On 480 negation pairs, Jev's probabilities for "the label is X" and "the label is not X" miss summing to one by 0.064 on average (95% CI 0.055 to 0.072). Qwen3.8-27B, run from its official BF16 weights, misses by 0.293 with first-token probabilities and by 0.122 with verbalized probabilities. The gap to the first-token readout persists on pairs where both systems give similar probabilities, without double-negation labels, and after averaging Jev's repeated calls. Jev is not coherent either: its violations are about five times its repeat noise, and it over-endorses statements about single labels, so that its three single-label probabilities sum to 1.14 on average. The two systems also fail differently. Qwen3.8-27B's first-token readout under-endorses the complement of a label whether or not the question contains "not", rejecting both a statement and its negation in 196 of 480 pairs, and it does not become more coherent where it is more confident, whereas Jev's violations concentrate where its answer is uncertain. Because the checks need no labels, they expose biases that appear only when question forms are compared, and inconsistencies within items.

The generative nature of Large Language Models (LLMs) is reflected in the conditional probabilities they compute to sample each response token given the previous tokens. These probabilities encode the distributional structure that the model learns in training and exploits in inference. In this work, we use these probabilities to situate LLMs within the mathematical theory of stochastic processes. We use this framework to design a model-agnostic probabilistic token attribution measure, using Bayes rule to invert the next-token log-probabilities so as to capture the models internal representation of the distribution over token sequences. The representation is independent of the models computational structure. This representation yields the conditional probability of the response given the prompt, and of the response given the prompt with a token marginalized away. Our attribution score is the log of the ratio of these probabilities. We further compute the entropies of a single prompts token distributions, conditioned on the remaining context. The interplay between entropy and attribution score sheds light on LLM behavior. We evaluate 8 models across 7 prompts and investigate anomalies, token sensitivity, response stability, model stability, and training convergence, thereby improving interpretability and guiding users to focus on uncertain or unstable parts of the generation.

The rapid capability gains of frontier language models are widely attributed to improved reasoning abilities, yet this cannot be verified as raw CoT traces in closed-source systems are hidden. By registering a simple custom tool through a standard API feature, we induce frontier models to externalize intermediate reasoning. Because these traces may reflect post-hoc rationalization rather than genuine reasoning, we first evaluate against native CoT on open-source models and extend to closed-source frontier models including GPT-6 Astra. We find that the extracted reasoning matches native reasoning performance and substantially outperforms no-reasoning baselines, across competition mathematics, science, and code generation. We then characterize how frontier models structure their intermediate reasoning. Across token efficiency, reasoning-step types, and induced reasoning trees, we identify systematic differences in how models externalize, compress, and organize reasoning. We find that Astra exhibits token-efficient directed reasoning, selecting a correct trajectory earlier, while resolving elementary steps internally and externalizing only crucial reasoning. These findings provide a behavioral lens on frontier-model reasoning beyond benchmark scores.