# Foundations of Stochastic Lexical Calculus: Semantic Descent and Random Dynamics on Probability Simplices

arXiv: https://arxiv.org/abs/2609.20207 · HTML: https://mlsanity.com/paper/2609.20207/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Linguistic Calibration of Long-Form Generations](https://mlsanity.com/paper/2404.00474/) (2024)
- [Confidence Under the Hood: An Investigation into the Confidence-Probability Alignment in Large Language Models](https://mlsanity.com/paper/2405.16282/) (2024)
- [Calibrating Semantic Uncertainty from Observable Language-Model Probabilities](https://mlsanity.com/paper/2607.17447/) (2026)
- [Language Invariant Properties in Natural Language Processing](https://mlsanity.com/paper/2109.13037/) (2021)
- [Generalizing determinization from automata to coalgebras](https://mlsanity.com/paper/1302.1046/) (2013)
- [Coalgebraic Behavioral Metrics](https://mlsanity.com/paper/1712.07511/) (2017)
- [Up-To Techniques for Behavioural Metrics via Fibrations](https://mlsanity.com/paper/1806.11064/) (2018)
- [Probability, valuations, hyperspace: Three monads on Top and the support as a morphism](https://mlsanity.com/paper/1910.03752/) (2019)
- [Measuring Fine-Grained Semantic Equivalence with Abstract Meaning Representation](https://mlsanity.com/paper/2210.03018/) (2022)
- [Probabilistic Attribution For Large Language Models](https://mlsanity.com/paper/2605.21726/) (2026)

## Abstract (arXiv)

Large language models produce prompt-dependent probabilities over words, whereas scientific systems require uncertainty over meaningful states that can be updated as evidence arrives. We develop an observable framework for determining when language-derived probabilities support such a sequential state representation. Theoretically, we define typed measurable transformations of contextual language, construct a minimal closed representation, and give necessary and sufficient conditions for semantic updates to exist uniquely. We bound irreducible nonclosure and accumulated error, and under average contraction prove existence, uniqueness and stability of an external random recursion on a probability simplex. These results define a stochastic lexical calculus without attributing an internal calculus to the language model. Empirically, frozen experiments test the observable implications. Raw prompt-conditioned probabilities fail the prespecified invariance gate; after prompt-specific calibration, a common three-state representation passes the stability gates and covers 28 of 30 untouched eight-step paths, or 0.933 at nominal level 0.90. Accordingly, language probabilities support a stochastic state only conditionally on verified closure, stability and coverage within a declared operating domain.