Foundations of Stochastic Lexical Calculus: Semantic Descent and Random Dynamics on Probability Simplices
Abstract
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.
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BibTeX
@misc{dixon2026foundations,
title = {{Foundations of Stochastic Lexical Calculus: Semantic Descent and Random Dynamics on Probability Simplices}},
author = {Matthew F Dixon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20207},
eprint = {2609.20207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20207},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Foundations of Stochastic Lexical Calculus: Semantic Descent and Random Dynamics on Probability Simplices
AU - Matthew F Dixon
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20207
UR - https://arxiv.org/abs/2609.20207
AB - 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.
ER -