Finite Certificates for In-Context Determinacy and a Threshold Theory of Emergence in Language Models
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Computer Science > Machine Learning
arXiv:2606.07623 (cs)
[Submitted on 30 May 2026]
Title:Finite Certificates for In-Context Determinacy and a Threshold Theory of Emergence in Language Models
View a PDF of the paper titled Finite Certificates for In-Context Determinacy and a Threshold Theory of Emergence in Language Models, by Faruk Alpay and 1 other authors
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Abstract: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.
| Comments: | 40 pages; ancillary files provided |
| Subjects: | Machine Learning (cs.LG); Logic in Computer Science (cs.LO) |
| MSC classes: | 03C13, 03C98, 03B70, 68Q15 |
| Cite as: | arXiv:2606.07623 [cs.LG] |
| (or arXiv:2606.07623v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.07623
arXiv-issued DOI via DataCite (pending registration)
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View a PDF of the paper titled Finite Certificates for In-Context Determinacy and a Threshold Theory of Emergence in Language Models, by Faruk Alpay and 1 other authors
Ancillary-file links:
Ancillary files (details):
- certificate_benchmark_protocol.txt
- certificate_benchmark_responses.jsonl
- certificate_benchmark_summary.csv
- fixed_query_curve.csv
- identification_curve.csv
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