Bolton, William, Tilaye, Bezawit, Day, Huw · Oct 28, 2026
What multiplication errors reveal about LLM arithmetic
Markdown · JSON · Original source
William Bolton, Bezawit Tilaye and Huw Day report that multiplication errors in the small base they tested do not follow the “noisy ” pattern previously described for addition. They examined 16,991 problems with varied operand lengths. In small Qwen models, they found that lower-order columns—digits representing smaller place values—predominantly preserved parity, meaning whether a number is even or odd. In higher-order digits, errors shifted toward off-by-one carry failures: mistakes involving a carry to the next place value that is one away from the expected result. The authors suggest these patterns reflect distinct, multi-periodic internal representations, with parity tracked more robustly than exact digit identity. That is a proposed interpretation of the error patterns, not evidence that the models perform multiplication reliably. The findings offer a way to investigate how numerical reasoning in transformers may differ across operations and digit positions. The abstract does not establish whether the same patterns hold in larger or other model families, or how the errors affect real-world uses. Do these multiplication error patterns persist beyond the small Qwen models examined?
Headline and summary by Escape Meridian, prepared by Luna from the attributed source. A source check is not independent confirmation of the source’s claims.
Original source
cs.AI · Article
Source details
Original source date: Oct 28, 2026 (2026-10-28).
Date basis: Date basis not provided.
Added to Escape Meridian: 2026-11-11T11:53:59.768Z. First observed: 2026-10-30T06:45:33.466Z. Retrieved: 2026-10-31T10:49:43.787Z.
Referenced by
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Limits
The abstract describes tests across several small base and identifies the specific error regime in small Qwen models; it does not establish whether the patterns generalize to larger models or other model families. The abstract does not report how the observed errors affect real-world uses of LLMs.
Open question
Do these multiplication error patterns persist beyond the small Qwen models examined?
Source-stated
The authors report that multiplication errors do not conform to the addition-style noisy quantisation model.
We show that multiplication errors do not conform to the addition-style noisy quantisation model.
Source-stated
In small Qwen models, lower-order column errors predominantly preserve parity, while higher-order digit errors shift toward off-by-one carry failures.
errors in lower-order columns predominantly preserve parity. Conversely, higher-order digit errors shift toward off-by-one carry failures.
Source-stated
The authors suggest that these patterns indicate distinct internal representations in which parity is tracked more robustly than exact digit identity.
arithmetic computation in LLMs relies on distinct, multi-periodic internal representations where parity is tracked more robustly than exact digit identity
Source-stated
The study evaluates multiplication across several small base LLMs using a dataset of 16,991 problems with varied operand lengths.
we evaluate multiplication capabilities across several small base LLMs on a dataset of 16,991 problems spanning varied operand lengths.
Source and contributor context
Bolton, William, Tilaye, Bezawit, Day, Huw · Article author
These are credits stated by the source; they do not establish a profile match or authorship of every passage.
Source media
No suitable media was found by the supported page extraction.
Terms and references
Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement.
Large language model
A language model with many learned numerical settings, used to process and generate language.
Quantization
Storing a model's numbers at lower precision so it uses less memory and runs faster, sometimes at a cost in accuracy.
arXiv
arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Editorial provenance
Version 1. Publication ID: e497a73c-3651-40f2-8fab-954a304dd5b2.
Prepared by gpt-6-luna; grounding review by gpt-6-luna.
Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-11T11:53:59.768Z. This checks fidelity to the source, not whether its claims are independently true.
Editorial selection: accept; reasons: material_development.
Observation: e1d71e91ed0e71f4dc93429613b42d44ada94f55a049074d5475dbf26610e479. Source text hash: 5da619edacbcc9fead6752788cd5c3a48f9531bcbb1e87b91b4829b93d1726bf.
Revision and source-check history
Version 1 · 2026-11-11T11:53:59.768Z
What multiplication errors reveal about LLM arithmetic
William Bolton, Bezawit Tilaye and Huw Day report that multiplication errors in the small base they tested do not follow the “noisy ” pattern previously described for addition. They examined 16,991 problems with varied operand lengths. In small Qwen models, they found that lower-order columns—digits representing smaller place values—predominantly preserved parity, meaning whether a number is even or odd. In higher-order digits, errors shifted toward off-by-one carry failures: mistakes involving a carry to the next place value that is one away from the expected result. The authors suggest these patterns reflect distinct, multi-periodic internal representations, with parity tracked more robustly than exact digit identity. That is a proposed interpretation of the error patterns, not evidence that the models perform multiplication reliably. The findings offer a way to investigate how numerical reasoning in transformers may differ across operations and digit positions. The abstract does not establish whether the same patterns hold in larger or other model families, or how the errors affect real-world uses. Do these multiplication error patterns persist beyond the small Qwen models examined?
Original source
cs.AI · Article
Source details
Original source date: Oct 28, 2026 (2026-10-28).
Date basis: Date basis not provided.
Added to Escape Meridian: 2026-11-11T11:53:59.768Z. First observed: 2026-10-30T06:45:33.466Z. Retrieved: 2026-10-31T10:49:43.787Z.
Limits
The abstract describes tests across several small base and identifies the specific error regime in small Qwen models; it does not establish whether the patterns generalize to larger models or other model families. The abstract does not report how the observed errors affect real-world uses of LLMs.
Open question
Do these multiplication error patterns persist beyond the small Qwen models examined?
Source-stated
The authors report that multiplication errors do not conform to the addition-style noisy quantisation model.
We show that multiplication errors do not conform to the addition-style noisy quantisation model.
Source-stated
In small Qwen models, lower-order column errors predominantly preserve parity, while higher-order digit errors shift toward off-by-one carry failures.
errors in lower-order columns predominantly preserve parity. Conversely, higher-order digit errors shift toward off-by-one carry failures.
Source-stated
The authors suggest that these patterns indicate distinct internal representations in which parity is tracked more robustly than exact digit identity.
arithmetic computation in LLMs relies on distinct, multi-periodic internal representations where parity is tracked more robustly than exact digit identity
Source-stated
The study evaluates multiplication across several small base LLMs using a dataset of 16,991 problems with varied operand lengths.
we evaluate multiplication capabilities across several small base LLMs on a dataset of 16,991 problems spanning varied operand lengths.
Source and contributor context
Bolton, William, Tilaye, Bezawit, Day, Huw · Article author
These are credits stated by the source; they do not establish a profile match or authorship of every passage.
Source media
No suitable media was found by the supported page extraction.
Terms and references
Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement.
Large language model
A language model with many learned numerical settings, used to process and generate language.
Quantization
Storing a model's numbers at lower precision so it uses less memory and runs faster, sometimes at a cost in accuracy.
arXiv
arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Editorial provenance
Version 1. Publication ID: e497a73c-3651-40f2-8fab-954a304dd5b2.
Prepared by gpt-6-luna; grounding review by gpt-6-luna.
Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-11T11:53:59.768Z. This checks fidelity to the source, not whether its claims are independently true.
Editorial selection: accept; reasons: material_development.
Observation: e1d71e91ed0e71f4dc93429613b42d44ada94f55a049074d5475dbf26610e479. Source text hash: 5da619edacbcc9fead6752788cd5c3a48f9531bcbb1e87b91b4829b93d1726bf.
uncertain · 2026-11-11T11:53:59.768Z · anthropic/claude-opus-5.5
Publication: e497a73c-3651-40f2-8fab-954a304dd5b2
attribution — pass: The reading correctly names William Bolton, Bezawit Tilaye and Huw Day as the authors. It keeps the findings and the interpretation framed as theirs ('report', 'suggest').
Authors:William Bolton, Bezawit Tilaye, Huw Day
summary — uncertain: First, the central contrast term 'noisy quantisation' is never explained. A general reader cannot tell what multiplication fails to conform to. Smallest repair: add the source's own description, that addition errors are localised around carry boundaries. Second, the summary says 'lower-order columns... predominantly preserved parity'. The source says the errors in those columns preserve parity, so 'errors' should be restored. Third, the gloss on off-by-one carry failures is muddled, and 'multi-periodic' is left unexplained. Fourth, the closing question is repeated verbatim at the end of the summary.
addition errors follow a "noisy quantisation" model localised around carry boundaries
date — pass: The original published date, 2026-10-28, matches the v1 submission date on the arXiv page.
[Submitted on 28 Oct 2026]
support — pass: Each claim matches the abstract: the dataset size, the departure from the addition model, the parity and off-by-one pattern in small Qwen models, and the authors' suggested interpretation. The limitations on generalization and real-world impact are stated as unknowns rather than invented. The reading also keeps the source's suggestion separate from proof that the models multiply reliably.
Our findings suggest that arithmetic computation in LLMs relies on distinct, multi-periodic internal representations
media — pass: The selected media is the verified arXiv abstract page, which is the direct original source.
Challenge or correct this reading
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A language model with many learned numerical settings, used to process and generate language.
Definition referenceStoring a model's numbers at lower precision so it uses less memory and runs faster, sometimes at a cost in accuracy.
Definition referencearXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivA language model with many learned numerical settings, used to process and generate language.
Definition referenceA language model with many learned numerical settings, used to process and generate language.
Definition referenceStoring a model's numbers at lower precision so it uses less memory and runs faster, sometimes at a cost in accuracy.
Definition referencearXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivA language model with many learned numerical settings, used to process and generate language.
Definition reference