Skip to content
AI News HubLIVE
Original source4 min read

Same Quantity, Different Answer: Numerical Representation Invariance in Language Models

Summary

A new arXiv paper by Ephraim Atta-Duncan tests whether open-weight language models give the same canonical answer to word problems when quantities are rewritten in numerically equivalent forms such as decimals, fractions, percentages, number words, scientific notation, or exactly converted units. Across 3,600 exact-rational problems and 8,600 prompts spanning five transformation families, five open-weight systems show high canonical accuracy (0.969–0.996) after a fixed syntax audit, but orbit correctness and invariance drop to roughly 0.85–0.98. Much of the apparent strict-parser collapse traces to multiplication-form scientific notation falling outside the evaluator's number grammar, while Mistral Small 4 exhibits a separate semantic failure on unit conversions, with 265 errors off by ex…

SourcearXiv Computational LinguisticsAuthor: Ephraim Atta-Duncan
Same Quantity, Different Answer: Numerical Representation Invariance in Language Models
Report an error

The correction channel is not available yet. You can copy the article reference below for later.

Correction instructions
Read article

[Submitted on 27 Jul 2026]

Title:Same Quantity, Different Answer: Numerical Representation Invariance in Language Models

View a PDF of the paper titled Same Quantity, Different Answer: Numerical Representation Invariance in Language Models, by Ephraim Atta-Duncan

View PDF HTML (experimental)

Abstract:Numerically equivalent word problems should yield the same canonical answer whether a quantity is written as a decimal, fraction, percentage, number word, scientific notation, or an exactly converted unit. We generate 3,600 exact-rational problems and 8,600 prompts spanning five identity-preserving transformation families, and evaluate five open-weight systems. After a fixed syntax audit that normalizes common answer forms without an LLM judge, canonical accuracy is 0.969-0.996, but orbit correctness falls to 0.848-0.981 and orbit invariance to 0.851-0.981; invariant-but-wrong orbits account for at most 0.003. Most of the broad strict-parser collapse arises because multiplication-form scientific notation lies outside the implemented number grammar, illustrating how evaluator interfaces can masquerade as reasoning failures. A distinct semantic pathology remains: Mistral Small 4 scores 0.699 on unit-converted inputs and produces 265 errors differing from the label by exact powers of ten. In a separate 9,000-call experiment that allocates equal calls to the compared arms, representation consensus does not outperform paraphrase consensus on a low-error subset and produces substantially more false alarms. The accompanying ancillary archive contains the frozen benchmark, evaluation and audit records, consensus raw responses, manifests, analysis code, and a one-command paper build.

Comments: 15 pages, 2 figures. Ancillary archive includes the frozen benchmark, evaluation and audit records, consensus raw responses, manifests, analysis code, and tests

Subjects:

Computation and Language (cs.CL); Machine Learning (cs.LG)

Cite as: arXiv:2609.25009 [cs.CL]

(or arXiv:2609.25009v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2609.25009

arXiv-issued DOI via DataCite

Submission history

From: Ephraim Atta-Duncan [view email] [v1] Mon, 27 Jul 2026 13:33:38 UTC (3,800 KB)

Full-text links:

Access Paper:

View a PDF of the paper titled Same Quantity, Different Answer: Numerical Representation Invariance in Language Models, by Ephraim Atta-Duncan

View PDF

HTML (experimental)

TeX Source

view license

Ancillary-file links:

Ancillary files (details):

MANIFEST.json

Makefile

README.md

artifacts/consensus/gpt-oss-20b-mxfp4/consensus_metrics.json

artifacts/consensus/gpt-oss-20b-mxfp4/manifest.json

artifacts/consensus/gpt-oss-20b-mxfp4/responses.jsonl

artifacts/consensus/mistral-small4-q4/consensus_metrics.json

artifacts/consensus/mistral-small4-q4/manifest.json

artifacts/consensus/mistral-small4-q4/responses.jsonl

artifacts/consensus/qwen35-4b-q4/consensus_metrics.json

artifacts/consensus/qwen35-4b-q4/manifest.json

artifacts/consensus/qwen35-4b-q4/responses.jsonl

artifacts/consensus/qwen35-9b-q4/consensus_metrics.json

artifacts/consensus/qwen35-9b-q4/manifest.json

artifacts/consensus/qwen35-9b-q4/responses.jsonl

artifacts/consensus/qwen35-9b-q8/consensus_metrics.json

artifacts/consensus/qwen35-9b-q8/manifest.json

artifacts/consensus/qwen35-9b-q8/responses.jsonl

artifacts/full/gpt-oss-20b-mxfp4/analysis.json

artifacts/full/gpt-oss-20b-mxfp4/audit.json

artifacts/full/gpt-oss-20b-mxfp4/eval.jsonl

artifacts/full/gpt-oss-20b-mxfp4/manifest.json

artifacts/full/gpt-oss-20b-mxfp4/metrics.json

artifacts/full/mistral-small4-q4/analysis.json

artifacts/full/mistral-small4-q4/audit.json

artifacts/full/mistral-small4-q4/eval.jsonl

artifacts/full/mistral-small4-q4/manifest.json

artifacts/full/mistral-small4-q4/metrics.json

artifacts/full/qwen35-4b-q4/analysis.json

artifacts/full/qwen35-4b-q4/audit.json

artifacts/full/qwen35-4b-q4/eval.jsonl

artifacts/full/qwen35-4b-q4/manifest.json

artifacts/full/qwen35-4b-q4/metrics.json

artifacts/full/qwen35-9b-q4/analysis.json

artifacts/full/qwen35-9b-q4/audit.json

artifacts/full/qwen35-9b-q4/eval.jsonl

artifacts/full/qwen35-9b-q4/manifest.json

artifacts/full/qwen35-9b-q4/metrics.json

artifacts/full/qwen35-9b-q8/analysis.json

artifacts/full/qwen35-9b-q8/audit.json

artifacts/full/qwen35-9b-q8/eval.jsonl

artifacts/full/qwen35-9b-q8/manifest.json

artifacts/full/qwen35-9b-q8/metrics.json

configs/consensus_benchmark_gpt_oss_20b.json

configs/consensus_benchmark_mistral_small_4.json

configs/consensus_benchmark_qwen35_4b.json

configs/consensus_benchmark_qwen35_9b.json

configs/consensus_benchmark_qwen35_9b_q8.json

configs/families_benchmark_gpt_oss_20b.json

configs/families_benchmark_mistral_small_4.json

configs/families_benchmark_qwen35_4b.json

configs/families_benchmark_qwen35_9b.json

configs/families_benchmark_qwen35_9b_q8.json

data/generated/base_benchmark.jsonl

data/generated/views_benchmark.jsonl

data/manifests/model_hashes.json

paper/figs/consensus_dissent.pdf

paper/figs/fingerprint.pdf

paper/figs/source_data.json

paper/main.tex

paper/references.bib

paper/tmlr.bst

paper/tmlr.sty

pyproject.toml

scripts/build_supplement.py

src/num_equiv/init.py

src/num_equiv/analyze.py

src/num_equiv/audit.py

src/num_equiv/consensus.py

src/num_equiv/dataset.py

src/num_equiv/evaluate.py

src/num_equiv/generate.py

src/num_equiv/paper_figures.py

src/num_equiv/parse.py

src/num_equiv/render.py

src/num_equiv/reproduce.py

src/num_equiv/run.py

src/num_equiv/units.py

tests/test_audit.py

tests/test_consensus.py

tests/test_evaluate.py

tests/test_generate.py

tests/test_parse.py

tests/test_render.py

tests/test_roundtrip.py

tests/test_units.py

uv.lock

(82 additional files not shown)

Current browse context:

cs.CL

new | recent | 2026-09

Change to browse by:

cs cs.LG

References & Citations

NASA ADS

Google Scholar

Semantic Scholar

Loading...

Data provided by:

Bibliographic Tools

Bibliographic and Citation Tools

Bibliographic Explorer Toggle

Bibliographic Explorer (What is the Explorer?)

Connected Papers Toggle

Connected Papers (What is Connected Papers?)

Litmaps Toggle

Litmaps (What is Litmaps?)

scite.ai Toggle

scite Smart Citations (What are Smart Citations?)

Code, Data, Media

Code, Data and Media Associated with this Article

alphaXiv Toggle

alphaXiv (What is alphaXiv?)

Links to Code Toggle

CatalyzeX Code Finder for Papers (What is CatalyzeX?)

DagsHub Toggle

DagsHub (What is DagsHub?)

GotitPub Toggle

Gotit.pub (What is GotitPub?)

Huggingface Toggle

Hugging Face (What is Huggingface?)

ScienceCast Toggle

ScienceCast (What is ScienceCast?)

Demos

Demos

Replicate Toggle

Replicate (What is Replicate?)

Spaces Toggle

Hugging Face Spaces (What is Spaces?)

Spaces Toggle

TXYZ.AI (What is TXYZ.AI?)

Related Papers

Recommenders and Search Tools

Link to Influence Flower

Influence Flower (What are Influence Flowers?)

Core recommender toggle

CORE Recommender (What is CORE?)

Author

Venue

Institution

Topic

About arXivLabs

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)

Key points and analysis

Article intelligence

InvestorsAdvanced

Key points

  • Five open-weight models were tested on 3,600 exact-rational problems and 8,600 prompts across five identity-preserving transformations.
  • Canonical accuracy reached 0.969–0.996, but orbit correctness and invariance fell to 0.848–0.981 and 0.851–0.981.
  • Much of the strict-parser collapse came from multiplication-form scientific notation missing from the evaluator's number grammar, not from reasoning failure.
  • Mistral Small 4 scored 0.699 on unit-converted inputs and made 265 errors differing by exact powers of ten; representation consensus underperformed paraphrase consensus and produced more false alarms.

Highlights and analysis are generated automatically and may contain errors. Check the original source.

More recent stories on this topic · Models