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This study reveals a critical distinction between achieving correct answers and maintaining internal representation invariance when mathematical rules are reordered. The research demonstrates a strong positive correlation across 16 models between the distinctness of internal representations for different rule orderings and overall accuracy in synthetic function-composition tasks. These findings suggest that evaluating mathematical reasoning capabilities requires looking beyond simple answer correctness to analyze how models encode structural information.
Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning
arXiv:2609.28442v1 Announce Type: cross Abstract: Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.
Abstract:Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computation and Language (cs.CL); Symbolic Computation (cs.SC)
arXiv:2609.28442 [cs.LG]
(or arXiv:2609.28442v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.28442 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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