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arxiv: 2606.23959 · v1 · pith:LS7FLLFEnew · submitted 2026-06-22 · 💻 cs.CL · cs.LG

Does My Embedding Reflect That A = B? Evaluating Mathematical Equivalence in Embedding Models

Jiaying Ye , Samarth Rao , Leo Carlin , Kedar Chintalapati , Saharsh Bhargava , Rachit Jaiswal , Michael Zhou , Jared Darlington
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Jarod Alper Vasily Ilin Henry Kvinge
This is my paper

Pith reviewed 2026-06-26 07:55 UTC · model grok-4.3

classification 💻 cs.CL cs.LG
keywords embedding modelsmathematical equivalenceMELD datasetcontrastive learninginformal mathematicsformal mathematicsretrieval tasks
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The pith

Embedding models group mathematical statements by terminology rather than underlying equivalence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether embedding models recognize that the same mathematical idea can appear in very different wording across subfields. It introduces the MELD dataset of mathematically equivalent statements written in lexically distinct ways and demonstrates that current state-of-the-art models cluster the statements by the words used instead of the math they express. This limitation matters for connecting ideas across mathematical languages, including natural language and formal systems. The authors therefore train new embeddings with a contrastive objective that aligns informal statements to different formalizations of the same content. The resulting models improve both on informal-to-formal retrieval and on the MELD evaluation itself.

Core claim

Current state-of-the-art embedding models tend to group statements by the terminology used to make them instead of the underlying math. The MELD dataset of mathematically equivalent but lexically different pairs exposes this behavior. A contrastive training approach that aligns informal statements with different formalizations produces embeddings that perform better on both informal-formal retrieval tasks and on MELD, which contains only natural-language statements.

What carries the argument

The MELD dataset of mathematically equivalent but lexically different statement pairs, together with a contrastive objective that pulls together informal statements and varied formalizations.

If this is right

  • Embeddings trained this way retrieve informal statements from formal versions and vice versa more reliably.
  • Navigation between different mathematical languages, such as natural language and systems like Lean, becomes more accurate.
  • Search and discovery of related results across subfields improves because equivalent ideas expressed differently are placed closer together.
  • Downstream applications that depend on recognizing mathematical identity across surface forms gain a measurable boost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same contrastive alignment idea could be applied to other domains where semantic equivalence is hidden by differing terminology, such as legal documents or scientific literature.
  • If the method scales without needing large paired informal-formal corpora, it might reduce the annotation burden for building math-aware search tools.
  • Models that succeed on MELD may still require separate checks to confirm they handle deeper structural equivalence beyond lexical variation.

Load-bearing premise

The lexical differences in MELD pairs do not systematically correlate with embedding distances in a manner that would turn the observed grouping failure into an artifact of dataset construction.

What would settle it

Construct a fresh set of mathematically equivalent statement pairs whose lexical variations are randomized or balanced across topics, then check whether the same models still fail to group by equivalence rather than wording.

Figures

Figures reproduced from arXiv: 2606.23959 by Henry Kvinge, Jared Darlington, Jarod Alper, Jiaying Ye, Kedar Chintalapati, Leo Carlin, Michael Zhou, Rachit Jaiswal, Saharsh Bhargava, Samarth Rao, Vasily Ilin.

Figure 1
Figure 1. Figure 1: A UMAP visualization of embedded statements from the “vector spaces vs module theory” subset of MELD. Blue points correspond to statements framed in terms of vector space terminology, brown points correspond to statements framed in terms of module theory. Small triangles are distractor statements. MELD pairs are large dots connected by lines. Embedded statements cluster by subfield rather than mathematical… view at source ↗
read the original abstract

Because mathematics is highly abstract, a single statement can take very different forms depending on what subfield it is framed in. There are many examples where breakthroughs occurred after researchers discovered that a question had already been answered in a different field. At the same time, the growth of new resources related to formalization has increased the need for tools that enable efficient and reliable navigation between mathematical 'languages' (e.g., from Lean to natural language). In this paper, we investigate whether current embedding models capture mathematical equivalence. To do this, we introduce the Mathematically Equivalent but Lexically Different Pairs (MELD) Dataset, a collection of mathematically equivalent statements that are expressed in very different language. We show that current state-of-the-art embedding models tend to group statements by the terminology used to make them instead of the underlying math. Motivated by this, we propose a contrastive approach to learning embeddings of mathematical text that focuses on aligning informal statements with different formalizations. Our experiments demonstrate that this leads to improvements not only on informal-formal retrieval tasks but also on MELD, which only contains natural language statements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces the MELD dataset of mathematically equivalent but lexically divergent statements to test whether embedding models capture mathematical equivalence. It claims that current SOTA models group statements by terminology rather than underlying math, and proposes a contrastive training procedure to align informal statements with different formalizations, reporting improvements on informal-formal retrieval and on MELD itself.

Significance. If the MELD construction is free of lexical confounds and the reported improvements are reproducible with proper baselines, the work would usefully document a limitation in existing embeddings for mathematical text and supply a targeted training method. The introduction of new data and a contrastive objective focused on math equivalence is a concrete contribution to mathematical NLP.

major comments (2)
  1. [Abstract] Abstract: the claim of demonstrated improvements on MELD and informal-formal retrieval is unsupported because the text supplies no quantitative results, dataset statistics, training details, or baseline comparisons.
  2. [MELD Dataset] MELD Dataset: the construction protocol, verification of mathematical equivalence, and controls for lexical confounds (length, syntactic complexity, subfield markers) are not described. This is load-bearing for the central claim that observed distances reflect terminology bias rather than other encoded features.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of demonstrated improvements on MELD and informal-formal retrieval is unsupported because the text supplies no quantitative results, dataset statistics, training details, or baseline comparisons.

    Authors: We agree that the abstract, as a concise summary, does not include quantitative results. The body of the paper reports the experimental outcomes, but to make the abstract self-contained we will revise it to include key metrics (e.g., retrieval accuracy gains and MELD performance deltas), dataset statistics, and a brief note on the contrastive training setup and baselines. revision: yes

  2. Referee: [MELD Dataset] MELD Dataset: the construction protocol, verification of mathematical equivalence, and controls for lexical confounds (length, syntactic complexity, subfield markers) are not described. This is load-bearing for the central claim that observed distances reflect terminology bias rather than other encoded features.

    Authors: We acknowledge that the provided manuscript text does not detail the MELD construction. In the revised version we will expand the dataset section to describe the generation protocol, the verification of mathematical equivalence (via expert review and cross-checks against formal statements), and the controls applied for lexical confounds, including length matching, syntactic complexity balancing, and subfield diversity without marker leakage. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical dataset construction and contrastive training are independent of fitted inputs

full rationale

The paper introduces the MELD dataset of mathematically equivalent but lexically divergent statements and evaluates existing embedding models on it, then trains a contrastive model to align informal and formal math text. No equations, fitted parameters renamed as predictions, or self-citation chains appear in the provided abstract or description. The central claims rest on new data and a new training procedure rather than any quantity defined in terms of itself or prior author work. This is the expected non-finding for an empirical methods paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical NLP study focused on dataset creation and model training; it contains no theoretical derivations, fitted constants in a mathematical sense, background axioms, or postulated entities.

pith-pipeline@v0.9.1-grok · 5769 in / 1070 out tokens · 29417 ms · 2026-06-26T07:55:20.651238+00:00 · methodology

discussion (0)

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Reference graph

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