The Tool Desk
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For reproducible mathematical research, no single Claude alternative is best for every task. Use OpenAI reasoning models for tool-using workflows and published proof artifacts, Gemini 2.5 Pro for very large papers, codebases, and datasets, DeepSeek R1 for open-weight general reasoning, and DeepSeek-Prover-V2 when the deliverable is a machine-checked Lean 4 theorem. In every case, reproducibility comes from pinning versions, saving prompts and raw outputs, executing code, and independently checking proofs—not from the model name alone.
What makes a Claude alternative suitable for mathematical research?
A useful research model must do more than produce a plausible explanation. A reproducible result should let another mathematician reconstruct the inputs, rerun the computation, inspect every transformation, and verify the final claim independently. That requirement changes how alternatives are evaluated.
- Mathematical reasoning quality: Can the model form correct conjectures, manipulate definitions, find counterexamples, and carry a multi-step derivation without silently changing assumptions?
- Long-context handling: Can it keep a large paper, source tree, bibliography, or dataset in scope without losing notation and dependencies?
- Reproducible access: Can you record a stable model identifier, generation settings, prompts, source files, tool calls, and returned artifacts?
- Machine-checked proof support: Can the output be translated into Lean 4 or another proof assistant and accepted by a checker?
- Tool integration: Can the workflow run Python or other code, inspect files, and preserve the exact environment used for calculations?
Natural-language confidence is not evidence of correctness. Treat every generated derivation as a draft until numerical calculations, executable code, and formal proof terms pass independent checks.
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Quick comparison
| Option | Best fit | Context and openness | Formal-proof position | What to preserve |
|---|---|---|---|---|
| OpenAI reasoning models | Tool-using scientific workflows, coding, and multi-step research | Use the exact model identifier and settings for each run; the OpenAI mathematics repository publishes manuscripts and supporting proof artifacts | Strong when Lean formalizations or other checkable artifacts are included in the workflow | Model ID, reasoning settings, prompts, tool calls, returned files, verification code, and environment |
| Gemini 2.5 Pro | Large papers, codebases, and datasets that must remain in context | Model code gemini-2.5-pro; documented limit of 1,048,576 input tokens and 65,536 output tokens |
Can draft proofs, but a proof assistant must still validate the result | Model code, date, generation settings, complete source bundle, and all outputs |
| DeepSeek R1 | Open-weight general mathematical reasoning experiments | Open-weight access is the distinguishing advantage; deployment details vary by implementation | Useful for drafts and reasoning comparisons, not a correctness guarantee | Exact checkpoint, inference software, sampling parameters, hardware, prompts, and outputs |
| DeepSeek-Prover-V2 | Generating formal theorem statements and Lean 4 proof scripts | Separate open-source model focused on formal proving, reinforcement learning, and subgoal decomposition | Designed for Lean 4; compile every generated script in a pinned environment | Checkpoint, Lean version, libraries, imports, build configuration, proof attempts, and compiler output |
| Lean 4 proof assistant | Independent correctness boundary for a formal claim | Not a conversational model; it checks proof terms against formal definitions and axioms | Rejects invalid proof terms, exposing errors that fluent prose can hide | Source files, toolchain version, package lock, build logs, and the exact theorem statement |
OpenAI reasoning models: the strongest general research workflow
OpenAI’s reasoning documentation positions its reasoning models for complex problem solving, coding, scientific reasoning, and multi-step workflows. That combination is valuable when the research process includes browsing a local corpus, writing scripts, executing calculations, and producing a structured record rather than just an answer.
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Why choose them
- They fit workflows in which the model proposes an argument, writes verification code, interprets the result, and revises the argument after a failed check.
- The OpenAI mathematics repository publishes mathematical manuscripts and supporting proof artifacts, including Lean formalizations and abridged reasoning summaries. These artifacts provide examples of how a model-assisted result can be packaged for inspection.
- The repository states that, on average, each result used three hours of ChatGPT Pro thinking compute with that model. That is an average reported by the repository, not a guaranteed time or a benchmark applicable to every problem.
How to make an OpenAI run reproducible
Record the model identifier exactly as returned by the API or interface, the reasoning setting, system and user prompts, attached files, tool calls, tool outputs, and every generated file. Save the code used to check algebra, numerical values, simulations, or data transformations. If a response changes after a retry, keep both the accepted and rejected attempts and explain which independent test decided between them.
For a paper, store a manifest containing hashes of the source documents and a chronological log of model requests. A later reader should be able to distinguish a model-generated claim from a result produced by an executable script or a proof assistant.
Gemini 2.5 Pro: best fit for very large mathematical corpora
Google describes Gemini 2.5 Pro as a thinking model for complex problems in code, mathematics, and STEM, with the ability to analyze large datasets, codebases, and documents using long context. Its documentation lists the model code gemini-2.5-pro, a 1,048,576-token input limit, and a 65,536-token output limit.
When the context window matters
A million-token input limit can accommodate a substantial collection of papers, source files, appendices, and experimental data in one request. That is useful for tasks such as tracing a definition through several chapters, checking whether a lemma is reused consistently, or locating all code paths that depend on a mathematical invariant. The limit is a documented capacity, not a promise that every problem will be solved correctly at that size; retrieval, chunking, and explicit file manifests still help prevent omissions.
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Reproducibility precautions
Save the complete source bundle rather than only a compressed summary. Record the model code, request date, generation settings, ordering of files, and any truncation or preprocessing. If the corpus changes, produce a new manifest and treat the run as a new experiment. Long context can make an answer look comprehensive while hiding a single missed definition, so test important statements against the original file and cite the exact section in your internal record.
DeepSeek R1: open-weight general reasoning
The official DeepSeek-R1 repository says that DeepSeek-R1 achieves performance comparable to OpenAI o1 across mathematics, code, and reasoning tasks. Its open-weight orientation makes it attractive for researchers who need to inspect or control more of the inference stack than a hosted API exposes.
What open weights enable
- Running a fixed checkpoint locally or on a controlled server, reducing dependence on a provider’s silent model updates.
- Recording the inference engine, quantization, hardware, temperature, sampling method, and random seeds alongside the prompt.
- Repeating an experiment after the hosted service has changed, provided the checkpoint and runtime remain available.
Limits of benchmark claims
DeepSeek AI’s DeepSeekMath-V2 preprint reports 118/120 on Putnam 2024, plus gold-level results on IMO 2025 and CMO 2024. These are the authors’ reported results. Before comparing them with another model, check the task version, number of samples, selection or verification procedure, compute budget, and whether the result was produced with tools. A headline score does not establish that a model will produce a valid proof for your research problem.
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Use R1 to generate conjectures, alternative derivations, code drafts, and competing explanations. Require an executable check or formal proof before accepting any of them. If you run a local checkpoint, publish the exact checkpoint identifier and runtime configuration; “DeepSeek R1” alone is not enough to identify a reproducible experiment.
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DeepSeek-Prover-V2: when the output must be Lean code
DeepSeek-Prover-V2 is a separate open-source model aimed at formal theorem proving in Lean 4. Its approach uses reinforcement learning and subgoal decomposition, making it a more direct fit than a general chat model when the deliverable is a theorem and a proof script that a checker can compile.
Use it for formal deliverables
Start with a precise Lean theorem statement, including all hypotheses, types, definitions, and imported libraries. Ask the model for a proof script, but consider the script only a candidate until Lean accepts it. Preserve failed attempts too: they show which subgoals were difficult and prevent a later reader from mistaking an edited final script for the first output.
Pin the formal environment
A proof can fail or succeed because of changes in Lean, Mathlib, compiler behavior, or imported lemmas. Record the Lean version, library revision, package lock, operating-system image or container, build command, and compiler output. A proof that compiles only after an undocumented library update is not reproducible in the strong sense.
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Lean 4 is the reproducibility boundary
Formalization turns an informal claim into definitions, hypotheses, and a proof term that a small trusted checker can inspect. Lean can reject an invalid term even when the accompanying explanation sounds convincing. This makes it particularly valuable for theorem proving, but it does not remove the need for judgment: the formal statement may be weaker, stronger, or simply different from the theorem a researcher intended.
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What a machine-checked package should contain
- The exact theorem statement and all definitions used by it.
- Imports and library revisions, with a lock file or equivalent dependency record.
- The generated proof source and the command that compiles it.
- Compiler output showing a successful check, plus logs for rejected attempts when they influenced the final proof.
- A plain-language explanation mapping the formal hypotheses to the mathematical paper.
Numerical experiments require a similar boundary. Include executable code, fixed input data, dependency versions, random seeds where applicable, expected outputs, and an independent implementation or sanity check for results that drive the theorem.
A reproducible AI-assisted mathematics protocol
Use the following sequence for any model, hosted or local. The goal is an auditable chain from source material to checked result.
- Define the claim. Write the research question, domain, hypotheses, notation, and acceptance criteria before asking a model for a solution. State whether the target is an informal argument, executable calculation, Lean theorem, or all three.
- Freeze the model. Record the exact model identifier or checkpoint, provider or runtime, date, reasoning mode, temperature, sampling parameters, and any system-level instructions. For
gemini-2.5-pro, record that model code rather than a generic “Gemini” label. - Freeze the source bundle. Save the papers, datasets, source trees, and appendices supplied to the model. Create a manifest with file names, versions, and hashes so that a future run uses identical inputs.
- Publish the prompt. Keep the complete prompt, including formatting instructions, definitions, examples, and requests for tool use. Do not publish only a paraphrase of the question.
- Run at least twice. Repeat the same task at least twice and save raw outputs. If the model is stochastic, record seeds or sampling settings; if the service does not expose them, report that limitation.
- Separate discovery from verification. Let the model propose lemmas, algorithms, or proof strategies, then run independent code and formal checks without allowing the model to silently edit the test.
- Execute every code block. Use a clean, documented environment. Capture standard output, errors, package versions, and generated files. A code fragment that looks correct but was never run is not evidence.
- Compile formal proofs. Check every Lean script with the pinned Lean and library versions. Include the exact command and successful compiler output.
- Investigate disagreements. When two runs differ, identify the first divergent assumption or calculation. Do not select the more persuasive prose; use an independent derivation, counterexample search, numerical test, or proof assistant to decide.
- Report failures. Keep discarded attempts, failed tests, timeouts, and unresolved claims in the experiment log. State which result was accepted and why.
- Archive the package. Bundle prompts, outputs, source data, code, proof files, environment specifications, and a README that explains how to reproduce the result from a clean checkout.
How to choose between the candidates
| Your primary constraint | First choice | Reason | Required safeguard |
|---|---|---|---|
| Large collection of papers, code, and data in one analysis | Gemini 2.5 Pro | Its documented 1,048,576-token input and 65,536-token output limits are the clearest fit for long-context work | Preserve the complete source bundle and verify important claims against original files |
| Interactive coding, tool calls, and a structured research trail | OpenAI reasoning models | Designed for complex problem solving, coding, scientific reasoning, and multi-step workflows | Log model settings, tool calls, returned files, and independently executed checks |
| Open-weight general reasoning | DeepSeek R1 | Supports controlled local or self-managed experiments and is reported by its authors as comparable to OpenAI o1 on math, code, and reasoning tasks | Pin checkpoint, runtime, hardware, sampling, and seeds; validate every result |
| Lean theorem or proof script as the deliverable | DeepSeek-Prover-V2 plus Lean 4 | Targets formal proving with Lean 4, while Lean supplies the machine-checking boundary | Compile in a pinned environment and publish the theorem, imports, libraries, and logs |
| Highest confidence in a mathematical claim | Any model paired with independent code and Lean where appropriate | Reproducibility is a workflow property, not a vendor property | Use repeated runs, executable calculations, formal proofs, and a record of failures |
Common failure modes and fixes
The model changes a definition halfway through
Put definitions and notation in a source file or a clearly delimited prompt, then ask the model to restate them before deriving anything. Compare the restatement with the canonical version and reject a response that introduces an unannounced convention.
A proof sounds complete but has a hidden gap
Translate the key claim into Lean or test it with a counterexample generator. If formalization is not practical, split the argument into lemmas and require an independently checked calculation for each nontrivial algebraic or numerical step.
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Repeated runs produce incompatible answers
Keep both raw outputs, identify the divergent step, and lower the problem to a testable subclaim. A changed answer is a signal to improve the specification or verification, not a reason to average the prose.
A long-context run omits a crucial passage
Use a file manifest and ask for references to exact sections, definitions, or line ranges. Re-run the claim with only the relevant excerpts and compare the result. Large context reduces manual copying but does not prove that every passage was used correctly.
A Lean proof works on one machine but not another
Pin the Lean toolchain and library revision, include the package lock, and build in a container or documented environment. Record imports explicitly; an accidental transitive import can make a proof appear more portable than it is.
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Inspect the benchmark protocol, sampling count, tool access, and compute budget. The reported DeepSeekMath-V2 figures, for example, are authors’ results with conditions that must be checked before comparison. Use benchmark evidence to choose experiments, not to waive proof obligations.
What to publish with an AI-assisted result
A concise reproducibility record should let a reader answer five questions: What was asked? Which model and version answered? What data did it see? What was executed or formally checked? What failed before acceptance?
- Model record: identifier or checkpoint, date, provider or runtime, generation settings, and hardware when locally hosted.
- Input record: complete prompts, system instructions, source manifest, file hashes, and preprocessing scripts.
- Output record: raw responses, tool calls, generated code, proof scripts, and any human edits clearly marked.
- Verification record: commands, dependency versions, numerical outputs, Lean compiler logs, and independent checks.
- Decision record: repeated-run differences, discarded attempts, unresolved limitations, and the reason the final claim was accepted.
This separation prevents a polished narrative from obscuring which parts were generated, which were executed, and which were machine-checked.
Bottom line
Choose Gemini 2.5 Pro when corpus size dominates, OpenAI reasoning models when the work depends on integrated tools and an inspectable artifact trail, DeepSeek R1 when open-weight general reasoning is the priority, and DeepSeek-Prover-V2 with Lean 4 when a formally checked theorem is the required output. The dependable alternative to Claude is not a single model: it is a pinned, repeatable workflow that preserves inputs and outputs, executes every calculation, compiles formal proofs, and reports failures honestly.
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