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There is no universal winner. For the hardest mathematics, independent results vary sharply by task: Gemini Deep Think led the cited 2025 proof benchmarks, while ChatGPT 4o led one calculus study. For software work, Claude Opus 4.6 is positioned for long-running repository agents, OpenAI’s reasoning models emphasize tool-using workflows, and Gemini 4 Argon combines deep reasoning with a unified million-token window. The reliable choice is the model that passes your own representative proofs, tests, and debugging tasks under the exact access plan you will use.
Quick verdict by workload
Choose by the work you need to finish, not by a single leaderboard.
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| Workload | Best starting point | Why | Important qualification |
|---|---|---|---|
| Graduate-level proofs and olympiad-style problems | Gemini Deep Think, where available | It scored 80.0% on IMO-AnswerBench and 65.7% on IMO-ProofBench in the cited EMNLP 2025 study. | Those are benchmark-specific results, and the same paper reports that even the strongest systems remain far from perfect proof reasoning. |
| Interactive calculus and symbolic manipulation | Test ChatGPT, Claude and Gemini on your own problem set | ChatGPT 4o scored 94.71% in one cross-evaluated differentiation study, ahead of Claude Pro at 85.74% and Gemini Advanced at 84.42%. | The study tested differentiation tasks, not general theorem proving, and found conceptual and algebraic weaknesses across all models. |
| Large, long-running coding agents | Claude Opus 4.6 | Anthropic highlights repository-scale coding, debugging, agent teams and compaction for extended API jobs. | This is Anthropic’s product positioning; it is not an independent head-to-head pass-rate measurement. |
| Tool-driven, multi-step coding and scientific workflows | OpenAI reasoning models, including GPT-6 Astra | OpenAI’s guide focuses on internal reasoning, tool use, ambiguity recovery, Responses API workflows and Codex CLI. | Actual reliability depends on your tools, prompts, model settings and test harness. |
| A repository or generated artifact approaching a million tokens | Gemini 4 Argon or Claude Opus 4.6 beta | Gemini documents a unified 1M-token input/thinking/output window; Claude Opus 4.6 has a 1M-token context window in beta. | Window size is not the same as useful attention, affordable throughput or accurate recall. |
What the current models provide
Model names, limits and availability change quickly. The following snapshot reflects the announcements and documentation cited for this comparison and should be checked in your account before committing to a production workflow.
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|---|---|---|---|
| Claude Opus 4.6 | Anthropic announced it on February 5, 2026, describing improved coding, debugging, longer agentic tasks, agent teams and compaction for extended API work. | A 1M-token context window is available in beta. | Available through claude.ai, the API and major cloud platforms. No subscription or API price is established in the cited material. |
Gemini 4 Argon (gemini-4-argon) |
Google calls it its latest and most capable Gemini model for complex coding, agentic, mathematics and science tasks. | Unified 1M-token input, thinking and output window; documented long decode up to 1M tokens; low, medium and high thinking levels. | Introductory API pricing is $2 per 1M input tokens and $10 per 1M output tokens through November 30, 2026. From December 1, 2026, the listed prices become $4 and $20 respectively. |
| OpenAI reasoning models, including GPT-6 Astra | OpenAI describes internal reasoning tokens for planning, tool use, inspecting alternatives, ambiguity recovery and difficult multi-step work. Its guide recommends GPT-6 Astra for most reasoning workloads. | The cited guide does not establish a single context or output limit for every reasoning model. | OpenAI access, limits and prices vary by product and model; the cited material does not provide a comparable figure. |
A million-token allowance is a ceiling, not a guarantee that every token will influence the answer equally. Retrieval strategy, file ordering, repeated symbols, truncation rules, latency and budget still determine whether a model can use a large repository effectively.
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What the independent mathematics evidence actually says
Proof-oriented benchmarks
The paper Towards Robust Mathematical Reasoning (EMNLP 2025) separates answer production from formal-style proof tasks. On IMO-AnswerBench, Gemini Deep Think (IMO Gold) scored 80.0%, GPT-5 65.6%, Claude Sonnet 4 23.0% and Claude Opus 4 22.3%. On the more demanding IMO-ProofBench, Gemini Deep Think scored 65.7%, GPT-5 20.0% and Claude Opus 4 2.9%.
| Benchmark | Gemini Deep Think (IMO Gold) | GPT-5 | Claude Sonnet 4 | Claude Opus 4 |
|---|---|---|---|---|
| IMO-AnswerBench | 80.0% — Google DeepMind, 2025 | 65.6% — OpenAI, 2025 | 23.0% — Anthropic, 2025 | 22.3% — Anthropic, 2025 |
| IMO-ProofBench | 65.7% — Google DeepMind, 2025 | 20.0% — OpenAI, 2025 | 2.9% — Anthropic, 2025 |
These figures must not be turned into a single “best math model” ranking. They use different tasks and grading criteria, and the paper reports substantial variation by problem type. A model can find a plausible numerical answer yet fail to justify every implication in a proof. Conversely, a model that writes elegant exposition can make an unnoticed algebraic substitution that invalidates the result.
Calculus results
A separate study dated March 31, 2025 cross-evaluated differentiation tasks. ChatGPT 4o achieved 94.71%, Claude Pro 85.74% and Gemini Advanced 84.42% in that study. The authors found that all three systems were better at procedural differentiation than at conceptual understanding and algebraic manipulation.
The apparent conflict with the proof benchmarks is expected: differentiation exercises, olympiad answers and proof verification measure different capabilities, and they used different model generations, prompts and graders. Treat the scores as evidence about those test setups, not as a permanent ordering of Claude, ChatGPT and Gemini.
Rank #2
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Which model should you try for graduate-level proofs?
Start with Gemini Deep Think if your account offers it and your problems resemble the cited olympiad benchmarks. Its benchmark lead is the strongest independent signal in the supplied evidence. Then test the same problems with an OpenAI reasoning model and Claude Opus 4.6. Do not assume that the older Claude Opus 4 result predicts Opus 4.6: the benchmark table evaluates Opus 4, while Anthropic’s 2026 announcement concerns a newer model.
Use a proof protocol, not a one-shot prompt
- State the domain and allowed tools. Specify whether real, complex, integer or measure-theoretic assumptions apply, and whether a calculator, computer algebra system or numerical check is permitted.
- Request a plan before the proof. Ask for lemmas, edge cases and the intended invariant. This exposes an invalid strategy before pages of prose accumulate.
- Require explicit assumptions. Ask the model to identify divisions by expressions that could be zero, hidden continuity or differentiability requirements, quantifier changes and uses of a theorem whose hypotheses have not been shown.
- Separate discovery from verification. Have one model propose a solution, then ask another model—or a fresh session of the same model—to audit every line without seeing the proposed conclusion first.
- Check computationally where possible. Substitute random admissible values into algebraic identities, differentiate both sides of an antiderivative, or use a computer algebra system for expansion and simplification. Numerical agreement is a sanity check, not a proof.
- Demand a concise final proof. After the audit, ask for a self-contained version with definitions, lemmas and the exact point at which each assumption is used.
A useful prompt skeleton is:
Prove the statement over [domain]. First list assumptions and a proof plan. Then give the proof line by line. For every theorem used, state its hypotheses and show they hold here. Mark any step that is heuristic or requires computational verification. End with a short list of possible counterexamples to the strategy.This workflow reduces hallucinated lemmas but cannot eliminate them. For a thesis, paper or graded submission, you remain responsible for checking the argument against the original definitions and authoritative references.
Which model produces the most reliable code?
“Best at coding” covers several different jobs: generating a function from a specification, navigating an unfamiliar repository, diagnosing a failing test, performing a risky migration, and keeping a long agent run coherent. The providers emphasize different strengths.
The Tool Desk
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Anthropic’s February 5, 2026 announcement says Claude Opus 4.6 improves coding and debugging, handles larger codebases, supports agent teams and uses compaction for longer-running API work. That positioning makes it a sensible first trial for a repository where the agent must inspect many modules, make coordinated edits and return repeatedly after failed tests. The announcement also reports the highest score on Terminal-Bench 2.0 and leadership among frontier models on Humanity’s Last Exam; those are vendor-reported claims and should not be treated as independently replicated pass rates.
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OpenAI reasoning models: tools, recovery and agent loops
OpenAI’s reasoning guide emphasizes internal reasoning tokens, tool calls, inspecting alternatives and recovering from ambiguity. It recommends GPT-6 Astra for most reasoning workloads and highlights complex problem solving, coding, scientific reasoning, multi-step agentic workflows and Codex CLI. This is a strong fit when your process depends on a test runner, repository search, issue tracker, calculator or other explicit tools rather than on a single generated answer.
Gemini 4 Argon: large inputs and long output
Google’s documentation describes Gemini 4 Argon as its latest, most capable model for hard coding, agentic, mathematics and science tasks. Its unified 1M-token input/thinking/output window and documented decode length up to 1M tokens are useful when a task requires a large codebase or a very long generated artifact in one interaction. The practical question is whether the model can locate the relevant files and preserve invariants, not merely whether the files fit inside the advertised window.
Measure code quality with executable outcomes
| Measure | How to test it | What a strong result looks like |
|---|---|---|
| Compilation and formatting | Run the project’s compiler, formatter and static checks after every change. | No new errors, stable formatting and no unexplained suppression comments. |
| Test correctness | Use hidden or held-out tests in addition to the visible suite. | Tests pass without weakening assertions or deleting difficult cases. |
| Debugging | Give the model a reproducible failure, logs and the relevant commit range. | It identifies the root cause, explains why the fix works and adds a regression test. |
| Repository navigation | Ask for a change that crosses modules, configuration and documentation. | All call sites, interfaces, migrations and documentation remain consistent. |
| Recovery | Allow one deliberate failed attempt, then provide the new test output. | The agent revises its hypothesis instead of repeating the same patch. |
Never score an agent on whether its explanation sounds confident. Record the patch, commands run, test results, elapsed time, token usage and manual corrections. A model that writes less code but passes more hidden tests is the better engineering choice.
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Gemini 4 Argon documents a unified million-token input, thinking and output window. Claude Opus 4.6 offers a million-token context window in beta. In both cases, “handle” requires more than ingestion:
Rank #4
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- Index first. Provide a file tree, build instructions, dependency versions and architectural notes before sending every source file.
- Send high-signal slices. Include the modules named by stack traces and their direct interfaces; add the rest only when the model needs it.
- Preserve boundaries. Label generated code, third-party code, tests and configuration so the model does not edit vendored material accidentally.
- Use checkpoints. Ask for a plan and affected-file list, then approve changes in small batches.
- Watch budget and latency. A context that fits technically can still be too slow or expensive for interactive debugging.
For a million-token task, compare retrieval accuracy and test pass rate against a smaller, curated context. If the curated version performs better, the larger window is not helping your workload.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Cost, access and subscription choices
Known API pricing
| Model | Input price | Output price | Validity stated in the documentation |
|---|---|---|---|
| Gemini 4 Argon | $2 per 1M tokens through November 30, 2026; $4 from December 1, 2026 | $10 per 1M tokens through November 30, 2026; $20 from December 1, 2026 | Introductory API pricing; dates and rates are from Google’s cited documentation. |
The supplied material does not establish comparable API or subscription prices for Claude Opus 4.6 or GPT-6 Astra. Consumer plans also impose message, rate, feature or geography limits that can differ from API access. Before paying, verify the model name, reasoning setting, context allowance, tool permissions, rate limits, data-retention terms and whether your country is supported.
One subscription or several APIs?
- Choose one subscription when you mostly need interactive help, have a stable workload and value a single interface over programmatic control.
- Use an API when you need reproducible prompts, automated tests, batch evaluation, tool calls, logging or a controlled data pipeline.
- Use more than one provider when correctness is critical enough to justify independent verification, or when one provider’s context, geography or rate limits block your workflow.
Run a one-week bake-off before committing. Include the cost of retries, human review, failed deployments and context re-uploading—not just the advertised token rate.
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How to run a private bake-off
- Build a representative set. Include proof, algebraic manipulation, numerical computation, conceptual explanation, code generation, bug fixing and a repository task. Use problems you are allowed to share.
- Freeze the prompts. Keep wording, files, tool permissions, temperature or reasoning settings and stopping rules identical where the products allow it.
- Define pass criteria first. For math, require a valid derivation and checked assumptions. For code, require compilation, existing tests, held-out tests and review of the diff.
- Run multiple trials. One lucky answer is not a reliability estimate. Record failures by category: arithmetic, missing case, invented API, regression, timeout or refusal.
- Measure practical cost. Capture latency, input and output tokens, tool calls, retries and human editing time.
- Blind the review when possible. Remove model names from answers so reviewers judge correctness and maintainability rather than style or brand expectations.
- Re-test after updates. Model versions and reasoning modes change; keep a small regression suite and rerun it whenever a provider changes the default model.
A simple scorecard can weight what matters to you: proof validity, code test pass rate, debugging recovery, long-context recall, latency and total cost. Do not average unlike tasks into one decorative score; report separate math and software results.
Best Value
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Common failure modes and fixes
A plausible proof has a hidden gap
Ask for theorem hypotheses and counterexamples, then verify each transformation independently. If the model cannot identify where continuity, compactness, positivity or invertibility is used, treat the proof as unverified.
The model changes code outside the requested scope
Require an affected-file list and a minimal diff. Run the test suite before accepting refactors that are not necessary for the stated bug.
A long context produces confident omissions
Provide a repository map and ask targeted questions about symbol definitions and call paths. Compare the answer with a smaller context containing only relevant files.
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The agent repeats a failed fix
Return the exact command, exit code and new stack trace. Ask it to state what hypothesis was falsified and propose a different diagnostic step before editing again.
Benchmark expectations do not match daily work
Classify your tasks before choosing a model. A calculus score says little about category-theory proofs; an agent benchmark says little about your regulated production repository. Your private test set is the deciding evidence.
Bottom line
For proof-heavy mathematics, Gemini’s cited Deep Think results are the strongest independent signal, but they do not establish universal superiority. For long-running repository work, Claude Opus 4.6 is the logical first trial based on Anthropic’s current positioning. For tool-rich, multi-step coding and scientific agents, OpenAI’s reasoning workflow is a strong fit. For million-token inputs or outputs, Gemini 4 Argon has the clearest documented allowance, while Claude Opus 4.6 offers a beta million-token context.
Make the final decision with a fixed bake-off that measures valid proofs, hidden-test pass rates, recovery from failure, context usefulness and total cost. Re-run it whenever model versions, pricing or access conditions change.
Quick Recap
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