Stop treating the output as a solution and preserve it as evidence. Copy the original problem, the solver’s exact proof, its assumptions, and the model or tool version if available. Then verify the theorem statement before checking any algebra, locate the first unsupported inference, and rebuild the argument independently. For a result that affects research, publication, engineering, grading, or a business decision, put the intended statement into Lean or have a qualified mathematician review it.
Start by freezing the original proof
Do not let the solver rewrite its answer before you have recorded what went wrong. Save the complete prompt, every displayed equation, hidden assumptions, tool calls, attached files, and the model or version when that information is available. Number the proof’s claims and list the assumptions it uses. A later “correction” can silently change the theorem or remove the step that caused the failure.
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- Keep the problem exactly as it was asked, including definitions and diagrams.
- Save the solver’s response verbatim, not just its final result.
- Record whether the answer used a calculator, computer algebra system, code, or external source.
- Mark each claim as an assumption, definition, calculation, lemma, or conclusion.
This record gives you a reproducible object to audit. It also lets another person check whether the claimed repair fixes the original proof rather than a cleaner version invented afterward.
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Check the theorem before checking the steps
A formally valid argument can still answer the wrong question. Lean’s verification guidance warns that a proof may be correct for its formal statement while failing to prove the result the author intended. Translate the proposed theorem into ordinary language before inspecting the derivation.
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Write down the domain
State whether each variable is an integer, rational, real number, complex number, vector, matrix, function, or another object. An operation that is harmless over the reals may be invalid over integers or modular arithmetic. A symbol such as “x” may also have a different definition from the one in the original problem.
Expand every quantifier
Distinguish “for every,” “there exists,” and “for a particular example.” Check which variables are bound by each quantifier and whether a conclusion is meant to hold simultaneously for all inputs. Testing several examples cannot establish a universal statement.
List side conditions
Write down requirements such as a denominator being nonzero, a quantity being positive, a function being continuous, or a matrix being invertible. Check whether those conditions are part of the theorem or merely assumed in a later line.
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An implication such as A implies B does not permit the reverse conclusion without a separate proof. Likewise, proving that one expression is sufficient for a result does not show that it is necessary. Ask, in words, exactly what each arrow in the proof claims.
At the end of this pass, write a one-sentence target: “For every [specified input] satisfying [specified conditions], prove [specified conclusion].” If that sentence differs from the solver’s apparent target, stop and correct the statement first.
Find the first failing inference
Audit the proof from the first nontrivial transformation onward. Do not begin with the final contradiction: once one step is invalid, every later line is unsupported, even if the final answer happens to be true.
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- Recompute each algebraic transformation. Expand products, simplify fractions, and check signs independently rather than trusting a plausible-looking line.
- Inspect divisions and cancellations. For every denominator or cancelled factor, identify where nonzero status was proved. If it was not proved, test the zero case separately.
- Check inequality direction. Multiplying or dividing by a negative quantity reverses an inequality. Squaring can introduce solutions or discard sign information.
- Test boundary cases. Try zero, one, negative values, equal endpoints, empty sets, singular matrices, and the smallest allowed inputs. Choose cases near any condition that changes the definition.
- Verify theorem hypotheses. If a line invokes a lemma, confirm that every hypothesis holds in the current context and that the lemma’s conclusion has the needed direction.
- Check limits and continuity claims. A limit cannot be substituted into an expression until the relevant existence and continuity conditions have been established.
- Record the first failure precisely. Quote the line, state the missing condition or invalid implication, and give a counterexample when one exists.
A small counterexample near a boundary is often more informative than a long symbolic calculation. It identifies the exact assumption the solver forgot and prevents you from repairing the wrong line.
Common ways AI proofs fail
| Failure pattern | What to test | Why the later proof cannot be trusted |
|---|---|---|
| Dividing by an expression that may be zero | Set the expression to zero and check whether that case is allowed. | The transformed equation may no longer be equivalent to the original one. |
| Cancelling a factor without proving it is nonzero | Identify the cancelled factor and prove nonzero status or split into cases. | A solution branch can disappear. |
| Reading a numerical pattern as a universal proof | Search for a counterexample outside the tested sample, especially at boundaries. | Finite evidence does not establish a statement over an infinite domain. |
| Swapping “if” and “only if” | Write both implications separately and test the reverse direction. | Sufficiency has been mistaken for necessity, or vice versa. |
| Using a theorem outside its hypotheses | Compare every stated hypothesis with the current variables and domain. | The cited result may not apply at all. |
| Losing a branch in case analysis | List all sign, absolute-value, parity, or ordering cases before simplifying. | The conclusion may hold only in the retained branch. |
| Changing a definition | Compare the solver’s definition symbol by symbol with the problem’s definition. | The proof may establish a different mathematical object. |
| Proving a weaker or differently quantified statement | Place the original and formal statements side by side. | A true weaker claim does not answer the requested theorem. |
| Citing an unchecked lemma or computation | Reproduce the lemma and run the calculation independently. | An unsupported external step can hide the actual error. |
Ask the solver for a critique—but do not count it as verification
After you have frozen the output and written down your own suspected failure, ask the model to identify the first unsupported step. Give it the numbered proof and require it to state all assumptions, edge cases, and theorem hypotheses. You can also ask for a counterexample to each implication and for a proof that does not use the disputed transformation.
This is a debugging aid, not independent evidence. The same model can repeat the original mistake, invent a persuasive explanation, or agree with a false critique. Two language models that produce the same answer are not independent checkers when they rely on similar training and reasoning patterns. OpenAI’s Help Center explicitly cautions that a language model “can produce incorrect or misleading outputs.”
Repair the argument independently
Once the first defect is documented, separate diagnosis from repair. Do not let a proposed alternative overwrite the failure record.
Restore missing cases
If a cancellation required a nonzero condition, split the proof into the zero and nonzero cases. If an absolute value or sign was simplified, handle each possible sign explicitly. If the original theorem is false, state the counterexample and formulate the strongest corrected version you can justify.
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Re-derive the disputed result from definitions, an independently checked lemma, or a separate calculation. A repair that merely repeats the same algebra in different words has not removed the original dependency.
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State the exact scope of the repair
Say whether you proved the original theorem, a theorem with additional assumptions, a one-way implication, or only a set of examples. Do not label a conditional or empirical result as a universal proof.
Use Lean when the claim needs machine-checked evidence
Lean is a functional programming language and theorem prover. Its interactive workflow asks the user for enough information to construct and check a formal derivation, reporting errors while proof scripts are edited. A successful check can provide strong evidence that the encoded proposition follows from the encoded definitions and axioms.
Formalize the intended statement, not just the solver’s code
Begin with the one-sentence theorem you wrote during the statement audit. Encode its domains, quantifiers, definitions, and side conditions. Translating the solver’s proof verbatim can preserve a wrong specification; the formal target must represent the mathematical claim you actually need.
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Build a reproducible project
- Keep a correctly formatted Lean repository with the project files and imports needed to compile it.
- Place the main theorem in a file that the project actually imports and checks.
- Run the project build and treat any error as unresolved, not as a warning to work around.
- Confirm that the checked target theorem is the one you intended, rather than an easier auxiliary lemma.
- Inspect the theorem’s dependencies with
#print axiomsand record what axioms the result uses.
The Lean community verification checklist emphasizes these repository, build, target, and axioms checks. They make the result reproducible for another person instead of tying it to a single conversation.
Understand what a green build proves—and what it does not
A successful Lean check establishes the formal proposition under the definitions, imported results, and axioms in that project. It does not automatically establish that your formalization matches the English problem. A type, quantifier, boundary condition, or definition can be translated incorrectly while the resulting file still compiles.
Review the correspondence in both directions: every requirement in the prose should appear in the formal statement, and every formal assumption should have a clear mathematical interpretation. Check that the file containing the main theorem is imported, that no unproved placeholder has replaced the intended argument, and that the axioms report is acceptable for your use case.
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Choose the right level of checking
| Approach | Speed | Coverage | Trust boundary | Reproducibility | Expertise |
|---|---|---|---|---|---|
| Manual spot-check | Fast for a short proof | Local algebra, definitions, and boundary cases | Human interpretation and arithmetic | Depends on saved notes and calculations | School or university mathematics |
| Independent derivation | Moderate | Can expose a wrong strategy or missing case | Human choice of a second method | Reproducible if the derivation is documented | Subject-matter knowledge |
| Lean formalization | Slower initially | All cases represented by the encoded proposition | Machine-checked derivation plus human specification | Buildable repository and axioms report | Lean and mathematical formalization skills |
| Mathematician review | Depends on availability | Can assess novelty, definitions, strategy, and interpretation | Qualified human judgment | Review record can be archived | Relevant domain expertise |
Use manual checks for routine exercises, add an independent derivation when the proof is unfamiliar, and formalize or seek review when an error would have professional, scientific, financial, or safety consequences.
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Ask a qualified mathematician to review the work when the statement is novel, the proof is long, the English-to-formal translation is nontrivial, the argument depends on an unfamiliar field, or the result will be published or used in a consequential decision. Computer-assisted and research-level arguments can be difficult to establish as correct without expert review. OpenAI’s Our First Proof submissions, published February 20, 2026, reported that some research-level AI proof attempts were later judged incorrect.
Give the reviewer the frozen prompt and output, your precise theorem statement, the numbered failure analysis, any repaired proof, and the Lean project or independent calculations. Ask specifically whether the formal statement matches the intended claim and whether any hidden assumptions remain.
A compact verification checklist
- Original prompt, solver output, model information, and tool calls are preserved.
- Domain, definitions, quantifiers, equality or implication, and side conditions are written explicitly.
- Zero denominators, cancelled factors, sign changes, limits, and boundary cases are checked.
- The first unsupported step is identified before any rewrite is accepted.
- Solver self-critique is treated as a hypothesis, not proof.
- The repaired argument is derived independently and its scope is stated accurately.
- Important claims are formalized in Lean or reviewed by a qualified mathematician.
- A Lean project builds, imports and checks the target theorem, and its axioms are inspected.
FAQ
What should I do if ChatGPT gives me a wrong proof?
Save the exact conversation, restate the theorem with all assumptions, locate the first invalid step, and re-derive the result independently. Use the model’s critique only to suggest checks; for consequential work, formalize or obtain expert review.
Can Lean verify an AI-generated proof?
Lean can check a formal proof script for the proposition encoded in its project. You must still verify that the encoded proposition matches the original mathematical claim and inspect the project’s imports and axioms.
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Why did the solver prove the wrong theorem?
Language models can silently change a definition, drop a quantifier, assume a denominator is nonzero, or prove a weaker implication. Comparing the prose statement with the formal target exposes these specification changes.
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How do I find the first mistake quickly?
Number the proof, recompute the first nontrivial transformation, and test boundary cases immediately. Check divisions, cancellations, inequality directions, and theorem hypotheses before reading the final conclusion.
Is a compiling formal proof automatically true?
It is evidence for the encoded proposition under the project’s definitions and axioms. It is not evidence that the encoding captured the English statement, so specification and translation still require human review.
Frequently Asked Questions
What should I do if ChatGPT gives me a wrong proof?
Save the exact conversation, restate the theorem with all assumptions, locate the first invalid step, and re-derive the result independently. Use the model’s critique only to suggest checks; for consequential work, formalize or obtain expert review.
Can Lean verify an AI-generated proof?
Lean can check a formal proof script for the proposition encoded in its project. You must still verify that the encoded proposition matches the original mathematical claim and inspect the project’s imports and axioms.
Why did the solver prove the wrong theorem?
Language models can silently change a definition, drop a quantifier, assume a denominator is nonzero, or prove a weaker implication. Comparing the prose statement with the formal target exposes these specification changes.
How do I find the first mistake quickly?
Number the proof, recompute the first nontrivial transformation, and test boundary cases immediately. Check divisions, cancellations, inequality directions, and theorem hypotheses before reading the final conclusion.
Is a compiling formal proof automatically true?
It is evidence for the encoded proposition under the project’s definitions and axioms. It is not evidence that the encoding captured the English statement, so specification and translation still require human review.
Quick Recap
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