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Treat an AI-generated proof as a draft, not as evidence. Preserve the exact problem and response, restate the theorem with all domains and quantifiers, then locate the first unsupported inference. Test definitions and boundary cases before attempting a repair. For an important result, encode the intended theorem in Lean, build it in a reproducible project, inspect its axioms, and obtain qualified human review.
Start by freezing what the solver actually said
Save the original prompt, the complete solver response, the model and version if available, and every tool call or calculator result. Do not let the solver silently rewrite its argument while you investigate. Copy the proof into a document and number each claim, assumption, equation, and appeal to a lemma. Mark which statements came from the problem and which were introduced by the solver.
This record matters because an explanation generated after you report an error is a new hypothesis, not evidence that the original proof was sound. Keeping both versions lets you identify exactly where the argument changed.
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- Record unstated assumptions the solver appears to use, such as positivity, continuity, invertibility, or a nonzero denominator.
- Keep numerical experiments separate from deductive steps; examples can expose a false claim but cannot establish a universal one.
Check the theorem before checking the algebra
A formally valid proof of the wrong formalization does not prove the theorem you intended. Lean’s verification guidance emphasizes that users can confuse a result’s statement with its proof. Apply that warning before examining any manipulation.
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Translate every symbol into words
Write down the domain of every variable: integers, real numbers, vectors, functions, matrices, or something else. Then expand the quantifiers. “For every” is not the same as “there exists,” and an implication is not an equivalence. Check whether an equality is claimed for all inputs or only under a condition.
List side conditions explicitly
Look for requirements that are easy to lose in notation:
- A denominator, cancelled factor, or matrix must be nonzero or invertible.
- A square root may require a nonnegative argument, while a logarithm requires a positive one.
- An inequality may reverse when multiplying or dividing by a negative quantity.
- A limit, derivative, integral, or infinite sum needs the hypotheses that justify the operation.
- An absolute value, sign, parity, or piecewise definition may create separate cases.
Ask whether it is the theorem you needed
Compare the solver’s formal-looking statement with the original prose. A solver may prove a weaker result, change a strict inequality to a non-strict one, restrict the domain without saying so, or establish a conditional result when you asked for an unconditional one. If the statement is wrong, stop there: repairing later steps cannot fix a specification error.
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Find the first failing step, not merely the bad conclusion
Work from the beginning and recompute each nontrivial transformation. The first invalid inference invalidates everything that follows, even if the final conclusion happens to be true.
- Check substitutions and definitions against the original problem.
- Recalculate the first algebraic transformation by hand or with an independent computer algebra system.
- Inspect every denominator and cancelled factor at the moment it is used.
- Verify the direction and hypotheses of each cited theorem or lemma.
- Test a small example near a boundary: zero, one, a negative value, a repeated root, or a degenerate dimension.
- For a case split, confirm that the branches cover all possibilities and do not overlap in a way that hides a missing case.
Do not jump directly to a contradiction at the end. A later contradiction may be caused by an earlier sign error, an omitted assumption, or a definition that changed halfway through.
Common defects in AI-generated proofs
| Pattern | What to inspect | Typical symptom |
|---|---|---|
| Illegal division or cancellation | Prove the divisor or cancelled factor is nonzero in the current domain. | An equation is transformed into a stronger one after a factor could be zero. |
| Numerical pattern treated as proof | Separate finite testing from a quantified argument. | Several computed cases are followed by “therefore, for all.” |
| Reversed implication | Read “if,” “only if,” and “if and only if” literally. | A necessary condition is used as though it were sufficient. |
| Theorem used outside its hypotheses | Match every assumption of the cited result to the current variables. | A continuity, independence, positivity, or invertibility requirement is absent. |
| Lost branch | Enumerate sign, absolute-value, parity, or domain cases. | The proof handles the positive case and silently claims the negative case. |
| Definition drift | Compare each symbol’s meaning with the problem statement. | The same notation denotes different objects in different paragraphs. |
| Changed quantifiers or target | Write the complete proposition in words. | The proof establishes a weaker, restricted, or differently ordered claim. |
| Unchecked citation or computation | Locate the exact lemma, calculation, or source and reproduce it. | A named result is invoked without a verifiable statement or conditions. |
Use the solver’s self-critique only as a search aid
After recording the failure, ask the solver to identify the first unsupported step and to state the missing assumption. This can help generate candidate locations to inspect, but it is not independent verification. A language model can confidently defend the same error, invent a justification, or produce a second proof with the same hidden assumption.
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Two models agreeing is not independent evidence when they share training data, conventions, or the same mistaken formalization. Re-derive the disputed step yourself, consult an independent textbook or specialist, or use a checker that does not accept prose as proof.
Repair the argument independently
Separate diagnosis from repair. First write a short failure report: the original claim, the exact first invalid inference, the condition that was missing or violated, and a counterexample if one exists. Only then decide whether the theorem can be fixed.
Choose the smallest legitimate repair
- Add a condition that was genuinely intended, such as
x ≠ 0, and state how it changes the theorem. - Split the proof into cases when a sign or piecewise definition requires it.
- Replace an overstrong conclusion with the strongest statement actually supported.
- Use a different method when the original algebra depended on an unavailable inverse, limit, or theorem.
- Reject the claim if a counterexample satisfies all of the original hypotheses.
Recheck the repaired statement from the beginning. Do not patch only the line that was flagged while leaving its dependent conclusions untouched.
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When and how to formalize the claim in Lean
For a consequential result, formalize the intended mathematical statement—not merely the solver’s code—in Lean. Lean is both a functional programming language and an interactive theorem prover. Its system constructs and checks a formal derivation, reporting errors as the proof script is edited. The official learning material includes Theorem Proving in Lean 4.
A reproducible verification workflow
- Create a correctly formatted Lean project and keep the source under version control.
- Write the theorem statement first, including the exact types, quantifiers, hypotheses, and conclusion you intend.
- Import the definitions and lemmas used by the theorem; avoid relying on undeclared local context.
- Place the main theorem in a file that is imported by the project entry point.
- Run the project build and treat every warning or error as part of the result, not as decoration.
- Confirm that the target theorem is actually checked by the build rather than sitting in an unimported file or an ignored example.
- Inspect
#print axiomsfor the theorem and understand which axioms or classical principles it uses. - Compare the checked proposition with the original prose line by line, including the meaning of overloaded notation.
A green build establishes what the encoded proposition proves under its definitions, imports, and axioms. It does not establish that you translated the original question correctly, chose the intended domain, or captured an unstated scientific assumption. Translation and specification remain human responsibilities.
What formalization can and cannot catch
- It can catch many invalid algebraic steps, missing cases, type mismatches, and misuse of lemmas once they are represented in the formal system.
- It can provide a repeatable artifact that others can build and inspect instead of trusting a chat transcript.
- It cannot decide whether your formal statement expresses the informal theorem you cared about.
- It cannot supply facts that were never encoded, such as an unstated physical constraint or the intended meaning of ambiguous notation.
- An axiom audit is necessary because a theorem may rely on assumptions that differ from the foundational guarantees you expected.
Choose the right level of checking
Different responses trade speed, coverage, reproducibility, and expertise. Use the lightest method that matches the consequence of being wrong, but escalate when the claim is novel or difficult.
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| Approach | Speed | Coverage | Trust boundary | Reproducibility | Expertise |
|---|---|---|---|---|---|
| Manual spot-check | Fast | Local algebra and obvious cases | Human interpretation | Usually a one-off transcript | School or university mathematics |
| Independent examples or counterexamples | Fast | Can disprove a universal claim; cannot prove it | Human choice of tests | Repeatable if inputs are saved | Basic computation and domain knowledge |
| Lean formalization | Slower setup | All cases represented by the encoded proposition | Machine-checked derivation plus human specification | Buildable repository and axioms record | Lean and mathematical modeling skills |
| Qualified human review | Depends on availability | Can assess interpretation, novelty, and proof strategy | Expert judgment | Strong when comments and revisions are recorded | Relevant mathematical specialist |
An illustrative failure: cancelling a possible zero
Suppose a solver starts with x² = x, subtracts x, and obtains x(x − 1) = 0. It then divides by x and concludes x = 1. The first unsupported step is the division: the original equation permits x = 0. The correct conclusion over the real numbers is x = 0 or x = 1. If the intended theorem separately assumes x ≠ 0, the division becomes legitimate, but that assumption must appear in the statement and be checked before use.
This small example illustrates the general procedure: identify the exact operation, state its side condition, test the boundary value that violates it, and revise the theorem or proof rather than explaining away the exception.
Know when human review is mandatory
Ask a qualified mathematician to review the work when the statement is novel, the proof is long or computer-assisted, the English-to-formal translation is subtle, or the result affects a publication, product, safety decision, legal position, or scientific conclusion. A reviewer should receive the original prompt, the unedited solver output, your failure analysis, the repaired statement, and any formal project—not only the polished final proof.
OpenAI’s Our First Proof submissions, published on February 20, 2026, documents research-level attempts that were later judged incorrect. The lesson is not that every AI proof fails; it is that fluency and sophistication do not replace verification, especially at the research frontier.
Quick Recap
A compact decision procedure
- Preserve: save the exact problem, response, model details, and tools.
- Specify: write the theorem with domains, quantifiers, definitions, and side conditions.
- Probe: test zero, boundary, sign, degenerate, and small counterexample cases.
- Audit: recompute each inference and stop at the first unsupported step.
- Challenge: ask the solver for a critique only after the failure is recorded.
- Repair: re-derive the corrected argument independently or reject the claim.
- Formalize: for important claims, build and check the intended theorem in Lean, confirm the target is imported, and inspect
#print axioms. - Escalate: obtain specialist review whenever interpretation, novelty, or consequences exceed your own expertise.
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