AI RNG: Practical Systems That Ship
Most proof mistakes are not algebra mistakes. They are assumption mistakes. Something that feels natural is smuggled in: a function is assumed continuous, a set is assumed nonempty, a limit is assumed to commute with an integral, a maximum is assumed to exist because it would be convenient.
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A good proof checker is not only someone who can follow steps. A good proof checker is someone who can surface what is being used without being said.
This routine is a practical way to find hidden assumptions, especially when AI has helped draft part of the argument.
Start by rewriting the claim with explicit quantifiers
Many hidden assumptions live inside vague quantifiers.
Rewrite the statement so that the order of quantifiers is explicit:
- For all x in X, there exists y in Y such that …
- There exists y in Y such that for all x in X …
These are different claims. A surprising number of proof gaps come from swapping them.
A simple check is to write the negation of the claim. If you cannot negate it cleanly, the statement is not pinned down enough to trust the proof.
Build a list of all objects and their required properties
Every object in the proof needs a minimal set of properties.
Make a short object table:
| Object | Where it is defined | Required properties | Where those properties are used |
|---|---|---|---|
| f | Definition section | continuous, bounded | limit interchange step |
| X | Theorem statement | compact metric space | maximum existence step |
| μ | Measure definition | finite, complete | dominated convergence step |
If a required property is not stated as a hypothesis or derived earlier, it is a hidden assumption.
Trace each major step back to a named theorem
If the proof uses a standard theorem, name it and write the conditions.
Examples that commonly hide assumptions:
- Extreme value theorem requires compactness and continuity.
- Interchanging limit and integral requires a theorem with conditions.
- Existence and uniqueness results often require completeness or Lipschitz conditions.
- Passing derivatives inside integrals requires uniform control.
When you name the theorem, you force the proof to pay the condition cost.
Run a boundary sweep
A boundary sweep is a fast attempt to break the proof with extreme cases.
Try:
- The smallest parameter values and degenerate cases
- Empty or singleton sets
- Zero functions, constant functions, or simple sequences
- Edge cases where inequalities become equalities
If the proof silently assumes something like nonemptiness or strict positivity, the boundary sweep will usually expose it.
Attempt to remove one hypothesis at a time
Hidden assumptions often appear because the proof is using more than the statement.
Take each hypothesis and ask:
- Do we actually use this
- If we drop it, does the argument still go through
- If it still goes through, is the theorem stronger than stated
- If it fails, where exactly does it fail
This exercise clarifies which hypotheses are essential and where they matter.
Use counterexample pressure
A powerful way to reveal hidden assumptions is to attempt a counterexample that violates a suspected missing condition.
If the proof seems to require compactness, try a noncompact space example.
If it seems to require continuity, try a function with a jump.
If it seems to require integrability, try a tail-heavy distribution.
You do not need to find a full counterexample every time. Often the attempt is enough to reveal the missing condition.
A detection table you can reuse
| Hidden assumption type | Typical clue in the proof | How to detect it fast | Typical repair |
|---|---|---|---|
| Regularity | Differentiation or limit swaps | Name the theorem and list conditions | Add hypothesis or prove regularity |
| Existence | Selecting a maximizer or minimizer | Check compactness or coercivity | Strengthen assumptions or use approximation |
| Uniqueness | Treating a solution as unique | Look for injectivity or strict convexity | Add condition or weaken conclusion |
| Nonemptiness | Choosing an element of a set | Test empty case in boundary sweep | Add nonemptiness hypothesis |
| Measurability | Integrating or applying expectations | Check measurability assumptions | Prove measurability explicitly |
| Uniformity | Moving quantifiers across limits | Write quantifiers and negate the claim | Add uniform bound or change claim |
Using AI as a checker, not a judge
AI is helpful when you ask it to perform constrained verification tasks.
Useful requests:
- List every theorem invoked implicitly and its conditions.
- Identify where each hypothesis is used.
- Negate the statement and explain what a counterexample would look like.
- Propose boundary cases that could break the claim.
Then you validate the output. The goal is not to outsource correctness. The goal is to increase the speed at which hidden assumptions are surfaced.
A proof that survives this routine is not guaranteed correct, but it is far less likely to be wrong for the most common reasons. That is a meaningful upgrade in reliability.
A quick quantifier slip test
Quantifier slips are common and subtle. A fast way to catch them is to translate the key claim into a game:
- One player chooses an x.
- The other player must respond with a y.
- If the statement says “for all x there exists y”, the responder may choose y after seeing x.
- If the statement says “there exists y for all x”, the responder must commit to one y before seeing x.
If the proof behaves like the first game but the claim is the second, there is a hidden assumption or a wrong direction.
Keep an invariant map for long proofs
Long proofs often have a few invariants that must remain true at every stage: bounds, domain membership, measurability, or monotonicity.
Write them as a short list and check them whenever the proof introduces a new object or changes variables. Many hidden assumptions are simply invariants that were true earlier but stopped being true after a substitution.
Keep Exploring AI Systems for Engineering Outcomes
• AI Proof Writing Workflow That Stays Correct
https://ai-rng.com/ai-proof-writing-workflow-that-stays-correct/
• AI for Building Counterexamples
https://ai-rng.com/ai-for-building-counterexamples/
• AI for Real Analysis Proofs: Epsilon Arguments Made Clear
https://ai-rng.com/ai-for-real-analysis-proofs-epsilon-arguments-made-clear/
• Root Cause Analysis with AI: Evidence, Not Guessing
https://ai-rng.com/root-cause-analysis-with-ai-evidence-not-guessing/
• AI Code Review Checklist for Risky Changes
https://ai-rng.com/ai-code-review-checklist-for-risky-changes/
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