AI RNG: Practical Systems That Ship
When a proof fails, it rarely fails everywhere. It usually fails at one step, a single hinge where an assumption was smuggled in, where a quantifier dependency was violated, where a lemma was used outside its scope, or where an algebraic manipulation was not legal. The pain is that the proof can look smooth while being wrong.
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A proof autopsy is a disciplined method for locating that hinge. It treats a proof like an engineered artifact: you trace dependencies, you test claims, and you find the first point where the chain is no longer justified.
AI can help with proof autopsies when it is used as a meticulous checker that asks for sources, not as a machine that produces a replacement proof. The goal is to keep your proof, fix the broken step, and learn the pattern that caused the break.
Why broken proofs are hard to debug
A proof can fail without obvious symptoms because:
- Each line is plausible on its own.
- The conclusion is true, but the argument is wrong.
- The argument works for special cases but not in general.
- A hidden assumption makes the chain appear valid.
The autopsy method makes these failure modes visible.
The proof autopsy workflow
Think of the proof as a sequence of claims, each requiring justification.
| Autopsy phase | Output | AI role |
|---|---|---|
| Claim segmentation | The proof split into small claims | Split and label claims, but do not change content |
| Dependency mapping | Each claim lists what it depends on | Build a dependency table and flag missing sources |
| Legality checks | Operations checked against domains | Ask for conditions that make each step legal |
| Quantifier checks | Dependencies of chosen parameters verified | Flag illegal dependence and quantifier swaps |
| Counterexample search | Test suspicious steps | Propose edge cases that stress the claim |
| Local repair | Fix only the broken hinge | Suggest minimal lemma or correction |
| Revalidation | Whole proof rechecked | Confirm chain now closes without new assumptions |
The point is to find the first unjustified step, not to rewrite everything.
Segment the proof into claims that can be checked
A proof is easier to debug when each line is a claim with a clear type:
- Definition application
- Theorem invocation
- Algebraic manipulation
- Inequality estimate
- Existence or uniqueness argument
Ask the AI to label each line by type and to ask, “What authorizes this line.” If the answer is vague, that line is a suspect.
Build a dependency table that exposes missing justifications
Most proof failures are missing justifications. A dependency table is a simple tool that makes this visible.
| Claim | Depends on | Justification source | Status |
|---|---|---|---|
| Claim k | earlier claims, definitions | theorem name or definition | justified or missing |
When AI fills this table, you should verify the sources. The moment a claim has no source, you have found a likely hinge.
Legality checks: domain is where many proofs die
Many manipulations are legal only under conditions.
Examples:
- Dividing by a quantity that could be zero
- Taking logarithms of nonpositive numbers
- Interchanging limits and integrals without conditions
- Differentiating under the integral sign without justification
- Using a theorem that requires completeness or compactness when you only have boundedness
AI can help by prompting you for the missing condition. If you cannot supply it, the step must be repaired.
Quantifier checks: the silent killer
Quantifier errors are common because they hide behind familiar words.
The autopsy approach is to rewrite the statement being proved with full quantifiers, then check whether the proof respects dependencies.
If the proof chooses something after seeing a variable it must not depend on, the proof is broken. AI is excellent at catching this if you ask it explicitly to track dependencies.
Counterexample-driven diagnosis
A powerful way to diagnose a suspicious step is to attempt to break it with a counterexample.
When a claim looks too strong, ask:
- Is this claim true for the smallest nontrivial case.
- Does it fail for a boundary case.
- Does it fail when a parameter is extreme.
- Does it fail when symmetry is broken.
AI can suggest candidates quickly. The purpose is not to disprove the theorem, but to stress-test the local claim.
If you find a counterexample, you have located the hinge. Now you can repair it by weakening the claim or adding the missing hypothesis.
Proof ledgers: make every step pay rent
A simple way to prevent and diagnose errors is to maintain a proof ledger, a running record of what you are allowed to use.
A proof ledger includes:
- Definitions written in full, not as slogans
- Theorems with their hypotheses stated explicitly
- A list of current assumptions and what they imply
- A record of where each new claim came from
AI can help maintain this ledger and can ask, for any step, “Which ledger entry authorizes this.” This reframes proof checking as bookkeeping rather than as subjective judgment.
Autopsy prompts that keep AI useful
AI is most helpful when you give it narrow tasks. These prompts are effective because they demand evidence.
- List each step and name the theorem, definition, or algebra rule that justifies it.
- Identify the first step with no valid justification and explain why.
- Rewrite the goal statement with explicit quantifiers and show which step violates dependencies.
- Propose a counterexample that would break the suspicious claim if it is false.
- Suggest the smallest lemma that would repair the hinge and integrate it into the chain.
This keeps the tool honest and keeps you in control.
Repairing the hinge without rewriting the proof
The best repair is local.
Common repair moves:
- Add a missing lemma that justifies the step.
- Replace a false statement with a true, weaker version.
- Add a missing hypothesis if the theorem statement allows it.
- Split a case argument that incorrectly merged distinct regimes.
- Replace a heuristic bound with a proven inequality.
AI can propose repair moves, but you should insist on minimality. A repair that adds new complexity everywhere is often a sign you have not truly found the hinge.
When to rebuild instead of repair
Sometimes the proof is so tangled that local repair becomes more expensive than rebuilding a clean chain. A good rule is:
- Repair when the hinge is a single missing lemma or condition.
- Rebuild when the proof contains multiple unjustified leaps that depend on each other.
AI can help with rebuilding by producing a high-level outline, but you should still perform the same autopsy discipline: every step must pay rent with a source.
Practicing proof autopsies as a skill
A proof autopsy gets easier when you practice it intentionally.
A simple routine:
- Take a proof you wrote last week.
- Remove one justification line and see if you can detect the gap.
- Ask AI to locate the first step that is unsupported and compare with your own diagnosis.
- Rewrite only the missing piece, not the entire proof.
- Record the gap type, so you can see patterns in your mistakes.
This is a fast way to become the kind of mathematician who can read proofs critically and write proofs that are hard to break.
Revalidate the full chain
After a repair, rerun the autopsy quickly:
- Every claim has a source.
- Every operation is legal.
- Quantifier dependencies are respected.
- The repaired hinge does not introduce a new hidden assumption.
At this point the proof either closes or it reveals the next hinge. Most proofs have one dominant hinge. Once it is fixed, the rest becomes routine.
The learning payoff
A proof autopsy is not only about fixing one proof. It trains your mind to notice the patterns that generate wrong steps:
- Overgeneralizing from an example
- Treating a definition as a slogan
- Using the right theorem in the wrong context
- Moving constants or limits without checking conditions
AI can accelerate this learning by tagging failure types across your proof attempts, so you see recurring errors. Over time, you begin to write proofs that are harder to break because they are built from explicit sources and disciplined dependencies.
Keep Exploring AI Systems for Engineering Outcomes
• How to Check a Proof for Hidden Assumptions
https://ai-rng.com/how-to-check-a-proof-for-hidden-assumptions/
• AI Proof Writing Workflow That Stays Correct
https://ai-rng.com/ai-proof-writing-workflow-that-stays-correct/
• Proofreading LaTeX for Logical Gaps
https://ai-rng.com/proofreading-latex-for-logical-gaps/
• Building a Personal Lemma Library
https://ai-rng.com/building-a-personal-lemma-library/
• Proof Outlines with AI: Lemmas and Dependencies
https://ai-rng.com/proof-outlines-with-ai-lemmas-and-dependencies/
Books by Drew Higgins
