adversarial-proof-review

Adversarial Proof Review

Rigorous, multi-pass adversarial review of mathematical proofs and formal arguments. Designed to catch subtle flaws that single-pass review misses - the technique that identified a fatal bug in a SNARG construction from LWE (Woodruff et al., 2026, Section 3.2).

When to use

• Raman asks you to "review," "check," "verify," or "audit" a proof, theorem, or derivation
• A new paper is ingested and contains novel proofs relevant to Raman's work
• Raman presents his own proof sketch and wants it stress-tested
• Any time a formal mathematical claim needs verification before being filed to the wiki

The protocol

Pass 1: Initial objective review

Focus exclusively on identifying potential errors. For each claim in the proof:

1. State the claim being made
2. Identify the proof technique (induction, contradiction, construction, etc.)
3. Check each logical step: Does the conclusion follow from the premises?
4. Flag gaps: Are there unstated assumptions? Missing cases? Unjustified steps?
5. Check definitions: Is every term used consistently with how it was defined?
6. Verify boundary conditions: Do edge cases (empty set, n=0, degenerate graphs) work?

Output format for Pass 1:

## Pass 1: Initial Review

### Claim 1: [statement]
- Technique: [proof technique]
- Status: VERIFIED / SUSPICIOUS / GAP FOUND
- Issues: [if any]

### Claim 2: ...

Pass 2: Self-correction (adversarial)

Now critique your own Pass 1 findings:

1. For each "VERIFIED" claim: Try harder to break it. Ask: "What if the hypothesis is false? What if I construct a specific counterexample?"
2. For each "SUSPICIOUS" claim: Is this a real issue or a false alarm? Can you construct a concrete instance where it fails?
3. For each "GAP FOUND": Is the gap genuine or did you misread the argument? Verify by re-reading the original text.
4. Check for hallucinations in your own review: Did you attribute claims to the proof that aren't there? Did you misstate definitions?

Pass 3: Distinction check

This is the critical step from the SNARGs case study. For every definition and construction pair:

1. Extract the exact definition (what is required)
2. Extract what the construction achieves (what is delivered)
3. Compare precisely: Does "achieved" imply "required"? Watch for:
   • "For all" vs "with high probability" gaps (perfect vs statistical)
   • "For all x" vs "for almost all x" gaps
   • Deterministic vs randomized guarantees
   • Worst-case vs average-case claims

Pass 4: Final assessment

Choose one output format:

COMPLETE VERIFICATION (use only if every step is 100% rigorous):

• Summary of the proof's correctness
• Any non-obvious but valid steps worth noting

STRUCTURED PARTIAL PROGRESS (use if ANY gaps or concerns remain):

• Verified lemmas: What you can confirm is correct
• The crux: Pinpoint exactly where the proof stalls or has issues
• Severity: Is this a fatal flaw, a fixable gap, or a presentation issue?
• Suggested fixes: Propose strategies to bridge the gap

Epistemic discipline

• Distinguish: "This step is wrong" vs "I cannot verify this step" vs "This step requires expertise I may lack"
• Never say "the proof is correct" by default. Absence of found errors is not proof of correctness.
• Use neutral prompting internally: Frame as "prove or refute" rather than "verify this is true"
• Flag confirmation bias: If the proof supports a result you expect to be true, be extra skeptical
• Cite specific lines/equations: Every critique must point to the exact location in the source

Common failure modes to watch for

From Woodruff et al. (2026), Section 9.2:

• Confident technical hallucinations: Subtle algebraic errors, dropped constraints, misapplied theorems
• Confirmation bias: Strong tendency to support the hypothesis presented in the prompt
• Alignment friction: Reluctance to critique a result that looks impressive

Integration with wiki

After completing a review:

1. If a significant flaw is found, create wiki/reviews/<paper-slug>-review.md documenting the issue
2. Update the corresponding source page with a "Known issues" section
3. If the proof is verified, note this in the source page's "Key claims" section with epistemic status
4. Log the review to log.md with op type review