1. Strip thinking before verifying — a verifier that sees the reasoning is biased toward agreement. Fresh context, cleaned proof only.
2. "Does this prove RH?" — if your theorem's specialization to ζ is a famous open problem, you have a gap. Most reliable red flag.
3. Short proof → extract the general lemma — try 2×2 counterexamples. If general form is false, find what's special about THIS instance.
4. Same gap twice → step back — the case split may be obscuring a unified argument. Three lines sometimes does what twelve pages couldn't.
5. Say "no confident solution" — wrong-and-confident is worse than honest abstain.

Read the problem statement. List 2-3 ways it could be interpreted. For each: is this reading TRIVIAL? If one reading makes the problem easy and another makes it hard, the hard one is almost certainly intended. State which interpretation you're solving and WHY you believe it's the intended one.

NO COMPUTATION. Do not use Bash, Python, WebSearch, Read, Write, or any tool that runs code or fetches data. Numerical verification is not a proof step. "I computed n=1..10 and the pattern holds" is not a proof.

(If your agent harness requires a StructuredOutput or similar return-mechanism tool call, that is NOT a computation tool — call it to return your answer. The restriction is on tools that DO work, not tools that REPORT work.)

Your internal process (iterate until done):
- Solve: Complete rigorous solution.
- Self-improve: Reread. Fix gaps before a grader sees it.
- Self-verify: Strict grader mode. Every step justified?
- Correct: Fix and re-verify. Up to 5 rounds.
- Stop: Self-verify passes twice clean, OR 5 rounds, OR approach fundamentally wrong.

A correct answer from flawed reasoning is a failure. If incomplete, say so honestly. Never hide gaps.

PROBLEM: <insert the problem statement here>
ANGLE: <insert one starting angle here>

references/solver_heuristics.md

• Work out small cases (test past n=3)
• Look for an invariant or monovariant
• Consider the extremal case
• Try induction
• What symmetries?
• Work backwards
• Drop a condition — where does it become trivially false?
• Generalize (inventor's paradox — more structure is sometimes easier)

**Verdict**: complete solution | partial result | no progress
**Rounds**: [how many verify→correct cycles]
**Method**: [key idea, one paragraph]
**Detailed Solution**: [full step-by-step, every step justified]
**Answer**: [if applicable]
**Self-verification notes**: [what you caught and fixed; remaining concerns]

• All thinking-block content
• All "Let me try..." / "Actually wait..." / "Hmm" prose
• All false starts and backtracking

• 2 say HOLE FOUND → refuted, stop (save the remaining 3 calls)
• 4 say HOLDS → confirmed, stop (save the 5th)
• After 3 verifiers: if 2 HOLDS + 1 HOLE, launch 2 more (outcome undecided). If 3 HOLDS + 0 HOLE, launch 1 more (could still hit 4-1).

• What does case B's hypothesis imply about f at other inputs?
• Is there a different pair (a,b) to plug into the governing equation?
• Are you proving too much? (A cleaner contradiction needs less machinery.)

• Unlimited time — no wall-clock pressure
• Targeted computation allowed — modular arithmetic checks, small-case enumeration, symbolic verification of identities. NOT exploratory brute force or unbounded recursion.
• The abstention reason as starting point — if verifiers found a specific gap, start there. If solvers never claimed complete, start from what they partially proved.

math-research

NO WEB ACCESS. Do not use WebFetch, WebSearch, or any tool that touches the internet. Do not look up this problem, its solution, or related problems. You are solving this yourself — the only allowed computation is local (Bash/Python for mod-k arithmetic, small-case enumeration n≤10, symbolic identity checks). If you invoke a web tool, the proof is void.

• The problem statement
• The best partial proof from tight-budget solvers
• The verifier gap descriptions (what specifically didn't close)
• The instruction: "NO WEB ACCESS — do not look up this problem or its solution. Bounded local computation allowed (mod 2^k, small cases n≤10, symbolic identity checks via Bash/Python only). 60-second computation limit. If n≤10 brute force reveals a pattern the tight-budget solvers missed, that pattern IS the proof structure."

references/presentation_prompts.md

• What's the simplest way to say this?
• Which lemmas should be inlined? Which deserve to stand alone?
• Is anything OVERKILL? (constructing a double exponential when linear suffices)
• Now that we know the answer, is there a 3-line hindsight proof?

pdflatex

scripts/check_latex.sh

scripts/compile_pdf.sh

• references/verifier_patterns.md

  — the 12 adversarial checks

• references/adversarial_prompts.md

  — ready-to-use verifier prompts

• references/presentation_prompts.md

  — beautification prompts + LaTeX template

• references/model_tier_defaults.md

  — per-model configuration

1. Dual context isolation: verifier is blind to (a) the solver's thinking trace — which biases toward agreement — and (b) other verifiers' verdicts — social proof also biases. Each verifier thinks it's first.
2. Pattern-specific attacks: not "is this correct?" but "does this make the #40 mistake? the #4 mistake?" Specific beats generic. The 7-category refutation taxonomy gives the verifier a checklist.
3. Asymmetric vote + pigeonhole exit: 4-to-confirm, 2-to-refute. One flaky verifier doesn't kill a correct proof; two dissents does. Stop launching verifiers once the outcome is decided — saves ~30% of verification cost on clear cases.
4. Specification-gaming check first: explicitly asks "is this the intended interpretation?" before solving. The #1 failure mode in prior work (50/63 "correct" answers solved the wrong reading).
5. Calibrated abstention: will say "no confident solution" with partial results. Optimizes conditional accuracy, not coverage.
6. Presentation pass: correctness and elegance are separate steps. The presentation agent gets the VERIFIED proof and finds the cleanest way to say it.