AI Proofs of First Proof

Problem Solver Problem {problemText} Instructions You are solving an open problem. These are unsolved problems — most have resisted expert attacks. A rigorous partial result or a well-documented failure is more valuable than a fake proof. What to do 1. Prior work and barriers (brief — 1-2 paragraphs) State the best-known bounds or partial results with attribution. Identify why the problem remains open: what specifically fails when you try standard approaches? If you recognize the problem, say what is known. Do NOT pad this section with textbook definitions. 2. Attack (the bulk of your response) Attempt a proof, disproof, or partial result. Requirements: • Number all major logical steps (Step 1, Step 2, …). • Cite every theorem you invoke by name and verify its hypotheses apply to your setting. • If a step requires an unproven claim, mark it with ⚠ and state it as a conditional: “If X holds, then…” • If you reach an impasse, document exactly where and why the argument breaks. Then try an alternative approach. Attempting 2-3 distinct approaches that each fail informatively is better than one long approach that trails off. • If you cannot improve on known results, say so explicitly. Do not repackage known bounds as new results. 3. Verification Before finalizing, re-examine your argument adversarially: • For each key step, ask: Would a skeptical referee accept this? Is any implication actually an equivalence? Are quantifiers in the right order? • Check: do boundary cases (n=1, 2, 3) or degenerate cases break the argument? • Check: Did you use any theorem whose hypotheses you did not verify? • Check: Does your claimed result contradict any known construction or lower/upper bound? • If you find an error, fix it or downgrade your claim. Do not leave a known error in the final output. Common failure modes to avoid • Survey masquerading as progress: Summarizing known results without contributing anything new. If all your content is attributable to prior work, classify as No Progress. • Wrong asymptotic regime: Proving O(f(n)) when the problem asks for o(f(n)), or proving an upper bound when a lower bound is needed. • Unverified theorem application: Invoking a theorem without checking that your objects satisfy its hypotheses. • Circular reasoning: Using the conclusion as a hidden assumption, especially in probabilistic arguments. • Overclaiming: Stating “this proves the conjecture” when what you proved is a weaker statement or a special case. Output Format Prior Work [Best known results, key references by author name, and why the problem is open.] Approach [Which strategy do you choose and why. If multiple approaches were attempted, describe each.] Solution / Analysis [The complete mathematical argument with numbered steps. If partial, clearly state what is proved vs. what remains open. Mark conditional steps with ⚠.] Verification [Your self-check. For each key deduction, state whether it survives scrutiny. If you found and fixed errors, document them.] Result • Status: [Solved / Major Partial Result / Minor Partial Result / Documented Failure / No Progress] • What is proved: [One sentence stating your strongest unconditional result, or “No new result beyond known bounds.”] • Open sub-problems: [If applicable, specific sub-problems whose resolution would complete the proof.] Repeat until solved Responding with the prompt “yes, and work on the open sub-problems.”