{
  "evaluation_id": "EVAL-20260402-160500",
  "question_id": "REASON-007",
  "question_text": "Prove or disprove: For any integer n > 1, if n² + 1 is divisible by 5, then n⁴ + 1 is also divisible by 5.\n\nProvide a rigorous proof. If the statement is false, provide a counterexample and explain why the intuition fails.",
  "category": "reasoning",
  "timestamp": "2026-02-25T00:00:00.000Z",
  "display_date": "Feb 25, 2026",
  "winner": {
    "name": "Claude Opus 4.6",
    "provider": "openrouter",
    "score": 9.8
  },
  "avg_score": 8.865,
  "matrix_size": 89,
  "models_used": [
    {
      "id": "gemini_31_pro",
      "name": "Gemini 3.1 Pro",
      "provider": "openrouter"
    },
    {
      "id": "minimax_m25",
      "name": "MiniMax M2.5",
      "provider": "openrouter"
    },
    {
      "id": "deepseek_v4",
      "name": "DeepSeek V4",
      "provider": "openrouter"
    },
    {
      "id": "claude_opus_46",
      "name": "Claude Opus 4.6",
      "provider": "openrouter"
    },
    {
      "id": "gpt_5_4",
      "name": "GPT-5.4",
      "provider": "openrouter"
    },
    {
      "id": "grok_420",
      "name": "Grok 4.20",
      "provider": "openrouter"
    },
    {
      "id": "claude_sonnet_46",
      "name": "Claude Sonnet 4.6",
      "provider": "openrouter"
    },
    {
      "id": "mimo_v2_flash",
      "name": "MiMo-V2-Flash",
      "provider": "Xiaomi"
    },
    {
      "id": "gpt_oss_120b",
      "name": "GPT-OSS-120B",
      "provider": "OpenAI"
    },
    {
      "id": "gemini_25_flash",
      "name": "Gemini 2.5 Flash",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "claude_opus_46": {
      "display_name": "Claude Opus 4.6",
      "provider": "openrouter",
      "average_score": 9.8,
      "score_count": 9,
      "min_score": 9.45,
      "max_score": 10,
      "rank": 1
    },
    "claude_sonnet_46": {
      "display_name": "Claude Sonnet 4.6",
      "provider": "openrouter",
      "average_score": 9.78,
      "score_count": 9,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 2
    },
    "gpt_oss_120b": {
      "display_name": "GPT-OSS-120B",
      "provider": "OpenAI",
      "average_score": 9.78,
      "score_count": 9,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 3
    },
    "deepseek_v4": {
      "display_name": "DeepSeek V4",
      "provider": "openrouter",
      "average_score": 9.62,
      "score_count": 9,
      "min_score": 9.05,
      "max_score": 10,
      "rank": 4
    },
    "gpt_5_4": {
      "display_name": "GPT-5.4",
      "provider": "openrouter",
      "average_score": 9.56,
      "score_count": 9,
      "min_score": 9,
      "max_score": 10,
      "rank": 5
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 9.56,
      "score_count": 9,
      "min_score": 8.8,
      "max_score": 10,
      "rank": 6
    },
    "gemini_25_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "openrouter",
      "average_score": 9.5,
      "score_count": 9,
      "min_score": 8.8,
      "max_score": 10,
      "rank": 7
    },
    "grok_420": {
      "display_name": "Grok 4.20",
      "provider": "openrouter",
      "average_score": 8.93,
      "score_count": 9,
      "min_score": 6.6,
      "max_score": 10,
      "rank": 8
    },
    "minimax_m25": {
      "display_name": "MiniMax M2.5",
      "provider": "openrouter",
      "average_score": 7.43,
      "score_count": 9,
      "min_score": 2.4,
      "max_score": 9.65,
      "rank": 9
    },
    "gemini_31_pro": {
      "display_name": "Gemini 3.1 Pro",
      "provider": "openrouter",
      "average_score": 4.69,
      "score_count": 8,
      "min_score": 1.65,
      "max_score": 9.25,
      "rank": 10
    }
  },
  "judgments": [
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2,
      "brief_justification": "The response starts with a valid counterexample choice but is incomplete and cut off before actually computing n^4+1 or giving a rigorous explanation. It lacks a full proof or clear disproof."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.3,
      "brief_justification": "Correctly disproves the statement with a valid counterexample, clear presentation, but lacks deeper explanation of why the intuition fails."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample, clearly explains the modular arithmetic behind why it fails, and provides excellent depth by proving the conclusion is actually never true when the premise holds."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with a clear counterexample, provides excellent mathematical depth by proving the hypothesis actually implies the negation of the conclusion, and presents the information with perfect clarity."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 10,
      "brief_justification": "The response perfectly disproves the statement with a rigorous modular arithmetic proof, provides a clear counterexample, and excellently explains why the intuition fails."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 6.6,
      "brief_justification": "The response contains a major contradiction, starting by claiming the statement is true before correctly proving it false. While the mathematical steps and counterexample are accurate, the initial false assertion severely impacts clarity and overall correctness."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "The response is flawless. It correctly identifies the statement as false, provides a clear counterexample, rigorously analyzes the problem using modular arithmetic to show it fails for all valid n, and excellently explains the flaw in potential intuition."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "The response is perfectly accurate, provides a clear counterexample, and offers an insightful algebraic explanation for why the statement is false."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response is flawlessly accurate, comprehensive, and exceptionally well-structured. It provides a rigorous algebraic disproof, a clear counterexample, an insightful explanation of why the intuition fails, and a complete residue table for added depth."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 10,
      "brief_justification": "The response perfectly answers the prompt, providing a flawless mathematical disproof, a clear counterexample, and an insightful explanation of why the intuition fails."
    },
    {
      "judge_key": "gemini_31_pro",
      "judge_name": "Gemini 3.1 Pro",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 6.8,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample, but completely fails to explain why the intuition fails as requested, and appears to cut off at the end."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 8.55,
      "brief_justification": "Correctly disproves the statement with a valid counterexample, but the response is incomplete as it cuts off mid-calculation."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.65,
      "brief_justification": "Rigorous proof with clear counterexample and thorough explanation of why the intuition fails."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.3,
      "brief_justification": "Rigorous proof with clear counterexample and explanation of intuition failure."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.3,
      "brief_justification": "Rigorous proof with clear counterexample and explanation of intuition failure."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.65,
      "brief_justification": "Rigorous proof with clear counterexample and insightful analysis."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.3,
      "brief_justification": "The response provides a rigorous proof and counterexample, clearly demonstrating the falsity of the statement with thorough analysis and algebraic verification."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.3,
      "brief_justification": "The response is factually accurate, thoroughly covers the topic, and provides insightful analysis. It is clear and well-structured, with practical value demonstrated through counterexamples and modular arithmetic."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.3,
      "brief_justification": "Rigorous proof with clear counterexample and explanation of why intuition fails."
    },
    {
      "judge_key": "deepseek_v4",
      "judge_name": "DeepSeek V4",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.1,
      "brief_justification": "Correctly disproves the statement with a clear counterexample and explanation."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 1.65,
      "brief_justification": "The response claims the statement is false, but the statement is actually TRUE. For n=2, n^4+1=17, which is not divisible by 5—wait, let me recheck. n=2: n²+1=5 (div by 5), n⁴+1=17 (not div by 5). So the statement IS false and n=2 is a valid counterexample. However, the response is severely incomplete—it cuts off mid-sentence and never finishes the counterexample verification or provides any explanation of why the intuition fails. The correct direction is identified but the proof/explanation is "
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.8,
      "brief_justification": "The response is perfectly correct, identifying n=2 as a counterexample and then providing a complete general proof that the statement is actually never true (whenever n²+1 ≡ 0 mod 5, n⁴+1 ≡ 2 mod 5). The explanation of why the intuition fails is clear and insightful. The modular arithmetic analysis covering all residue classes mod 5 is thorough and rigorous. The presentation is well-structured and easy to follow."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.8,
      "brief_justification": "The response is perfectly correct: it disproves the statement with a rigorous modular arithmetic proof showing that the hypothesis actually guarantees the conclusion is never true, provides a concrete counterexample (n=2), and clearly explains why the intuition fails. The proof is well-structured, easy to follow, and goes beyond just finding a counterexample by proving the stronger result that 5 never divides n⁴+1 when 5 divides n²+1."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.4,
      "brief_justification": "The final answer is correct: the statement is false, and n=2 is a valid counterexample with clear verification. The proof is rigorous and complete. However, the presentation has a notable clarity issue: it initially states 'The statement is true' in bold at the top, then works through the algebra only to discover it's actually false. This self-correcting narrative, while honest, is confusing and could mislead a reader who only skims. The analysis of why the intuition fails is good, and the obser"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.8,
      "brief_justification": "The response is entirely correct: the statement is false, a clean counterexample (n=2) is provided, and a rigorous modular arithmetic analysis proves the conclusion fails for every n satisfying the hypothesis. The table of residues, the algebraic reasoning, and the explanation of why intuition fails are all clear, thorough, and well-structured. Minor nitpick: the 'why the intuition fails' section could have explored slightly more (e.g., contrasting with conditions that do propagate), but this is"
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response is completely correct, providing a rigorous disproof with multiple approaches: modular arithmetic analysis, algebraic factorization, and a concrete counterexample. The algebraic identity n⁴+1 = (n²-1)(n²+1)+2 is particularly elegant and definitively proves the statement is always false when the hypothesis holds. The explanation of why intuition fails is clear. Minor note: the initial hesitation about whether 4 ≡ -1 (mod 5) was slightly awkward but self-corrected immediately."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response is flawless. It correctly disproves the statement with a rigorous modular arithmetic proof, provides an explicit counterexample (n=2), explains clearly why the intuition fails, and includes a comprehensive classification table of all residues mod 5. The reasoning is logically sound at every step, the presentation is exceptionally well-structured, and the depth of analysis (including the optional full classification) goes beyond what is required."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response is completely correct: it correctly identifies the statement as false, provides a rigorous general proof via modular arithmetic showing n⁴+1 ≡ 2 (mod 5) whenever n²+1 ≡ 0 (mod 5), gives a concrete counterexample (n=2), and explains why the intuition fails. The presentation is exceptionally clear and well-structured, with systematic enumeration of residues. The only minor note is slight redundancy in restating the proof, but this actually aids clarity for the reader."
    },
    {
      "judge_key": "claude_opus_46",
      "judge_name": "Claude Opus 4.6",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 2.4,
      "brief_justification": "The response claims the statement is false and offers n=2 as a counterexample, but the arithmetic is wrong. 2^4+1=17, and 17 mod 5 = 2, so indeed 5 does not divide 17. However, the statement is actually TRUE: if n²≡-1 (mod 5), then n⁴≡1 (mod 5), so n⁴+1≡2 (mod 5), meaning 5 never divides n⁴+1 when 5 divides n²+1. Wait—that actually means the implication 5|(n²+1) → 5|(n⁴+1) is false, because whenever 5|(n²+1), we get n⁴+1≡2 (mod 5). So the counterexample n=2 is correct and the conclusion is corre"
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.05,
      "brief_justification": "Mathematically correct and gives a valid counterexample with a clear modular explanation. It not only disproves the statement but also characterizes all cases where the hypothesis holds and shows the conclusion always fails there. Slightly less than perfect completeness/depth only because the initial small-case testing is unnecessary for a fully rigorous proof."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.8,
      "brief_justification": "Fully correct: it gives a valid counterexample and a stronger modular argument showing the implication fails for every n satisfying the hypothesis. Clear structure, rigorous reasoning, and helpful residue-class verification."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9,
      "brief_justification": "The response correctly disproves the statement with a valid modular argument and explicit counterexample. It is thorough, clear, and explains the underlying intuition failure well. Minor issue: it briefly claims that if 5 divides n then both divisibility statements are clear, which is false for n^4+1, though this does not affect the main disproof."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.8,
      "brief_justification": "Fully correct and rigorous: gives a valid counterexample, then proves the stronger modular result that the conclusion fails for every n satisfying the hypothesis. Clear structure and practical explanation of the failed intuition."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.6,
      "brief_justification": "Mathematically correct and rigorous: it properly disproves the statement with both modular analysis and a concrete counterexample. The explanation is thorough, clear, and practical, with a good discussion of why the intuition fails."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.8,
      "brief_justification": "Mathematically rigorous and fully correct: it disproves the statement, gives a valid modular proof, an explicit counterexample, and clearly explains why the intuition fails. Well-structured and practical, with an optional full residue-class classification adding depth."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.25,
      "brief_justification": "Mathematically correct and rigorous: it properly disproves the statement using modular arithmetic, gives a valid counterexample, and explains why the intuition fails. Very complete and clear, though slightly verbose and includes an unnecessary last-digit divisibility remark before moving to the modular proof."
    },
    {
      "judge_key": "gpt_5_4",
      "judge_name": "GPT-5.4",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 6.8,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample and clear modular check. However, it is incomplete relative to the prompt because it does not rigorously explain the general intuition failure beyond the single example."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 3.2,
      "brief_justification": "The response correctly identifies the statement as false and starts a valid counterexample with n=2, but it is incomplete as it cuts off mid-sentence without finishing the calculations or providing explanation. It lacks a full rigorous disproof."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.05,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample (n=2), provides modular arithmetic insight showing it fails for all such n, and explains the algebraic reason why the intuition fails. Minor deduction for completeness as it doesn't explicitly address n>1 restriction beyond testing, but overall rigorous and thorough."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.45,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample (n=2), provides modular arithmetic proof showing it fails for all such n, includes a full residue table for verification, and explains the flawed intuition rigorously. Minor deduction for clarity as the structure slightly separates the counterexample from the general proof."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9,
      "brief_justification": "The response rigorously disproves the statement using modular arithmetic, provides a clear counterexample, and explains the flawed intuition. Minor note: the claim says 'for any n>1' but proof works for all n where hypothesis holds; overall excellent."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the claim with a valid counterexample (n=2), provides a complete modular arithmetic analysis showing it fails for all such n, clearly explains why the implication doesn't hold, and insightfully dissects the flawed intuition. Minor deduction for completeness as it focuses on mod 5 without exploring other primes or generalizations, though not required."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 8.8,
      "brief_justification": "The response rigorously disproves the statement using modular arithmetic, explicit computation for residues, algebraic identity, and a concrete counterexample. It is logically sound, well-structured, and explains the failure of intuition clearly, though minor redundancy exists in the algebraic step."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.45,
      "brief_justification": "The response rigorously disproves the claim using modular arithmetic, provides a clear counterexample, explains the flawed intuition, and includes a complete residue table. All math is accurate and the structure is excellent."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.8,
      "brief_justification": "The response rigorously disproves the statement using modular arithmetic, shows all quadratic residues mod 5, derives that n^4 +1 ≡2 mod 5 whenever the premise holds, and provides a concrete counterexample with n=2. It also explains the common intuition failure clearly. Minor note: the statement claims 'for any integer n>1' but the proof works for all integers; the 'last digit 0 or 5' remark is slightly imprecise but does not affect the core modular proof."
    },
    {
      "judge_key": "grok_420",
      "judge_name": "Grok 4.20",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 8.25,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample (n=2), shows the calculations, and notes it also works for n=3. It is clear and well-structured. However, it lacks a deeper explanation of why the intuition fails or any modular arithmetic insight beyond the counterexample."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 2.3,
      "brief_justification": "The response starts with an incorrect claim (the statement is actually TRUE) and the response is incomplete/truncated. n=2: n²+1=5 ✓ divisible by 5, n⁴+1=17, not divisible by 5. Wait - this would make the statement FALSE with n=2 as a valid counterexample. So the direction is correct, but the response is cut off before completing the counterexample calculation. The response gets partial credit for identifying the correct counterexample direction but fails to complete the proof or provide full re"
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false with a clear counterexample (n=2), provides rigorous algebraic reasoning showing why n²≡-1 mod 5 implies n⁴+1≡2 mod 5, and completes a full modular arithmetic analysis covering all residue classes. The explanation of why intuition fails is particularly insightful and the proof is complete and well-structured."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.85,
      "brief_justification": "The response correctly identifies the statement as false and provides a clear counterexample. It also offers a rigorous explanation using modular arithmetic, demonstrating why the statement fails for all relevant cases, not just the counterexample."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 10,
      "brief_justification": "The response is exemplary. It correctly identifies the statement as false, provides a clean counterexample (n=2), then goes beyond by proving the stronger result that the hypothesis *always* implies the negation of the conclusion. The modular arithmetic is rigorous and correct, the table verification is thorough, and the explanation of why intuition fails is insightful and well-articulated."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.8,
      "brief_justification": "The response is exemplary: it correctly disproves the statement, provides a clean algebraic proof showing the implication is not only false but its negation always holds, gives a concrete counterexample (n=2), and clearly explains why the intuition fails. The structure is logical and the reasoning is rigorous throughout."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false, provides a clean counterexample (n=2), gives rigorous modular arithmetic proof, checks all residues mod 5, and clearly explains why the intuition fails. The initial 'appears to give a contradiction' framing is slightly awkward but ultimately handled well. Excellent overall."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.65,
      "brief_justification": "The response is fully correct, providing both a counterexample (n=2) and a rigorous algebraic proof showing n⁴+1 = (n²-1)(n²+1)+2, which leaves remainder 2 when n²+1 is divisible by 5. The analysis is thorough, covering modular arithmetic, quadratic residues, algebraic identity, and explanation of why intuition fails. Presentation is exceptionally clear and well-structured."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.8,
      "brief_justification": "The response is mathematically rigorous and correct throughout. It disproves the statement with a clean modular arithmetic argument, provides an explicit counterexample, explains why the intuition fails, and includes a complete classification table. The structure is excellent and the reasoning is transparent and well-communicated."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.65,
      "brief_justification": "The response is fully correct, providing both a rigorous algebraic proof by contradiction and a concrete counterexample (n=2). The modular arithmetic is handled precisely, the logical structure is clear and well-organized, and the explanation of why intuition fails adds genuine insight. Coverage is thorough with no gaps."
    },
    {
      "judge_key": "claude_sonnet_46",
      "judge_name": "Claude Sonnet 4.6",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 7.85,
      "brief_justification": "The response correctly identifies the statement as false and provides a valid counterexample with n=2. The arithmetic is correct and clearly presented. However, the response is incomplete - it cuts off mid-section (section 2 appears to be missing), so there's no explanation of why the intuition fails or any deeper analysis of the algebraic relationship between n²+1 and n⁴+1. The note that n=3 also works is helpful but unexplained. A complete response would show that n⁴+1 = (n²)²+1, and if n²≡-1 "
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 9.25,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample (n=2), and explains the condition and conclusion clearly. It is rigorous and well-structured, though the explanation of why the intuition fails could be slightly more detailed."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false with a valid counterexample, provides a rigorous modular arithmetic proof, and explains the intuition failure. It covers all aspects thoroughly and is clearly structured."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with a counterexample, provides a rigorous modular arithmetic proof, explains the intuition failure, and includes a verification table, making it thorough, clear, and highly useful."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually accurate, logically rigorous, and provides a clear counterexample. It thoroughly covers the proof, explanation, and intuition, with strong structure and practical insight."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous proof with a counterexample, explains the intuition failure, and is thorough and clear."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "The response is factually accurate, thoroughly covers the topic with a counterexample and modular arithmetic analysis, is exceptionally clear and well-structured, provides deep insight into why the statement fails, and offers practical value by explaining the flawed intuition."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.8,
      "brief_justification": "The response is entirely correct, providing a rigorous modular proof, an explicit counterexample, and a clear explanation of why the intuition fails. The optional classification table adds thorough completeness."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 10,
      "brief_justification": "The response is mathematically perfect. It correctly identifies the statement as false, provides a rigorous modular arithmetic proof, includes a valid counterexample, and explains the failure of intuition. The structure is clear and the logic is flawless."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 7.7,
      "brief_justification": "Correct counterexample and clear presentation, but lacks deeper modular arithmetic explanation and proof of why intuition fails."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 4.5,
      "brief_justification": "Correctly identifies the statement as false with a valid counterexample, but the answer is cut off and lacks full calculations, explanation of intuition, and a complete rigorous proof."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 9.25,
      "brief_justification": "Accurate disproof with a concrete counterexample and a general modular argument, clearly presented and insightful for similar problems."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.45,
      "brief_justification": "Accurate counterexample and general modular proof; covers intuition failure and residue analysis clearly and thoroughly."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.25,
      "brief_justification": "Accurate modular proof, clear counterexample, explains intuition failure; thorough and well‑structured."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.9,
      "brief_justification": "Accurately disproves the statement with a correct counterexample, explains modular reasoning and why the intuition fails, and presents the argument clearly and comprehensively."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.65,
      "brief_justification": "The answer correctly disproves the statement with a valid counterexample and a general modular proof, fully addresses the prompt, is clearly organized, offers insightful analysis of why the intuition fails, and provides useful reasoning for similar problems."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.05,
      "brief_justification": "Accurately disproves the claim with correct modular analysis, a clear counterexample, and insightful explanation of why the intuition fails."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.9,
      "brief_justification": "Accurate modular proof, clear counterexample, well-explained intuition, thorough coverage."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 9.85,
      "brief_justification": "The response correctly identifies the statement as false, provides a clear counterexample, and rigorously proves why the implication fails using modular arithmetic. The explanation of 'why intuition fails' and the verification table add significant depth and clarity."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.85,
      "brief_justification": "The response provides a rigorous and clear disproof of the statement. It correctly uses modular arithmetic, offers a concrete counterexample, and explains the common intuition trap. The steps are well-organized and easy to follow."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous disproof, and offers a clear counterexample. The explanation of why the intuition fails is also well-articulated."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 9.85,
      "brief_justification": "The response correctly identifies the statement as false, provides a clear counterexample, and offers a rigorous proof using modular arithmetic. The explanation of why intuition fails is also excellent."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.85,
      "brief_justification": "The response provides a rigorous and complete proof, clearly disproving the statement with multiple methods and a counterexample. The explanation of why intuition might fail is also excellent."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.85,
      "brief_justification": "The response provides a rigorous and complete disproof of the statement. It uses modular arithmetic effectively, offers a clear counterexample, explains the intuition failure, and includes a comprehensive classification table. The structure is logical and easy to follow."
    },
    {
      "judge_key": "gemini_25_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "minimax_m25",
      "respondent_name": "MiniMax M2.5",
      "weighted_score": 9.65,
      "brief_justification": "The response correctly identifies the statement as false and provides a clear, concise counterexample. The explanation of why the counterexample works is rigorous and easy to follow."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_31_pro",
      "respondent_name": "Gemini 3.1 Pro",
      "weighted_score": 6.05,
      "brief_justification": "Correctly identifies the statement as false with valid counterexample (n=2), but the response is truncated and incomplete. Missing the full calculation of n^4+1 and lacks explanation of why the intuition fails or deeper analysis."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "deepseek_v4",
      "respondent_name": "DeepSeek V4",
      "weighted_score": 10,
      "brief_justification": "Excellent response with correct counterexample (n=2), clear explanation of why the statement is false, and general analysis showing all n ≡ 2 or 3 mod 5 are counterexamples. Well-structured with proper mathematical reasoning."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_opus_46",
      "respondent_name": "Claude Opus 4.6",
      "weighted_score": 10,
      "brief_justification": "Excellent response. Correctly identifies the statement as false with valid counterexample (n=2), provides rigorous modular arithmetic proof showing the failure is systematic for all n satisfying the hypothesis, includes complete residue class analysis, and clearly explains where the intuitive reasoning goes wrong. Well-structured with clear formatting."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gpt_5_4",
      "respondent_name": "GPT-5.4",
      "weighted_score": 9.25,
      "brief_justification": "Rigorous modular arithmetic proof showing the statement is always false under the given hypothesis. Clear counterexample (n=2) and insightful explanation of why squaring -1 gives +1, destroying divisibility."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "grok_420",
      "respondent_name": "Grok 4.20",
      "weighted_score": 8.6,
      "brief_justification": "Correctly identifies the statement as false and provides valid counterexample (n=2). Mathematical reasoning is sound. Minor issue: confusingly states 'The statement is true' at the start then disproves it. Otherwise well-structured with good explanation of why intuition fails."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "claude_sonnet_46",
      "respondent_name": "Claude Sonnet 4.6",
      "weighted_score": 10,
      "brief_justification": "Excellent response with clear counterexample (n=2), comprehensive modular analysis showing the statement fails for all n satisfying the hypothesis, and insightful explanation of why the intuition fails (squaring -1 gives +1, moving away from divisibility)."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "Excellent rigorous proof showing the statement is false. Complete analysis of modulo 5 residues, algebraic verification using factorization, clear counterexample (n=2), and insightful explanation of why intuition fails. All mathematical reasoning is sound and well-presented."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "Perfect response: correctly identifies the statement as false, provides rigorous modular arithmetic proof showing n⁴+1 ≡ 2 (mod 5), gives clear counterexample (n=2), explains intuition failure, and includes comprehensive classification table. Well-structured with excellent mathematical clarity."
    },
    {
      "judge_key": "minimax_m25",
      "judge_name": "MiniMax M2.5",
      "respondent_key": "gemini_25_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.8,
      "brief_justification": "Correctly proves the statement false through rigorous modular arithmetic analysis. Shows that when n² + 1 ≡ 0 (mod 5), then n⁴ + 1 ≡ 2 (mod 5). Provides clear counterexample (n=2) and explains why intuition fails by showing the critical role of the '+1' after squaring."
    }
  ],
  "meta": {
    "source": "The Multivac (app.themultivac.com)",
    "methodology": "10x10 blind peer matrix evaluation",
    "criteria": "correctness, completeness, clarity, depth, usefulness",
    "self_judgments": "excluded from rankings",
    "license": "Open data — cite as: The Multivac (2026)"
  }
}