{
  "evaluation_id": "EVAL-20260207-134115",
  "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": "GPT-OSS-120B",
    "provider": "OpenAI",
    "score": 9.94
  },
  "avg_score": 9.684,
  "matrix_size": 90,
  "models_used": [
    {
      "id": "gemini_3_flash",
      "name": "Gemini 3 Flash Preview",
      "provider": "Google"
    },
    {
      "id": "claude_sonnet",
      "name": "Claude Sonnet 4.5",
      "provider": "Anthropic"
    },
    {
      "id": "deepseek_v3",
      "name": "DeepSeek V3.2",
      "provider": "DeepSeek"
    },
    {
      "id": "claude_opus",
      "name": "Claude Opus 4.5",
      "provider": "Anthropic"
    },
    {
      "id": "gemini_3_pro",
      "name": "Gemini 3 Pro Preview",
      "provider": "Google"
    },
    {
      "id": "gemini_2_5_flash",
      "name": "Gemini 2.5 Flash",
      "provider": "Google"
    },
    {
      "id": "gpt_oss_120b",
      "name": "GPT-OSS-120B",
      "provider": "OpenAI"
    },
    {
      "id": "olmo_think",
      "name": "OLMo Think",
      "provider": "Allen AI"
    },
    {
      "id": "grok_direct",
      "name": "Grok 3 (Direct)",
      "provider": "xAI"
    },
    {
      "id": "mimo_v2_flash",
      "name": "MiMo-V2-Flash",
      "provider": "Xiaomi"
    }
  ],
  "rankings": {
    "gpt_oss_120b": {
      "display_name": "GPT-OSS-120B",
      "provider": "OpenAI",
      "average_score": 9.94,
      "score_count": 9,
      "min_score": 9.65,
      "max_score": 10,
      "rank": 1
    },
    "gemini_3_flash": {
      "display_name": "Gemini 3 Flash Preview",
      "provider": "Google",
      "average_score": 9.78,
      "score_count": 9,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 2
    },
    "grok_direct": {
      "display_name": "Grok 3 (Direct)",
      "provider": "xAI",
      "average_score": 9.78,
      "score_count": 9,
      "min_score": 9.45,
      "max_score": 10,
      "rank": 3
    },
    "claude_opus": {
      "display_name": "Claude Opus 4.5",
      "provider": "Anthropic",
      "average_score": 9.76,
      "score_count": 6,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 4
    },
    "deepseek_v3": {
      "display_name": "DeepSeek V3.2",
      "provider": "DeepSeek",
      "average_score": 9.76,
      "score_count": 7,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 5
    },
    "olmo_think": {
      "display_name": "OLMo Think",
      "provider": "Allen AI",
      "average_score": 9.69,
      "score_count": 9,
      "min_score": 8.9,
      "max_score": 10,
      "rank": 6
    },
    "mimo_v2_flash": {
      "display_name": "MiMo-V2-Flash",
      "provider": "Xiaomi",
      "average_score": 9.66,
      "score_count": 6,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 7
    },
    "claude_sonnet": {
      "display_name": "Claude Sonnet 4.5",
      "provider": "Anthropic",
      "average_score": 9.64,
      "score_count": 6,
      "min_score": 9.3,
      "max_score": 10,
      "rank": 8
    },
    "gemini_2_5_flash": {
      "display_name": "Gemini 2.5 Flash",
      "provider": "Google",
      "average_score": 9.46,
      "score_count": 7,
      "min_score": 8.85,
      "max_score": 9.8,
      "rank": 9
    },
    "gemini_3_pro": {
      "display_name": "Gemini 3 Pro Preview",
      "provider": "Google",
      "average_score": 9.37,
      "score_count": 8,
      "min_score": 7.8,
      "max_score": 10,
      "rank": 10
    }
  },
  "judgments": [
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 10,
      "brief_justification": "The response is factually correct, provides a rigorous proof and counterexample, explains the intuition failure thoroughly, and is exceptionally clear and well-structured."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous proof with modular arithmetic, gives clear counterexamples, and explains the intuitive error, making it factually accurate, thorough, well-structured, insightful, and highly actionable."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with a rigorous modular arithmetic proof and a clear counterexample, thoroughly covering all aspects of the question with exceptional clarity and practical value."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous modular arithmetic proof, gives clear counterexamples, and explains the algebraic reason for the failure of the intuition."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false with a valid counterexample (n=2) and provides a rigorous modular arithmetic proof that the conclusion n^4+1 ≡ 2 mod 5, not 0, under the hypothesis. It is thorough, well-structured, and insightful."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample (n=2), and offers a rigorous modular arithmetic proof showing n^4+1 ≡ 2 (mod 5) under the given condition, with clear explanation of the intuition failure."
    },
    {
      "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 factually correct, thoroughly covers the proof and counterexample, is exceptionally clear and well-structured, provides deep insight into modular arithmetic and why the intuition fails, and is highly useful for understanding the logical flaw."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly identifies the statement as false with a valid counterexample (n=2), provides a rigorous modular arithmetic proof, and explains the intuition failure. It is factually accurate, thoroughly covers the proof and explanation, is clearly structured, offers insightful analysis of the modular arithmetic, and is highly actionable for understanding the problem."
    },
    {
      "judge_key": "mimo_v2_flash",
      "judge_name": "MiMo-V2-Flash",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.8,
      "brief_justification": "The response is factually correct, thoroughly covers the proof and counterexample, is exceptionally clear and well-structured, provides deep modular arithmetic analysis, and is highly useful for understanding the problem."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous proof using modular arithmetic, offers a valid counterexample (n=2), and explains the mathematical reason why the implication fails."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous modular arithmetic proof, offers concrete counterexamples, and explains the logical fallacy behind the initial claim."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 10,
      "brief_justification": "The model correctly disproves the statement using modular arithmetic, provides a clear counterexample (n=2), and explains the logical reason why the conclusion is impossible (n^4 + 1 will always be 2 mod 5 if n^2 + 1 is 0 mod 5)."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a rigorous modular arithmetic proof, offers concrete counterexamples, and explains the algebraic reason why the property fails."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample (n=2), and uses modular arithmetic to prove that n^4 + 1 will always be congruent to 2 mod 5 if the hypothesis holds."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.8,
      "brief_justification": "The model correctly identifies the statement as false, provides a rigorous proof using modular arithmetic, and offers a clear counterexample (n=2). It also insightfully explains that n^4 + 1 will always leave a remainder of 2 under the given conditions."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response provides a perfect rigorous disproof using modular arithmetic, correctly identifies that the conclusion is never true when the hypothesis is met, and includes a clear counterexample and intuitive explanation."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample (n=2), and offers a rigorous modular arithmetic proof showing that n^4 + 1 will always be 2 mod 5 if n^2 + 1 is 0 mod 5."
    },
    {
      "judge_key": "gemini_3_flash",
      "judge_name": "Gemini 3 Flash Preview",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement using both modular arithmetic and specific counterexamples (n=2, n=3). It provides a rigorous explanation of why the condition n^2 + 1 ≡ 0 (mod 5) leads to n^4 + 1 ≡ 2 (mod 5), and it explains the failure of intuition as requested."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous, mathematically correct proof using modular arithmetic, includes a concrete counterexample (n=2), and offers excellent insight into why the intuitive expectation fails. The explanation is clear, complete, and pedagogically valuable."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 10,
      "brief_justification": "Flawless mathematical proof with rigorous modular arithmetic analysis, clear counterexample, and exceptional insight into why the intuition fails, including connections to Fermat's Little Theorem and group theory."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.6,
      "brief_justification": "The response provides a rigorous, mathematically correct proof that the statement is false, using modular arithmetic and Fermat's Little Theorem. It includes a concrete counterexample (n=2), explains the underlying mathematics clearly, and addresses why intuition might fail, making it both complete and highly useful."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous, mathematically correct proof that the statement is false, includes concrete counterexamples, and offers excellent insight into why the intuition fails through algebraic analysis. The self-correction mid-proof demonstrates careful reasoning."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous disproof with both a concrete counterexample (n=2) and a general modular arithmetic proof showing n⁴+1≡2(mod 5) when the hypothesis holds. The logic is flawless, well-structured, and addresses all aspects of the question including why the intuition fails."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with rigorous modular arithmetic, provides a concrete counterexample (n=2), and offers excellent insight into why the intuition fails by showing n^4+1 ≡ 2 (mod 5) whenever the condition holds."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "Flawless mathematical proof with systematic modular arithmetic analysis, concrete counterexamples, and excellent explanation of why the intuitive reasoning fails. The response is rigorous, complete, and pedagogically excellent."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a valid counterexample (n=2), rigorous modular arithmetic proof showing n⁴+1≡2(mod 5), and excellent explanation of why the intuition fails. All mathematical reasoning is sound and complete."
    },
    {
      "judge_key": "claude_sonnet",
      "judge_name": "Claude Sonnet 4.5",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous, mathematically sound proof by systematically analyzing the problem using modular arithmetic, finding concrete counterexamples (n=2, n=3), and explaining why the intuition fails. The logic is flawless and the presentation is exceptionally clear and thorough."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.5,
      "brief_justification": "The response correctly disproves the statement with a rigorous modular arithmetic proof and a clear counterexample, thoroughly covering the topic and explaining the intuition failure, though the depth could slightly improve by generalizing the result for all such n."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.5,
      "brief_justification": "The response is mathematically flawless, provides a complete modular analysis and counterexample, explains the intuition failure insightfully, and is exceptionally clear, though the depth is slightly limited by not exploring the group-theoretic perspective more explicitly."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement with a clear counterexample and modular arithmetic, thoroughly explains the reasoning, and provides insightful analysis of why the intuitive approach fails."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with a rigorous modular arithmetic proof, provides explicit counterexamples, and insightfully explains why the intuitive expectation fails."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement with a clear counterexample and provides a rigorous modular arithmetic proof, though the conclusion is cut off mid-sentence in the final section."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.45,
      "brief_justification": "The response correctly disproves the statement with a rigorous modular arithmetic proof and a clear counterexample (n=2), thoroughly covering all aspects and providing insightful analysis on why the intuition fails."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with rigorous modular arithmetic, provides counterexamples, and explains the intuition failure, making it accurate, thorough, clear, insightful, and educational."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.65,
      "brief_justification": "The response provides a perfect counterexample, rigorous modular arithmetic proof, and insightful explanation of why intuitive reasoning fails, making it mathematically flawless and pedagogically valuable."
    },
    {
      "judge_key": "deepseek_v3",
      "judge_name": "DeepSeek V3.2",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.5,
      "brief_justification": "The response is mathematically flawless, thoroughly covering all necessary modular arithmetic cases and providing clear counterexamples, though the explanation of why intuition fails could be slightly more detailed."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.35,
      "brief_justification": "The response correctly disproves the statement using rigorous modular arithmetic, provides a concrete counterexample (n=2), and explains why the intuition fails. There's a minor inconsistency where the proof initially claims the statement is 'true' but then correctly concludes it's 'false' - this appears to be a formatting error that doesn't affect the mathematical validity of the work."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a flawless proof by contradiction using modular arithmetic, correctly identifies that n²≡4(mod 5) implies n⁴≡1(mod 5), gives concrete counterexamples (n=2,3), and offers excellent insight into why the intuition fails through multiple perspectives including Fermat's Little Theorem and group theory. The structure is impeccable with clear logical progression."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with rigorous modular arithmetic, provides multiple counterexamples (n=2 and n=3), and clearly explains why the intuition fails. The self-correction during the proof demonstrates careful verification, and the explanation of why squaring doesn't preserve the divisibility condition is insightful and mathematically precise."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous, correct proof that the statement is false. It systematically analyzes the problem using modular arithmetic, applies Fermat's Little Theorem appropriately, provides a concrete counterexample (n=2), and explains why intuition might fail. The structure is excellent with clear steps and a well-formatted conclusion."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 9.45,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample (n=2) and provides a rigorous general proof using modular arithmetic showing n⁴+1 ≡ 2 (mod 5) whenever n²+1 ≡ 0 (mod 5). The explanation is exceptionally clear and well-structured, though the final sentence appears truncated and could have included a brief note on why the intuition fails."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 9.6,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample (n=2), and gives a rigorous modular arithmetic proof. The self-correction process shown actually enhances understanding. The explanation of why intuition fails is insightful, showing that n²≡-1(mod 5) leads to n⁴≡1(mod 5), hence n⁴+1≡2(mod 5). Minor clarity deduction for the initially confusing 'true' claim before correction."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": ""
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample (n=2), provides a rigorous modular arithmetic proof showing n⁴+1 ≡ 2 (mod 5) whenever n²+1 ≡ 0 (mod 5), and clearly explains why the intuition fails. The proof is complete, well-structured, and demonstrates that the implication never holds, not just for one case."
    },
    {
      "judge_key": "claude_opus",
      "judge_name": "Claude Opus 4.5",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.8,
      "brief_justification": "The response provides a rigorous, complete proof that the statement is false. It systematically analyzes all residue classes mod 5, correctly identifies when n²+1 ≡ 0 (mod 5), computes n⁴+1 for those cases, provides concrete counterexamples (n=2 and n=3), and explains why the intuition fails. The modular arithmetic is flawless, the presentation is exceptionally clear and well-structured, and the verification is thorough."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample, and offers an excellent theoretical explanation using modular arithmetic and Fermat's Little Theorem."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement using modular arithmetic, provides a valid counterexample, and offers clear algebraic insight into why the statement fails."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The model correctly disproves the statement using rigorous modular arithmetic, provides a valid counterexample, and clearly explains the algebraic reason why the intuition fails."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 10,
      "brief_justification": "The model correctly disproved the statement with a valid counterexample and provided a rigorous algebraic proof showing that the conclusion is actually never true given the premise."
    },
    {
      "judge_key": "gemini_3_pro",
      "judge_name": "Gemini 3 Pro Preview",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement using rigorous modular arithmetic, provides valid counterexamples, and clearly explains why the implication fails."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 10,
      "brief_justification": "The model correctly disproved the statement, provided a rigorous proof using modular arithmetic, and gave a clear counterexample with an excellent explanation of why the intuition fails."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.65,
      "brief_justification": "The response correctly disproves the statement using modular arithmetic and provides a clear counterexample. The explanation of why intuition fails is insightful."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.65,
      "brief_justification": "The model correctly identifies the statement as false, provides a rigorous counterexample, and explains the mathematical reasoning clearly. The self-correction process is well-demonstrated."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with a rigorous proof and a clear counterexample. The explanation of why intuition might fail is also well-articulated."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.65,
      "brief_justification": "The model correctly identifies the statement as false, provides a rigorous proof for its falsity, and offers clear counterexamples and an explanation for the intuition's failure."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a valid counterexample, and then rigorously proves the general case using modular arithmetic, showing why the conclusion cannot be true given the hypothesis."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement, provides a rigorous proof using modular arithmetic, gives a clear counterexample, and explains the intuition behind the failure."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false, provides a clear counterexample, and rigorously proves the falsity using modular arithmetic. The explanation of why intuition might fail is also insightful."
    },
    {
      "judge_key": "gemini_2_5_flash",
      "judge_name": "Gemini 2.5 Flash",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.65,
      "brief_justification": "The response correctly disproves the statement using modular arithmetic and provides clear counterexamples. The explanation of why intuition might fail is also good."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.8,
      "brief_justification": "The response correctly disproves the statement with a solid modular arithmetic proof and a concrete counterexample, explains why the intuition fails, and does so in a clear, thorough, and insightful manner."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 8.6,
      "brief_justification": "The answer correctly disproves the statement with a valid counterexample and a solid modular arithmetic argument, though it ends abruptly, slightly affecting completeness and clarity."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.85,
      "brief_justification": "The response correctly disproves the statement with a valid modular proof and counterexample, fully addressing the prompt, though it repeats steps slightly, affecting clarity."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 8.9,
      "brief_justification": "The answer correctly disproves the statement with a valid counterexample and a clear modular arithmetic proof, explaining the intuition error; it covers all required points though depth could be slightly deeper."
    },
    {
      "judge_key": "gpt_oss_120b",
      "judge_name": "GPT-OSS-120B",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 9.45,
      "brief_justification": "The answer correctly disproves the statement with a general modular proof and concrete counterexamples, and clearly explains why the intuition fails."
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 10,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample and provides a rigorous modular arithmetic analysis, addressing both the mathematical proof and common intuitive misconceptions."
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 0,
      "brief_justification": ""
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 10,
      "brief_justification": "The response rigorously disproves the statement with a valid counterexample and modular arithmetic, clearly explaining the flaw in the implication."
    },
    {
      "judge_key": "olmo_think",
      "judge_name": "OLMo Think",
      "respondent_key": "grok_direct",
      "respondent_name": "Grok 3 (Direct)",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies the statement as false with valid counterexamples and thorough modular arithmetic analysis, demonstrating accurate reasoning and addressing the intuition failure."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "mimo_v2_flash",
      "respondent_name": "MiMo-V2-Flash",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement using modular arithmetic and a counterexample, with thorough and clear steps. The explanation of intuition failure adds depth, though practical usefulness is slightly limited."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_3_flash",
      "respondent_name": "Gemini 3 Flash Preview",
      "weighted_score": 9.3,
      "brief_justification": "The response accurately disproves the statement with a clear counterexample and rigorous modular arithmetic analysis. It is thorough, well-structured, and provides deep insight into why the intuition fails, though the practical usefulness is slightly limited by the abstract nature of the topic."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "claude_sonnet",
      "respondent_name": "Claude Sonnet 4.5",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement with a rigorous proof and counterexamples, covering all necessary aspects with clear reasoning. Minor improvement could be made in the intuitive explanation's conciseness."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "deepseek_v3",
      "respondent_name": "DeepSeek V3.2",
      "weighted_score": 9.3,
      "brief_justification": "The response provides a rigorous and correct proof by contradiction using modular arithmetic, includes a clear counterexample, and explains potential intuitive errors. It is thorough and well-structured, though minor redundancy slightly affects clarity."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "claude_opus",
      "respondent_name": "Claude Opus 4.5",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement with a rigorous proof and counterexamples, covers all aspects of the problem, and provides clear reasoning with minor stylistic redundancy; it offers deep algebraic insight but has limited practical application."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_3_pro",
      "respondent_name": "Gemini 3 Pro Preview",
      "weighted_score": 7.8,
      "brief_justification": "The response correctly disproves the statement with a counterexample and uses modular arithmetic for a general proof, though it is incomplete as the final conclusion is cut off. The explanation is clear and structured, but lacks full depth due to the unfinished modular proof."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gemini_2_5_flash",
      "respondent_name": "Gemini 2.5 Flash",
      "weighted_score": 8.9,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample and modular arithmetic, though it initially stumbles in tone before correcting itself. It provides a thorough explanation with minor clarity issues in the flow of reasoning."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "gpt_oss_120b",
      "respondent_name": "GPT-OSS-120B",
      "weighted_score": 9.65,
      "brief_justification": "The response is mathematically accurate, thoroughly disproves the statement with a clear counterexample and modular arithmetic, and explains the failure of intuition. It is well-structured and insightful, though slightly less actionable for non-mathematical contexts."
    },
    {
      "judge_key": "grok_direct",
      "judge_name": "Grok 3 (Direct)",
      "respondent_key": "olmo_think",
      "respondent_name": "OLMo Think",
      "weighted_score": 9.3,
      "brief_justification": "The response correctly disproves the statement with a valid counterexample and rigorous modular arithmetic proof. It is thorough, clear, and insightful, though slightly less actionable for practical use."
    }
  ],
  "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)"
  }
}