{ "evaluation_id": "EVAL-20260402-175350", "question_id": "REASON-023", "question_text": "Achilles gives a tortoise a 100-meter head start. Achilles runs at 10 m/s, the tortoise at 1 m/s. Zeno argues Achilles can never catch the tortoise because he must first reach where the tortoise was, but by then the tortoise has moved. (1) Resolve the paradox using limits. (2) Resolve it without calculus — using only physical reasoning. (3) Is there a version of Zeno's paradox that modern physics cannot fully resolve? (Hint: consider Planck length.)", "category": "reasoning", "timestamp": "2026-04-02T17:53:50.687Z", "display_date": "Apr 02, 2026", "winner": { "name": "GPT-OSS-120B", "provider": "OpenAI", "score": 9.38 }, "avg_score": 8.354, "matrix_size": 88, "models_used": [ { "id": "gemini_31_pro", "name": "Gemini 3.1 Pro", "provider": "openrouter" }, { "id": "deepseek_v4", "name": "DeepSeek V4", "provider": "openrouter" }, { "id": "gpt_5_4", "name": "GPT-5.4", "provider": "openrouter" }, { "id": "claude_opus_46", "name": "Claude Opus 4.6", "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" }, { "id": "minimax_m25", "name": "MiniMax M2.5", "provider": "openrouter" } ], "rankings": { "gpt_oss_120b": { "display_name": "GPT-OSS-120B", "provider": "OpenAI", "average_score": 9.38, "score_count": 9, "min_score": 8.65, "max_score": 10, "rank": 1 }, "claude_sonnet_46": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 9.29, "score_count": 9, "min_score": 8.45, "max_score": 10, "rank": 2 }, "mimo_v2_flash": { "display_name": "MiMo-V2-Flash", "provider": "Xiaomi", "average_score": 9.19, "score_count": 9, "min_score": 8.55, "max_score": 10, "rank": 3 }, "claude_opus_46": { "display_name": "Claude Opus 4.6", "provider": "openrouter", "average_score": 9.1, "score_count": 9, "min_score": 8.5, "max_score": 9.85, "rank": 4 }, "gpt_5_4": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 8.99, "score_count": 9, "min_score": 8.45, "max_score": 10, "rank": 5 }, "gemini_25_flash": { "display_name": "Gemini 2.5 Flash", "provider": "openrouter", "average_score": 8.9, "score_count": 9, "min_score": 8.4, "max_score": 9.65, "rank": 6 }, "grok_420": { "display_name": "Grok 4.20", "provider": "openrouter", "average_score": 8.89, "score_count": 9, "min_score": 8.45, "max_score": 9.2, "rank": 7 }, "deepseek_v4": { "display_name": "DeepSeek V4", "provider": "openrouter", "average_score": 8.8, "score_count": 9, "min_score": 8, "max_score": 9.65, "rank": 8 }, "gemini_31_pro": { "display_name": "Gemini 3.1 Pro", "provider": "openrouter", "average_score": 6.71, "score_count": 8, "min_score": 3.65, "max_score": 8.25, "rank": 9 }, "minimax_m25": { "display_name": "MiniMax M2.5", "provider": "openrouter", "average_score": 4.29, "score_count": 8, "min_score": 1.55, "max_score": 8.45, "rank": 10 } }, "judgments": [ { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.25, "brief_justification": "Accurate mathematical derivation and solid physical reasoning. Missing the final part regarding modern physics/Planck length." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9, "brief_justification": "Accurate and thorough explanation of all parts, clear structure, good depth on Planck length, practical insights." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.2, "brief_justification": "Accurate mathematical and physical resolutions; correctly identifies the Planck length as the modern unresolved variant." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9.65, "brief_justification": "The response accurately and comprehensively addresses all three parts of the prompt, providing clear mathematical, physical, and quantum-mechanical explanations." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.5, "brief_justification": "" }, { "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 addresses all three parts of the prompt with accurate mathematics, intuitive physical reasoning, and a highly nuanced explanation of the Planck length and quantum gravity." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.2, "brief_justification": "The response accurately addresses all parts of the prompt with clear explanations and good depth, though it misses the simple relative velocity calculation for the physical resolution." }, { "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 exceptionally accurate, comprehensive, and well-structured. It flawlessly addresses all three parts of the prompt, providing rigorous mathematical proofs, sound physical reasoning, and profound insights into quantum gravity and the Planck scale." }, { "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 perfectly addresses all three parts of the prompt with mathematical accuracy, clear physical reasoning, and deep philosophical insight into modern physics." }, { "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 perfectly addresses all parts of the prompt with accurate mathematics, clear physical reasoning, and deep insights into modern physics and the Planck scale." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9.65, "brief_justification": "The response accurately and comprehensively addresses all three parts of the prompt, providing clear mathematical and physical explanations, and delving deeply into the implications of Planck-scale physics." }, { "judge_key": "gemini_31_pro", "judge_name": "Gemini 3.1 Pro", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 1.55, "brief_justification": "The response cuts off mid-sentence and fails to address any of the three parts of the prompt, providing only the initial setup." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 8.1, "brief_justification": "Accurate and detailed mathematical resolution, though physical reasoning section is incomplete." }, { "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": "Comprehensive and accurate resolution of Zeno's paradox with both mathematical and physical reasoning, including a deep dive into modern physics implications." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9.3, "brief_justification": "Comprehensive and accurate resolution of the paradox with clear explanations and depth." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.65, "brief_justification": "Accurate and thorough with insightful analysis, though Planck-scale discussion could be slightly more detailed." }, { "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": "Comprehensive and accurate resolution of Zeno's paradox with both mathematical and physical reasoning, plus insightful discussion of Planck-scale implications." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.3, "brief_justification": "Accurate, thorough, and insightful with clear explanations." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.3, "brief_justification": "Comprehensive and accurate with insightful analysis, though slightly dense in presentation." }, { "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": "Comprehensive and accurate resolution of the paradox with both calculus and physical reasoning, plus insightful discussion on Planck length implications." }, { "judge_key": "deepseek_v4", "judge_name": "DeepSeek V4", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 7.9, "brief_justification": "Accurate and clear, but lacks depth in the modern physics aspect." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.25, "brief_justification": "The calculus resolution (Part 1) is excellent—correct geometric series calculation, clear explanation, and accurate numerical result. However, the response is clearly truncated: Part 2 is incomplete (cuts off mid-sentence) and Part 3 about Planck length and modern physics is entirely missing. Since two of the three requested parts are incomplete or absent, completeness and depth suffer significantly despite the high quality of Part 1." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8, "brief_justification": "The response correctly resolves the paradox using geometric series and limits, provides a clear physical reasoning approach, and addresses the Planck length question reasonably. The math is accurate throughout. However, the depth on part (3) could be stronger — it doesn't deeply engage with whether spacetime is actually discrete at the Planck scale vs. merely being epistemically inaccessible, nor does it mention the quantum Zeno effect or other related physics. The 'without calculus' section (pa" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 9, "brief_justification": "The response correctly resolves the paradox using limits (geometric series converging to 100/9), provides a clean physical reasoning resolution using relative velocity, and thoughtfully addresses the Planck length question with appropriate nuance—noting that the Planck length is not proven to be a minimum distance but rather a scale where current theories break down. The mathematical work is accurate, the explanations are exceptionally clear and well-structured, and the depth is good though coul" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.2, "brief_justification": "The response is exceptionally well-structured and addresses all three parts thoroughly. The geometric series calculation is correct (converging to 100/9 ≈ 11.111 s). The physical reasoning is sound and clearly articulated. The Planck-scale discussion is nuanced and appropriately hedged—it correctly notes that we lack a complete theory of quantum gravity and that the discreteness of spacetime at the Planck scale is hypothetical rather than established. Minor quibble: the claim that the Planck-sca" }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.4, "brief_justification": "Excellent response. Accurate, thorough, and insightful. The connection to Planck scale and quantum gravity is particularly strong and well-explained. Minor deduction in correctness for a slightly imprecise description of the quantum Zeno effect's relevance, but overall outstanding." }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "Excellent response across all three parts. Part 1 correctly sets up and sums the geometric series, identifies the convergence to 100/9 seconds, and clearly articulates Zeno's logical error. Part 2 provides both the algebraic solution and a genuinely insightful philosophical point about confusing description with mechanism. Part 3 thoughtfully engages with the Planck-scale version, correctly noting the open questions in quantum gravity while mentioning both LQG and string theory perspectives. The" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.2, "brief_justification": "The response is exceptionally well-structured and addresses all three parts thoroughly. Part 1 correctly derives the geometric series and its convergent sum. Part 2 uses relative velocity elegantly and provides clear physical intuition. Part 3 thoughtfully discusses the Planck length and the tension between continuous and discrete spacetime, though it slightly overstates the case that this constitutes an unresolvable version of Zeno's paradox (the Planck length is not definitively established as" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.4, "brief_justification": "The response is exceptionally well-structured, addressing all three parts of the question with mathematical rigor and clear physical reasoning. The calculus resolution correctly derives the geometric series and its convergent sum. The non-calculus resolution elegantly uses relative velocity and discrete-step reasoning. The Planck-scale discussion is nuanced, correctly noting that we lack experimental confirmation of spacetime discreteness and mentioning the tension with Lorentz invariance. Minor" }, { "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, "brief_justification": "The response is exceptionally well-structured and covers all three parts thoroughly. Part 1 correctly derives the geometric series and its convergent sum. Part 2 uses relative velocity cleanly without calculus. Part 3 thoughtfully discusses the Planck scale implications, though it slightly overstates the case for spacetime discreteness (it's speculative, not established) and could have been more careful to note that Planck length is not confirmed as a minimum distance but rather a scale where cu" }, { "judge_key": "claude_opus_46", "judge_name": "Claude Opus 4.6", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 2.45, "brief_justification": "The response is clearly truncated and incomplete. It sets up the problem well but fails to actually resolve any of the three parts of the question. There is no limit calculation, no physical reasoning resolution, and no discussion of Planck length or modern physics. The response cuts off mid-sentence, providing essentially no substantive answer to any of the three required parts." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 3.65, "brief_justification": "The calculus part is correct, but the response is incomplete: the non-calculus resolution is cut off and the modern-physics/Planck-length part is missing. What is present is fairly clear but shallow and not very useful overall because two of the three requested parts are not answered." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "Accurate and well-structured response that correctly resolves the paradox with limits and relative speed, and gives a sensible modern-physics caveat. Minor issue: it overstates the Planck length as the smallest meaningful length and could better distinguish speculative quantum-gravity limits from an actual unresolved paradox." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.75, "brief_justification": "Accurate and well-structured, with strong mathematical and physical resolutions. The discussion of Planck scale and quantum Zeno effect is insightful, though it slightly overstates the openness of the issue and treats Planck length/time as possible minimum scales more definitively than current physics warrants." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.8, "brief_justification": "Accurate and well-structured response that correctly resolves the paradox with limits and with physical reasoning. The Planck-scale discussion is thoughtful and appropriately cautious, though it slightly overstates the openness by tying it specifically to Planck discreteness, which is speculative rather than established." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.6, "brief_justification": "Accurately resolves the classical paradox with both geometric-series limits and elementary relative-motion reasoning, and clearly explains Zeno’s mistake. The modern-physics discussion is thoughtful and appropriately notes uncertainty, though it slightly overstates the status of Planck length as a direct unresolved 'version' of Zeno and mixes speculative quantum-gravity issues with established physics." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.55, "brief_justification": "Accurate and well-structured resolution of the standard paradox with correct calculations and a solid non-calculus explanation. The physics discussion is thoughtful, but it overstates Planck length as the smallest meaningful unit and suggests discreteness more strongly than current physics warrants." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9, "brief_justification": "Accurate and well-structured response that fully addresses all three parts, including the geometric-series limit, a non-calculus relative-speed explanation, and a nuanced discussion of Planck-scale discreteness as an open issue in quantum gravity. Minor caveat: the non-calculus section leans slightly on practical measurement resolution rather than purely idealized physical reasoning." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.4, "brief_justification": "Accurate and well-structured resolution of the classical paradox using both limits and relative speed. Strong coverage and clear exposition. Slightly overstated on Planck length/time as established smallest meaningful units and on physics being unable to order events below Planck time; these are speculative aspects of quantum gravity rather than settled facts." }, { "judge_key": "gpt_5_4", "judge_name": "GPT-5.4", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 1.85, "brief_justification": "The response is only an incomplete setup and does not actually resolve the paradox in any of the three requested ways. What is present is mostly accurate and clearly stated, but it lacks the substantive answers needed." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.5, "brief_justification": "Part 1 is fully correct with accurate math. Part 2 is incomplete as it cuts off mid-sentence without providing the promised physical reasoning. Part 3 is entirely missing despite the question asking for it. Overall response is clear and insightful where present but lacks full coverage." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.45, "brief_justification": "The response accurately resolves the paradox with limits via geometric series convergence, provides a solid non-calculus physical argument, and thoughtfully addresses the Planck length limitation in modern physics. Minor issues include slightly mislabeling the relative speed method as 'limits' and a small arithmetic error in the series (r=0.1 but first term should align precisely), but overall it is factually strong, clear, and insightful." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.85, "brief_justification": "The response accurately resolves the paradox with limits via geometric series and direct equations, provides clear physical reasoning without calculus using relative velocity, and insightfully discusses Planck-scale issues with quantum gravity implications. It is thorough, well-structured, and practically useful for understanding both classical and modern physics perspectives." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.65, "brief_justification": "The response accurately resolves the paradox with limits via equations and geometric series, provides clear physical reasoning without calculus using relative speed and analogy, and thoughtfully discusses the Planck length's implications for spacetime divisibility while correctly noting current physics limitations. Minor deduction for not explicitly using the term 'limit' in part 1 beyond the series sum, but overall thorough, insightful, and well-structured." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.65, "brief_justification": "The response accurately resolves the paradox with limits via geometric series and direct algebra for non-calculus part. It provides clear tables, equations, and insightful physical analogies. Part 3 thoughtfully addresses Planck-scale issues with current quantum gravity theories, though some claims like 'spacetime loses operational meaning' are interpretive rather than strictly factual. Overall thorough, well-structured, and useful." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.65, "brief_justification": "The response accurately resolves the paradox with limits via geometric series and relative velocity without calculus. It correctly identifies the Dichotomy paradox and Planck scale issues in modern physics. Thorough, clear structure with equations and explanations. Minor nitpick: the Planck length discussion is insightful but slightly overstates 'cannot fully resolve' as the kinematic paradox is resolved; quantum gravity is incomplete but not a direct failure." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 8.65, "brief_justification": "The response accurately resolves the paradox with limits via convergent series, provides clear non-calculus physical arguments using relative velocity and discrete steps, and insightfully discusses the Planck-scale version as an open question in quantum gravity, covering all parts thoroughly with good structure and practical explanations." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.65, "brief_justification": "The response accurately resolves the paradox with limits via geometric series and relative velocity, provides clear physical reasoning without calculus, and insightfully discusses Planck-scale implications for quantum gravity, covering all parts thoroughly with good structure and practical explanations." }, { "judge_key": "grok_420", "judge_name": "Grok 4.20", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 3.65, "brief_justification": "The response is incomplete as it cuts off mid-sentence without addressing any of the three required parts. It sets up the problem correctly but fails to resolve the paradox with limits, without calculus, or discuss modern physics/Planck length." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 7.85, "brief_justification": "The mathematical resolution in part 1 is excellent — clear, correct, and well-structured with proper geometric series application. The physical reasoning in part 2 appears to be cut off mid-sentence, leaving that section incomplete. Part 3 about Planck length and modern physics limitations is entirely missing. The response is strong where it exists but fails to deliver on two of the three required components, significantly hurting completeness scores." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.6, "brief_justification": "The response correctly resolves the paradox both mathematically (geometric series converging to finite sum) and physically (relative speed argument). The Planck length discussion is accurate and relevant, though it could go deeper into quantum gravity implications and the discreteness vs. continuity debate. All three parts are well-addressed with clear structure and accurate calculations." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 9.4, "brief_justification": "Exceptional response covering all three parts with rigor and insight. The mathematical resolution is correct and well-verified, the physical reasoning is elegant and accessible, and the Planck-scale discussion is genuinely sophisticated, touching on loop quantum gravity, spacetime foam, the quantum Zeno effect, and the limits of real analysis as a physical model. Minor quibble: the quantum Zeno effect connection is slightly tangential but adds interesting depth. Tables enhance clarity throughout" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.8, "brief_justification": "The response accurately resolves the paradox both mathematically and physically, with clear step-by-step reasoning. The geometric series approach is correct and well-explained. The physical reasoning section is intuitive and accessible. The third part thoughtfully addresses the Planck length hint while correctly noting the important caveat that space isn't proven to be discrete at that scale — showing genuine depth. Minor deduction for depth as the quantum gravity discussion, while accurate, cou" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9, "brief_justification": "Excellent response covering all three parts with accuracy and clarity. The calculus resolution is correct with proper geometric series computation. The physical reasoning is intuitive and well-articulated. The Planck-scale discussion is thoughtful and appropriately hedged. Minor limitation: the depth on the Planck-scale paradox could explore more (e.g., loop quantum gravity, discrete spacetime models, supertask literature), but overall this is a high-quality, well-structured answer." }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9, "brief_justification": "Excellent response covering all three parts thoroughly. The calculus resolution is correct with proper geometric series calculation. The physical reasoning using relative velocity is elegant and accessible. The Planck length discussion is well-reasoned, correctly identifying the tension between continuous and discrete spacetime. Minor quibble: the 'teleportation' framing for discrete space is slightly imprecise, and the response could have mentioned that even with a Planck length cutoff, the ser" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.4, "brief_justification": "Exceptionally thorough and well-structured response. The calculus resolution is mathematically precise with correct geometric series computation. The physical reasoning section offers two distinct non-calculus approaches (relative velocity and discrete-step reasoning). The Planck-scale discussion is sophisticated, correctly identifying the open questions around spacetime discreteness, Lorentz invariance conflicts, and experimental inaccessibility. Minor quibble: the 'discrete-step reasoning' in " }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.8, "brief_justification": "Excellent response covering all three parts thoroughly. The calculus resolution is correct and well-illustrated with the geometric series. The physical reasoning using relative speed is clean and accessible. The Planck scale discussion is accurate and thoughtfully connects to quantum gravity, loop quantum gravity, and string theory. Minor limitation: the third part could push deeper into why this remains genuinely unresolved rather than just describing the open problem, but overall this is a hig" }, { "judge_key": "claude_sonnet_46", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 3.1, "brief_justification": "The response is cut off mid-sentence and fails to actually resolve the paradox in any of the three requested ways. While the setup is correct and the framing of Zeno's argument is accurate, none of the three parts (limits, physical reasoning, Planck length) are addressed. The response is essentially incomplete, providing only an introduction before abruptly ending." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.2, "brief_justification": "Accurate and thorough resolution of the paradox using both calculus and physical reasoning. The discussion of the Planck scale is insightful and correctly identifies the open question in quantum gravity." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.2, "brief_justification": "Accurate and thorough with clear structure, insightful analysis, and practical explanations." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9, "brief_justification": "The response is highly accurate, covering all three parts of the question thoroughly. The mathematical derivation is correct, the physical reasoning is sound, and the discussion of the Planck-scale paradox is nuanced and up-to-date. The structure is clear and easy to follow." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 9.2, "brief_justification": "Accurate and thorough resolution of all three parts, with clear structure and insightful connection to modern physics." }, { "judge_key": "mimo_v2_flash", "judge_name": "MiMo-V2-Flash", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 8.45, "brief_justification": "Accurate and thorough, with clear physical reasoning and limits; the modern-physics part is brief but correct." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.75, "brief_justification": "The answer correctly applies limits to resolve the paradox and presents accurate calculations, but it stops short of providing the non-calculus physical reasoning and the modern physics discussion, making it incomplete and limiting its overall depth and usefulness." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "Accurately solves the paradox with limits and simple reasoning, covers all three prompts, explains series convergence clearly, and offers a thoughtful note on Planck-scale limits, though deeper quantum-gravity detail could improve depth." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.65, "brief_justification": "Accurate math and physics, fully addresses all parts, clearly presented, and adds insightful discussion of quantum limits." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.65, "brief_justification": "Accurately solves the paradox with limits and physical reasoning, covers all three parts, explains series and spacetime issues, and is clearly presented with useful insight." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.45, "brief_justification": "Accurate calculations and logical reasoning, clear step‑by‑step explanation, addresses all three prompts, adds insight about Planck‑scale limits, though the non‑calculus argument could be fleshed out further." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "claude_sonnet_46", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.45, "brief_justification": "Accurately resolves the paradox with limits and algebra, covers the Planck-scale twist, well-structured and insightful, though could elaborate more on modern physics limits." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 8.8, "brief_justification": "Accurately solves the paradox with correct calculus and relative motion reasoning, covers all three prompts, clear presentation, and offers thoughtful discussion of quantum limits." }, { "judge_key": "gpt_oss_120b", "judge_name": "GPT-OSS-120B", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.45, "brief_justification": "Accurately solves the paradox with correct calculations, covers all three prompts, presents clear explanations, adds insightful discussion of Planck-scale limits, and offers useful reasoning for learners." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 9.1, "brief_justification": "The response accurately and clearly resolves Zeno's paradox using both limits and physical reasoning. It also provides a thoughtful and insightful discussion on the Planck length's relevance to an unresolved version of the paradox, demonstrating good depth." }, { "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 is exceptionally thorough, accurate, and well-structured. It addresses all parts of the question with remarkable clarity and depth, especially the nuanced discussion of Zeno's paradox in the context of modern physics and Planck scales. The use of both mathematical limits and physical reasoning provides a comprehensive understanding. The only minor point is that the 'usefulness' could be slightly higher if it explicitly stated practical applications of understanding these paradoxes b" }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.85, "brief_justification": "The response provides a comprehensive and accurate resolution of Zeno's paradox using both mathematical limits and physical reasoning. It also thoughtfully addresses the modern physics implications, particularly regarding Planck length and the nature of spacetime, demonstrating good depth." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 9.1, "brief_justification": "The response provides a comprehensive and accurate resolution of Zeno's paradox using both mathematical and physical reasoning. It also introduces the Planck length concept effectively, adding depth to the discussion." }, { "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 is exceptionally thorough, accurate, and insightful. It addresses all parts of the question with clear explanations, mathematical rigor, and deep physical reasoning. The discussion of the Planck length and its implications for Zeno's paradox is particularly well-articulated and demonstrates a sophisticated understanding of modern physics." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "mimo_v2_flash", "respondent_name": "MiMo-V2-Flash", "weighted_score": 9.25, "brief_justification": "The response provides a comprehensive and accurate resolution of Zeno's paradox using both calculus and physical reasoning. It also delves into the more complex, unresolved aspects related to Planck length, demonstrating excellent depth and clarity." }, { "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 comprehensive and accurate resolution to Zeno's paradox using various methods, including a deep dive into modern physics implications. It is exceptionally clear and well-structured." }, { "judge_key": "gemini_25_flash", "judge_name": "Gemini 2.5 Flash", "respondent_key": "minimax_m25", "respondent_name": "MiniMax M2.5", "weighted_score": 5.4, "brief_justification": "The response only provides the setup and restates Zeno's argument, but does not answer any of the three questions posed in the prompt. It's a good start but incomplete." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_31_pro", "respondent_name": "Gemini 3.1 Pro", "weighted_score": 5.35, "brief_justification": "Part 1 (calculus resolution) is mathematically correct and well-explained. However, Part 2 (physical reasoning without calculus) is incomplete - the response cuts off mid-sentence. Part 3 (Planck length/modern physics) is completely missing, which is a significant gap given the explicit hint in the question. The response only addresses 2 out of 3 requested parts." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "deepseek_v4", "respondent_name": "DeepSeek V4", "weighted_score": 8.8, "brief_justification": "Accurate and comprehensive response with clear explanations for all three parts. Well-structured with correct mathematical derivations and appropriate discussion of Planck length limitations." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "claude_opus_46", "respondent_name": "Claude Opus 4.6", "weighted_score": 8.85, "brief_justification": "Excellent response covering all three parts thoroughly. Mathematical resolution is correct and well-presented with clear equations. Physical reasoning (reference frame argument) is elegant and non-calculus. Part 3 shows genuine depth by connecting to real physics questions about discrete spacetime, Planck scale, and quantum gravity - going beyond textbook answers." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_5_4", "respondent_name": "GPT-5.4", "weighted_score": 8.45, "brief_justification": "Accurate mathematical and physical resolutions with clear explanations. Good depth on Planck length discussion, acknowledging physics limitations. Well-structured with helpful analogy." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "grok_420", "respondent_name": "Grok 4.20", "weighted_score": 8.45, "brief_justification": "Mathematically accurate resolution with clear calculations, good physical reasoning about continuous motion, and thoughtful discussion of Planck-scale limitations. Well-structured and educational." }, { "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": "Comprehensive and accurate resolution of all three parts. Mathematical solution is correct, physical reasoning is insightful, and the Planck-scale discussion correctly identifies genuine open questions in physics. Well-structured with clear explanations and deep philosophical insight." }, { "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 response covering all three parts thoroughly. The mathematical resolution is correct with proper geometric series derivation. The physical reasoning using relative velocity is clear and intuitive. The discussion of Planck length and quantum gravity shows deep understanding of modern physics limitations, correctly identifying the unresolved tension between continuous spacetime (GR) and discrete quantization." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gpt_oss_120b", "respondent_name": "GPT-OSS-120B", "weighted_score": 9.8, "brief_justification": "Comprehensive and accurate response covering all three parts. Mathematical derivations are correct, physical reasoning is sound, and the Planck-scale discussion shows good depth while acknowledging open questions in physics. Well-structured with clear explanations and a helpful summary table." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "gemini_25_flash", "respondent_name": "Gemini 2.5 Flash", "weighted_score": 8.65, "brief_justification": "Accurate calculus solution with correct geometric series sum. Clear physical reasoning using relative speed. Excellent depth in part 3 discussing Planck scale and quantum gravity as unresolved frontier. Well-structured with clear explanations." } ], "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)" } }