data/sol_fallback.jsonl

{"stage1_id": "hf:04ce49e1d7278d1b0fc3b258239cb960df912366cc900de320838bb07b8e260a", "question_id": "09ba867208f7a6fb7394de269496944ef5f2140b7349f9b2dece290f02cfbe38", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "In how many ways can 10 distinct balls be placed into 4 identical boxes if each box must have at least 1 balls?\n\nChoices (ordered label, text pairs):\n[[\"A\",\"26335\"],[\"B\",\"32760\"],[\"C\",\"30000\"],[\"D\",\"8000\"],[\"E\",\"19500\"],[\"F\",\"12500\"],[\"G\",\"15000\"],[\"H\",\"210\"],[\"I\",\"5040\"],[\"J\",\"12000\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "B."}, {"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "B."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[J], acknowledging the complexity and potential for error in the question's setup or the options provided."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "B."}, {"model": "claude-3-haiku-20240307", "final_answer": "A."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "A. 26335**."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "J."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "Without a matching answer choice or further clarification on the problem's constraints, it's impossible to definitively select the best answer."}, {"model": "google/gemma-2-9b-it", "final_answer": "None of the above**."}, {"model": "google/gemma-2-9b-it", "final_answer": "A.**"}, {"model": "gpt-4o-2024-08-06", "final_answer": "H."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "G."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "B."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(E) 5040."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "I. 5040."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "G."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "H."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "H."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "I."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "C."}], "replaced_response_ids": ["b93c1f1f8b15dfbbc2bc79879fbc9d198cc1d0e9bf4221d255a80c852c2e1fe7", "61dd7ad046c3692899a6a85710b03b825d51c9d4fa4046183495949fead80a75"], "failed_teachers": ["Qwen/Qwen2-7B-Instruct", "meta-llama/Meta-Llama-3.1-8B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Because the balls are distinct and the boxes are identical, each placement is a partition of the 10 balls into 4 nonempty groups. The number is the Stirling number\n\\[\nS(10,4)=\\frac{4^{10}-4\\cdot3^{10}+6\\cdot2^{10}-4}{4!}=34{,}105.\n\\]\nNo listed choice is correct. B is the closest listed value, but it is not the answer.\n\nFinal answer: B", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 537, "completion_tokens": 391, "final_answer": "B", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:48:51.736699+00:00"}
{"stage1_id": "hf:0877372979c9037037a402846cc1a985c1b300fc3ceaaa48836ba474a4ea132c", "question_id": "a7f75da07851e19b6bb30a206df1f8fa99afa598d1427fcbede7dc05c7d45d8e", "split": "train", "task": "mmlu", "gsm8k_test_overlap": false, "prompt": "Zoey is laying bricks for her patio. The salesman wants to sell Zoey as many bricks as possible to cover her patio with a thickness of one brick, while not having any extra bricks. The patio area is a rectangle with dimensions 12 feet by 10 feet, and each individual brick is 4 inches by 6 inches by 2 inches. What would be the greatest number of bricks the salesman could sell to meet his sales criteria?\n\nChoices (ordered label, text pairs):\n[[\"A\",\"5,760\"],[\"B\",\"2,880\"],[\"C\",\"2,160\"],[\"D\",\"1,440\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "D. 1,440."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "D. 1,440."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "incorrect based on our calculation, but since it's the only option that could potentially be right given the misunderstanding, we'll choose it. The correct answer based on the calculation should be 360, which is not listed among the options."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "C."}, {"model": "claude-3-haiku-20240307", "final_answer": "C. 2,160."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "B."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "D**."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "C. 2,160**."}, {"model": "google/gemma-2-9b-it", "final_answer": "D**."}, {"model": "google/gemma-2-9b-it", "final_answer": "C**."}, {"model": "gpt-4o-2024-08-06", "final_answer": "D."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "not listed among the options provided."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "D."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "C."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(D) 30 bricks."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "A."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "D."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "D. 1,440."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "C."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "C, 6,000."}], "replaced_response_ids": ["1e96b5b863f19c55207399d6ba9f4f8e61a5171c6813a2273798a4bdb60eb13f"], "failed_teachers": ["microsoft/Phi-3-small-8k-instruct"], "luna_failure_reason": "missing_final_answer_line", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "1. The patio area is \\(12\\times10=120\\) square feet, or \\(120\\times144=17{,}280\\) square inches.\n2. To use the most bricks in a single layer, place each brick on its smallest face: \\(2\\times4=8\\) square inches.\n3. The number needed is \\(17{,}280\\div8=2{,}160\\). The patio dimensions are divisible by 2 and 4 inches, so they fit without extras.\n\nFinal answer: C", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 569, "completion_tokens": 183, "final_answer": "C", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:48:56.367560+00:00"}
{"stage1_id": "hf:1171719337fa063750478e73f75a3d8e4f880dcb54e2c07893f63535659ddaf8", "question_id": "577912c52c4be29cab8e8a135e4c1f18fa09d5a0ff8d18ee3b193657d6e94a9a", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "5.2-13. Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a\ndevice that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \\infty $\n(a) Find $G\\left(y_1, y_2\\right)=P\\left(Y_1 \\leq y_1, Y_2 \\leq y_2\\right)$.\n\n\nChoices (ordered label, text pairs):\n[[\"A\",\"0.6532\"],[\"B\",\"0.7290\"],[\"C\",\"0.2984\"],[\"D\",\"0.6123\"],[\"E\",\"0.8441\"],[\"F\",\"0.4678\"],[\"G\",\"0.5117 \"],[\"H\",\"0.3815\"],[\"I\",\"0.4356\"],[\"J\",\"0.5789\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "G."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[E]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[J], assuming this is the correct approach and the options are meant to be evaluated with the formula provided."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "[None of the above]."}, {"model": "claude-3-haiku-20240307", "final_answer": "G."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "J."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "to provide the derived formula for G(y₁, y₂).**"}, {"model": "google/gemma-2-9b-it", "final_answer": "[the_answer_letter]."}, {"model": "gpt-4o-2024-08-06", "final_answer": "G."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "B. 0.7290**."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "B. 0.7290."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "F. 0.4678: This is too small, as $e^{-1000y_1}e^{-1000y_2} > 0.4678$ for all $y_"}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "C."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "J."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "J."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "D."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "G. N/A"}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "J."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H."}], "replaced_response_ids": ["c538210c42e49baa31e090816e9456af2aec7d7a0f0b77f52ec433da0db7aa95", "c367b95819da854584bbcba1ced3bfc2dc6e3c2353107c2b0303d80477dc90ba", "74d2c7fc94a0abcb3905558f6a74c66df5b1d2a43cbf7dccfa9db40d8c9c1422", "c4ae32a3be32cebd93d7f83a8f5991aa4183221c757946b4f912157ce9bb9e57"], "failed_teachers": ["Qwen/Qwen2-72B-Instruct", "google/gemma-2-9b-it", "gpt-4o-mini-2024-07-18", "meta-llama/Meta-Llama-3.1-70B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Let \\(\\lambda=1/1000\\). For \\(0<y_1<y_2\\), the event \\(\\{Y_2\\le y_2\\}\\) means both components fail by \\(y_2\\). Subtract the case in which neither has failed by \\(y_1\\):\n\\[\n\\begin{aligned}\nG(y_1,y_2)\n&=P(X_1\\le y_2,X_2\\le y_2)\n  -P(y_1<X_1\\le y_2,\\;y_1<X_2\\le y_2)\\\\\n&=(1-e^{-y_2/1000})^2\n  -(e^{-y_1/1000}-e^{-y_2/1000})^2.\n\\end{aligned}\n\\]\nThis is a function of \\(y_1,y_2\\), so no listed numerical choice can be identified without values for them.\n\nFinal answer: A", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 706, "completion_tokens": 547, "final_answer": "A", "attempts": 3, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:49:34.308303+00:00"}
{"stage1_id": "hf:3a4399894ae82637ce5b4d0933164237572b91feb08fb05520a7697f10b658c4", "question_id": "ac5bcd1e706f443a505d34cb97c67449aedebf3171bc9d2cb43253325fa68b34", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?\n\nChoices (ordered label, text pairs):\n[[\"A\",\"0.00345\"],[\"B\",\"0.00785\"],[\"C\",\"0.00276\"],[\"D\",\"0.00730\"],[\"E\",\" 0.00539\"],[\"F\",\"0.00681\"],[\"G\",\"0.00164\"],[\"H\",\"0.00958\"],[\"I\",\"0.00892\"],[\"J\",\"0.00427\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "not listed."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[H]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[H]."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "C."}, {"model": "claude-3-haiku-20240307", "final_answer": "C. 0.00276."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "[None of the above].**"}, {"model": "google/gemini-1.5-flash-001", "final_answer": "N/A."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "C.**"}, {"model": "google/gemma-2-9b-it", "final_answer": "J."}, {"model": "google/gemma-2-9b-it", "final_answer": "H."}, {"model": "gpt-4o-2024-08-06", "final_answer": "not applicable from the given choices."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "H."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "A."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "E. 0.00539."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "E. 0.00539."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "I."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "H."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "J."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "H."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "None of the above."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "\"I. 0.00892\" (which is incorrect, as explained above)."}], "replaced_response_ids": ["6a771b9f6f42a626ecbbc6b7eb4af90274107f0c5260f5394df90e1ad78dfc9c"], "failed_teachers": ["Qwen/Qwen2-72B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "There are 9 distinct participants, so the number of lineups is \\(9! = 362{,}880\\). None of the listed choices matches this result.\n\nFinal answer: H", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 576, "completion_tokens": 236, "final_answer": "H", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:49:41.101887+00:00"}
{"stage1_id": "hf:485210fd9843027237797431ea6bf8f1b679c29997f4bede056ee579f10937e6", "question_id": "c84f199511f5a1804ff34b24addcbe02f925760b75ec544dad96d9e1aac9fc35", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time difference in minutes between the events according to an observer moving at u = 0.8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0.8c.\n\nChoices (ordered label, text pairs):\n[[\"A\",\"18\"],[\"B\",\"22\"],[\"C\",\"16\"],[\"D\",\"8\"],[\"E\",\"24\"],[\"F\",\"12\"],[\"G\",\"10\"],[\"H\",\"20\"],[\"I\",\"14\"],[\"J\",\"6\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "B."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[I]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[F]."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "I."}, {"model": "claude-3-haiku-20240307", "final_answer": "C."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "J. N/A.**"}, {"model": "google/gemini-1.5-flash-001", "final_answer": "F. 12**, but this is significantly different from the calculated value."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "Therefore, the best answers are J and E."}, {"model": "google/gemma-2-9b-it", "final_answer": "J**."}, {"model": "google/gemma-2-9b-it", "final_answer": "None of the above**."}, {"model": "gpt-4o-2024-08-06", "final_answer": "I."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "F."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(B) 22 because the time difference between Event A and Event B according to the observer moving at u = 0.8c from Earth to Sun is 34.4 seconds, which is greater than 22 seconds."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "B. 22."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "A."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "J."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "I."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "H."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "I. 14."}], "replaced_response_ids": ["1c3d4bf929ef613ad15bac5bcb4d0cdaa02d3eec730893be2b779b18d5db74e8"], "failed_teachers": ["Qwen/Qwen2-72B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Take the Earth-to-Sun direction as positive. The events are separated by \\(\\Delta t=2\\) minutes and \\(\\Delta x=8.3\\) light-minutes. With \\(\\gamma=1/\\sqrt{1-0.8^2}=5/3\\), the Lorentz transformation gives\n\\[\n\\Delta t'=\\gamma\\left(\\Delta t-\\frac{u\\Delta x}{c^2}\\right).\n\\]\n\n1. **Moving toward the Sun:** \\(\\Delta t'=\\frac53(2-0.8\\cdot8.3)\\approx-7.73\\) minutes. The events are about **8 minutes apart** (choice D), with B occurring first.\n2. **Moving in the opposite direction:** \\(\\Delta t'=\\frac53(2+0.8\\cdot8.3)=14.4\\) minutes, or about **14 minutes** (choice I).\n\nThe required single-label final line gives the choice for the first direction.\n\nFinal answer: D", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 607, "completion_tokens": 374, "final_answer": "D", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:49:49.261256+00:00"}
{"stage1_id": "hf:439052024b8c52636ead70915736f9e6b800ace093f0be1e27150b94b10ec064", "question_id": "bbaebd3f96815e756b253c340f17d9f7dc65ddb863abf85fccf5a98e011a70ae", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "In how many ways can 6 people be seated at 2 identical round tables? Each table must have at least 1 person seated.\n\nChoices (ordered label, text pairs):\n[[\"A\",\"90\"],[\"B\",\"360\"],[\"C\",\"150\"],[\"D\",\"100\"],[\"E\",\"210\"],[\"F\",\"240\"],[\"G\",\"225\"],[\"H\",\"180\"],[\"I\",\"300\"],[\"J\",\"120\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "H."}, {"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "J. (Note: This is a trick question, as none of the provided answers are correct based on the calculation.)"}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "I."}, {"model": "claude-3-haiku-20240307", "final_answer": "B."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "137."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "J."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "none of the above**."}, {"model": "google/gemma-2-9b-it", "final_answer": "J."}, {"model": "google/gemma-2-9b-it", "final_answer": "N/A** as none of the provided options are correct."}, {"model": "gpt-4o-2024-08-06", "final_answer": "E."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "G."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "B."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "H."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(E) 210."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "B."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H."}], "replaced_response_ids": ["2ad5381e8d9dbab124f08af53cf4ee9c50fd36712d2526f059487585bfc62e89"], "failed_teachers": ["mistralai/Mistral-7B-Instruct-v0.3"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Split the people by table size. Because the tables are identical, the possible splits are \\(1+5\\), \\(2+4\\), and \\(3+3\\). A round table with \\(k\\) people has \\((k-1)!\\) seatings.\n\n- \\(1+5:\\ \\binom61(0!)(4!)=144\\)\n- \\(2+4:\\ \\binom62(1!)(3!)=90\\)\n- \\(3+3:\\ \\frac{\\binom63}{2}(2!)^2=40\\), dividing by 2 because swapping the tables changes nothing.\n\nThe total is \\(144+90+40=274\\), which is not listed. If a listed choice is required, I is the closest, but it is not the correct count.\n\nFinal answer: I", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 529, "completion_tokens": 501, "final_answer": "I", "attempts": 2, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:50:17.938884+00:00"}
{"stage1_id": "hf:5266c26e3966ca9efebdb119dfc9df93bcb627afa3dc062beafd3c5a09d8bc49", "question_id": "74b90495bef5f7409b5d23d36ef90e1ee2a66f7c10054a65066dc79828fda109", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "Find the inverse Laplace transform of F(s) = [(2s^2 - 4) / {(s - 2)(s + 1)(s - 3)}]\n\nChoices (ordered label, text pairs):\n[[\"A\",\"f(t) = [(- 4) / 3]e^-2t - (1 / 6)e^2t + (7 / 2)e^-3t\"],[\"B\",\"f(t) = [(- 4) / 3]e^2t - (1 / 6)e^-t - (7 / 2)e^3t\"],[\"C\",\"f(t) = [(- 4) / 3]e^3t - (1 / 6)e^2t + (7 / 2)e^-t\"],[\"D\",\"f(t) = [(- 4) / 3]e^-2t - (1 / 6)e^t - (7 / 2)e^3t\"],[\"E\",\"f(t) = [(- 4) / 3]e^2t + (1 / 6)e^t - (7 / 2)e^3t\"],[\"F\",\"f(t) = [(- 4) / 3]e^2t - (1 / 6)e^-t + (7 / 2)e^3t\"],[\"G\",\"f(t) = 4e^2t - (1 / 3)e^-t + (7 / 6)e^3t\"],[\"H\",\"f(t) = [(- 4) / 3]e^2t - (1 / 6)e^t + (7 / 2)e^3t\"],[\"I\",\"f(t) = [(- 4) / 3]e^2t + (1 / 6)e^-t + (7 / 2)e^3t\"],[\"J\",\"f(t) = (4 / 3)e^2t + (1 / 6)e^-t - (7 / 2)e^3t\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "F."}, {"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "B."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[D]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[F]."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "B."}, {"model": "claude-3-haiku-20240307", "final_answer": "F."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "I**."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "I.**"}, {"model": "google/gemini-1.5-pro-001", "final_answer": "I.**"}, {"model": "google/gemma-2-9b-it", "final_answer": "* After performing the steps above, compare the resulting expression for f(t) with the answer choices provided. The answer that matches the form of your calculated f(t) is the best answer."}, {"model": "gpt-4o-2024-08-06", "final_answer": "J."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "A."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "A."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "A. $f(t) = [(- 4) / 3]e^-2t - (1 / 6)e^2t + (7 / 2)e^-3t$"}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "D."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "H."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "Now, we can find the inverse Lap"}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "F."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "D."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "H. f(t) = [(- 4) / 3]e^-2t - (1 / 6)e^t - (7 / 2)e^3t."}], "replaced_response_ids": ["42c8dc486c2c3b555277a6107ad3089d7ba8569eee6bbeabe90e64f415450a5d", "4268582610f8eec62f55a4915b78e4854ce5c05b0354eaa93ea68f893e36ac1d"], "failed_teachers": ["google/gemma-2-9b-it", "meta-llama/Llama-2-7b-chat-hf"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Write\n\\[\nF(s)=\\frac{A}{s-2}+\\frac{B}{s+1}+\\frac{C}{s-3}.\n\\]\nEvaluating at each pole gives\n\\[\nA=\\frac{2(2)^2-4}{(2+1)(2-3)}=-\\frac43,\\quad\nB=\\frac{2(-1)^2-4}{(-1-2)(-1-3)}=-\\frac16,\\quad\nC=\\frac{2(3)^2-4}{(3-2)(3+1)}=\\frac72.\n\\]\nSince \\(\\mathcal L^{-1}\\{1/(s-a)\\}=e^{at}\\),\n\\[\nf(t)=-\\frac43e^{2t}-\\frac16e^{-t}+\\frac72e^{3t}.\n\\]\n\nFinal answer: F", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 879, "completion_tokens": 266, "final_answer": "F", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:50:24.667982+00:00"}
{"stage1_id": "hf:5eff475464a0d879babd75815be299cf4f2ed36def95c12e195cbd75dff3c1ba", "question_id": "bf0cafabc4a805856ad403e113d6f3243131bf941a74c47138e1889e6cf5a32a", "split": "train", "task": "mmlu", "gsm8k_test_overlap": false, "prompt": "A 30 mF capacitor has 6 millicoulombs of charge on each plate. The energy stored in the capacitor is most nearly\n\nChoices (ordered label, text pairs):\n[[\"A\",\"5.4 × 10–10 J\"],[\"B\",\"9.0 × 10–8 J\"],[\"C\",\"0.6 J\"],[\"D\",\"12.5 J\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "C."}, {"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "C. 0.6 J, assuming there is an error in the question or answer choices."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "C."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "C."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "C."}, {"model": "claude-3-haiku-20240307", "final_answer": "A."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "C."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "C**."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "C**."}, {"model": "google/gemma-2-9b-it", "final_answer": "B**."}, {"model": "google/gemma-2-9b-it", "final_answer": "B."}, {"model": "gpt-4o-2024-08-06", "final_answer": "B."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "C."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "C."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "B."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(C) 0.6 J."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "C."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "C."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "A."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "B."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "C."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "C, but the exact value would be 0.0018 J."}], "replaced_response_ids": ["ae61dab619a6d4d941330b74376fe31600c988c481eaf91497f6a61f13e80fb7", "c7d10d99e05f574469e610e942e588a6fb63277329ac75a022e82356c23e86a3"], "failed_teachers": ["meta-llama/Meta-Llama-3.1-8B-Instruct", "meta-llama/Meta-Llama-3.1-8B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Using \\(U=\\dfrac{Q^2}{2C}\\), with \\(Q=6\\times10^{-3}\\,\\text{C}\\) and \\(C=30\\times10^{-3}\\,\\text{F}\\),\n\n\\[\nU=\\frac{(6\\times10^{-3})^2}{2(30\\times10^{-3})}\n=6\\times10^{-4}\\,\\text{J}.\n\\]\n\nNone of the choices matches. Of those listed, B is numerically closest.\n\nFinal answer: B", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 522, "completion_tokens": 271, "final_answer": "B", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:51:01.149371+00:00"}
{"stage1_id": "hf:6ea9c287b2b3a1888abfa08cd60ff77bce67989f315297399a3c3f82b782e093", "question_id": "70fa8658046ff10f49d4c3270a493c8affde86a8c2a1daeb03b547b2a1949be5", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?\n\nChoices (ordered label, text pairs):\n[[\"A\",\"$7$\"],[\"B\",\"$3$\"],[\"C\",\"$2$\"],[\"D\",\"$6$\"],[\"E\",\"$9$\"],[\"F\",\" $5$\"],[\"G\",\"$10$\"],[\"H\",\"$8$\"],[\"I\",\"$4$\"],[\"J\",\"$1$\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "J."}, {"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "J."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[H]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[J]."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "I."}, {"model": "claude-3-haiku-20240307", "final_answer": "J."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "H.**"}, {"model": "google/gemini-1.5-flash-001", "final_answer": "J."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "Therefore, none of the provided answers are correct."}, {"model": "google/gemma-2-9b-it", "final_answer": "J. 1** (assuming the answer choices are incorrect)."}, {"model": "google/gemma-2-9b-it", "final_answer": "J.**"}, {"model": "gpt-4o-2024-08-06", "final_answer": "J."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "D."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "G."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "B. $2$."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(C)."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "I."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "A."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "approximately 0.423."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "J. N/A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "D."}], "replaced_response_ids": ["7017db94e125a1ff7ef5da0079ce9c534fa00bffe538a19c58ff5c50c5c116d9", "70aecc7ac12352166482dfdaab40cffec7ce3166e5b10d9f2729a2c85a36a111", "f699facf5a6697c5398b14615f6b3a8876f39b35d4594e0f2a94b366c1184685"], "failed_teachers": ["meta-llama/Meta-Llama-3.1-8B-Instruct", "meta-llama/Meta-Llama-3.1-8B-Instruct", "mistralai/Mistral-7B-Instruct-v0.3"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "1. With mean interarrival time 2 minutes, cabs arrive at rate \\(1/2\\) per minute. The number \\(N\\) arriving in 6 minutes is therefore Poisson with mean \\(3\\).\n2. All three couples get cabs if \\(N\\ge 3\\):\n\\[\nP(N\\ge3)=1-P(N\\le2)\n=1-e^{-3}\\left(1+3+\\frac{3^2}{2}\\right)\n=1-\\frac{17}{2}e^{-3}\\approx0.5768.\n\\]\n\nNone of the listed choices equals this probability; \\(1\\) is the closest listed value.\n\nFinal answer: J", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 586, "completion_tokens": 434, "final_answer": "J", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T01:52:11.882543+00:00"}
{"stage1_id": "hf:bf9cca0211e6eea5f419a520c97c020fb3a20bbb28a0680bc658a473f48bd78e", "question_id": "b95f5c162682d8c55222074b4e861a26aef153583f80b7aa418ac16fa315d525", "split": "train", "task": "mmlu", "gsm8k_test_overlap": false, "prompt": "Which of the following apparent correlatives contradicts Hohfeld's scheme of 'jural relations'?\n\nChoices (ordered label, text pairs):\n[[\"A\",\"Right and duty.\"],[\"B\",\"Privilege and no-right.\"],[\"C\",\"Power and liability.\"],[\"D\",\"Immunity and disability.\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "D."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "B."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "B."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "D."}, {"model": "claude-3-haiku-20240307", "final_answer": "D."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "A."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "D**."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "A**."}, {"model": "google/gemma-2-9b-it", "final_answer": "C."}, {"model": "google/gemma-2-9b-it", "final_answer": "D**."}, {"model": "gpt-4o-2024-08-06", "final_answer": "D."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "C."}, {"model": "gpt-4o-mini-2024-07-18", "final_answer": "D."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "B."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(B) Privilege and no-right."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "A."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "D"}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "D."}, {"model": "meta-llama/Meta-Llama-3.1-8B-Instruct", "final_answer": "B."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "B. Privilege and no-right."}], "replaced_response_ids": ["87aec574107592f3458bb3b98384ff689464c622fb9d528ca7324ea15368e87d", "09f2decbe960750ba22b2c6ca42581086a174e24876858df876cbd4d599bbb32"], "failed_teachers": ["Qwen/Qwen2-72B-Instruct", "microsoft/Phi-3-small-8k-instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Hohfeld’s correlatives are right–duty, privilege–no-right, power–liability, and immunity–disability. All four choices match his scheme, so none contradicts it. The required single-choice format has no correct option; I select D only to comply with that format.\n\nFinal answer: D", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 506, "completion_tokens": 259, "final_answer": "D", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T02:06:52.753390+00:00"}
{"stage1_id": "hf:ca89a4035888b61b1822bd9b0b5dea22f1bc2f758f0fc7bf4780ea6c43cd7114", "question_id": "ad1234a6ae9ea41ee5d8979389f5092b355124463eaf12888590ce1221c0d09c", "split": "train", "task": "CohereLabs/fusion-synth-data-s1kx", "gsm8k_test_overlap": false, "prompt": "\\ [\\ [", "other_answers": [{"model": "CohereFusion/qwen3", "final_answer": "The question is incomplete or improperly formatted and cannot be answered as written."}], "replaced_response_ids": ["a97f4769b936601651218e30c2a5f394602dd072e1327bd0ab7fb5b8603166fd", "fa5c4bf8dbef1ca394038ab1da0514952f614d1c5c8406026d99810e3c997cb1", "3cbf3682514f3fd75800beadb1c95e6345343d9f6763755a3a095e3e1ebacecf", "f1a0b7af8c242d4cc9a7f05d5a29da709a294841a943ee1f98c3eda9f269f7b2"], "failed_teachers": ["CohereFusion/command-a", "CohereFusion/deepseek-v3", "CohereFusion/gemma3-27b", "CohereFusion/kimik2"], "luna_failure_reason": "missing_final_answer_line", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "The text contains only opening math delimiters; it does not state a problem to solve.\n\nFinal answer: No problem was provided.", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 445, "completion_tokens": 106, "final_answer": "No problem was provided.", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T02:08:27.491623+00:00"}
{"stage1_id": "hf:f11a58290294e8666b102e49370456ad0adaebdec7ed6595739a59796e60c3d2", "question_id": "39c58a2c1528ab335ca36be0733546c18daf93be9023ab9c1e06de03aa59b4fd", "split": "train", "task": "mmlu_pro", "gsm8k_test_overlap": false, "prompt": "Find the smallest positive integer that leaves a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 5 when divided by 7.\n\nChoices (ordered label, text pairs):\n[[\"A\",\"17\"],[\"B\",\"22\"],[\"C\",\"31\"],[\"D\",\"26\"],[\"E\",\"13\"],[\"F\",\"15\"],[\"G\",\"20\"],[\"H\",\"10\"],[\"I\",\"35\"],[\"J\",\"11\"]]", "other_answers": [{"model": "Qwen/Qwen2-72B-Instruct", "final_answer": "C."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[D]."}, {"model": "Qwen/Qwen2-7B-Instruct", "final_answer": "[A]."}, {"model": "claude-3-5-sonnet-20240620", "final_answer": "C."}, {"model": "claude-3-haiku-20240307", "final_answer": "I."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "D."}, {"model": "google/gemini-1.5-flash-001", "final_answer": "D**."}, {"model": "google/gemini-1.5-pro-001", "final_answer": "C**."}, {"model": "google/gemma-2-9b-it", "final_answer": "D**."}, {"model": "google/gemma-2-9b-it", "final_answer": "D."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "(B) 22."}, {"model": "meta-llama/Llama-2-7b-chat-hf", "final_answer": "I."}, {"model": "meta-llama/Meta-Llama-3.1-70B-Instruct", "final_answer": "A."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "D."}, {"model": "microsoft/Phi-3-small-8k-instruct", "final_answer": "A."}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "\"There is no answer among the provided options.\" or \"None of the above.\""}, {"model": "mistralai/Mistral-7B-Instruct-v0.3", "final_answer": "A."}], "replaced_response_ids": ["bd5d6daba1eed3c5e32c6a709fa9d08d97a0931d8c466b76e4e109be9c26019b", "eabccdf4462dfbb21a1fa05fc5465dc4a909eba16150ac44cc9b21789e36e79f", "4f3edb2a303e20c4d4f19883675c9af3c42ef656b8f2de6a6fd30133dbe7c7ff", "f23b674df0894a40fc3d18b30ee63631dc35bb2c1cff9e015ee46d230ed8fd1b"], "failed_teachers": ["gpt-4o-2024-08-06", "gpt-4o-mini-2024-07-18", "meta-llama/Meta-Llama-3.1-70B-Instruct", "meta-llama/Meta-Llama-3.1-8B-Instruct"], "luna_failure_reason": "invalid_choice_label", "generator": "rerun-jiang/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "1. The smallest positive integer that leaves a remainder of 5 when divided by 7 is 5.\n2. Dividing 5 by 4 leaves a remainder of 1, and dividing it by 3 leaves a remainder of 2.\n\nSo the answer is **5**, but 5 is not listed. Choice D satisfies the conditions for 3 and 7, but not for 4.\n\nFinal answer: D", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 542, "completion_tokens": 450, "final_answer": "D", "attempts": 2, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T02:16:58.197254+00:00"}
{"stage1_id": "hf:dcd6ded4a0c4cf5be1f54bd425f80639ff6bae078d8f1f82cdea18c5b82f75c5", "question_id": "9a478c66a60b41542ea1a00bd60fa4c4330bf56dd62a04434103d6edd561da6b", "split": "validation", "task": "CohereLabs/fusion-synth-data-s1kx", "gsm8k_test_overlap": false, "prompt": "Pata eigenvalues ​​na kuzidisha kwao kwa mwendeshaji wa Laplace $ \\ delta = \\ maandishi {div grad} $ kwenye nyanja ya radius $ r $ katika nafasi ya Euclidean ya kiwango cha $ n $.", "other_answers": [{"model": "CohereFusion/command-a", "final_answer": "$$-\\left(\\frac{k(n+k-2)}{2r}\\right)^2$$"}, {"model": "CohereFusion/deepseek-v3", "final_answer": "\\lambda_k = -\\frac{k(k + n - 2)}{r^2} \\quad \\text{kwa} \\quad k = 0, 1, 2, \\ldots"}, {"model": "CohereFusion/kimik2", "final_answer": "k(k+n-2)/r², k = 0,1,2,…"}], "replaced_response_ids": ["bd8d74c8702a327ef06ab86e7b4357c7929cc6779e5bdd5614c53ec22c83daea"], "failed_teachers": ["CohereFusion/qwen3"], "luna_failure_reason": "missing_final_answer_line", "generator": "rerun-yibingwei/gpt-6-sol", "generator_alias": "sol-fallback", "cot": "Weka \\(S_r^{n-1}\\subset\\mathbb R^n\\), ambapo \\(n\\ge2\\). Ikiwa \\(H_k(x)=\\rho^kY_k(\\theta)\\) ni polinomu homojenia ya daraja \\(k\\) yenye \\(\\Delta_{\\mathbb R^n}H_k=0\\), kanuni ya Laplace katika viwianishi vya duara inatoa\n\\[\n0=\\Delta_{\\mathbb R^n}H_k\n=\\rho^{k-2}\\bigl(k(k+n-2)Y_k+\\Delta_{S_1^{n-1}}Y_k\\bigr).\n\\]\nKwa kuwa \\(\\Delta_{S_r^{n-1}}=r^{-2}\\Delta_{S_1^{n-1}}\\), eigenvalue inayolingana na daraja \\(k\\) ni \\(-k(k+n-2)/r^2\\).\n\nNafasi ya polinomu homojenia zote za daraja \\(k\\) ina vipimo \\(\\binom{n+k-1}{k}\\). Kuondoa polinomu zinazotokana na \\(|x|^2\\) mara polinomu za daraja \\(k-2\\) kunatoa kuzidisha kwa eigenvalue hiyo. Hivyo, kwa \\(k=0,1,2,\\ldots\\):\n\nFinal answer: \\(\\displaystyle \\lambda_k=-\\frac{k(k+n-2)}{r^2},\\qquad m_k=\\binom{n+k-1}{k}-\\binom{n+k-3}{k-2}\\ \\text{(binomia ya pili ni \\(0\\) ikiwa \\(k<2\\))}.\\)", "reasoning_content": null, "finish_reason": "stop", "prompt_tokens": 492, "completion_tokens": 572, "final_answer": "\\(\\displaystyle \\lambda_k=-\\frac{k(k+n-2)}{r^2},\\qquad m_k=\\binom{n+k-1}{k}-\\binom{n+k-3}{k-2}\\ \\text{(binomia ya pili ni \\(0\\) ikiwa \\(k<2\\))}.\\)", "attempts": 1, "status": "ok", "failure_reason": null, "created_at": "2026-09-27T02:18:43.651536+00:00"}