answers.json

[
  {
    "task_id": "set_missing_element-9223",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "136hk81o-q6h8-3g2e-j1r9-56u103s3y13m",
        "name": "Exact answer",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "",
        "weight": 100,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "xd"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "0sL35u",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "090ca17d-b4b6-4f4d-a1f8-33d102f8f98b",
        "name": "Exact answer",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "",
        "weight": 100,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "350"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "15yKc6",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "7bc66417-5fe2-48c8-990f-57d5e97ceacd",
        "name": "High-capacitance component role",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that a 1000 pF capacitor at a 1.5 T MRI coil frequency is too large to be part of RF tuning, matching, LC traps, or filters because its RF impedance is very low."
      },
      {
        "id": "3a7cde17-394f-548c-b76c-1c0256b20f5b",
        "name": "DC-blocking function",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission identifies the capacitor's primary role as blocking DC while passing RF in the signal path, or an equivalent coupling/DC-blocking function."
      },
      {
        "id": "4b2cb3e2-0b6d-5875-8aea-08020d96cd41",
        "name": "Replacement conclusion",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission concludes that replacing the capacitor with a comparable or larger capacitance such as 2000 pF is technically acceptable and should not materially affect coil performance."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "1ncrzI",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "514136b5-1bd9-491d-87cf-637e26e8fb57",
        "name": "Three-channel mode-mixing concept",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that the three physical coil channels are combined with fixed phase shifts into three mixed modes or mixed signals, i.e. a mode-matrix-style combination."
      },
      {
        "id": "5d4598ea-67b5-50f2-8df7-e60d27630609",
        "name": "Fundamental or sum mode purpose",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that one of the mixed outputs is the fundamental or sum mode that combines signal from all three antenna elements."
      },
      {
        "id": "b6fccba4-f48c-529a-8f22-f6108f258b8f",
        "name": "Receive-channel reduction benefit",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that this combination reduces the number of hardware receive channels needed, or frees channels for other coils while covering the desired field of view."
      },
      {
        "id": "c1617b25-d5f7-5aaa-8a10-7861a015c563",
        "name": "Recoverability of individual channels",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that the original individual channel signals can be reconstructed from the three mixed outputs, or states an equivalent invertibility property."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "7iy4wr",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "27c65202-1802-408c-87cc-479754edd669",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "293.76 nanotesla"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "93I7Dw",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "53843861-3385-4e89-8766-60b537bb4bd7",
        "name": "High-inductance component role",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that a 5.6 uH inductor at a 1.5 T MRI coil frequency is too large to be part of RF tuning, matching, LC traps, or filters because its RF impedance is very high."
      },
      {
        "id": "b91d74a8-bdd9-5152-9015-912e9fa3ce1e",
        "name": "RF-choke function",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission identifies the inductor's primary role as blocking RF current from entering a DC path, or an equivalent RF-choke function."
      },
      {
        "id": "34a1aba9-aa1f-5162-8e56-4e9086d4ce6b",
        "name": "Replacement conclusion",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission concludes that replacing the inductor with a comparable or larger inductance such as 8 uH is technically acceptable and should not materially affect coil performance."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "CNODK8",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "26c0b5bb-dbd9-4fea-83b2-b7d9e3fe32d7",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "llm-judge",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "$e−\\frac{2161​}{1440}$"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "GKcq0m",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "statement-01-exact",
        "name": "Statement 1 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "1. False"
      },
      {
        "id": "statement-02-exact",
        "name": "Statement 2 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "2. True"
      },
      {
        "id": "statement-03-exact",
        "name": "Statement 3 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "3. True"
      },
      {
        "id": "statement-04-exact",
        "name": "Statement 4 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "4. False"
      },
      {
        "id": "statement-05-exact",
        "name": "Statement 5 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "5. False"
      },
      {
        "id": "statement-06-exact",
        "name": "Statement 6 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "6. False"
      },
      {
        "id": "statement-07-exact",
        "name": "Statement 7 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "7. False"
      },
      {
        "id": "statement-08-exact",
        "name": "Statement 8 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "8. True"
      },
      {
        "id": "statement-09-exact",
        "name": "Statement 9 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "9. True"
      },
      {
        "id": "statement-10-exact",
        "name": "Statement 10 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "10. True"
      },
      {
        "id": "statement-11-exact",
        "name": "Statement 11 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "11. False"
      },
      {
        "id": "statement-12-exact",
        "name": "Statement 12 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "12. True"
      },
      {
        "id": "statement-13-exact",
        "name": "Statement 13 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "13. False"
      },
      {
        "id": "statement-14-exact",
        "name": "Statement 14 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "14. False"
      },
      {
        "id": "statement-15-exact",
        "name": "Statement 15 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "15. True"
      },
      {
        "id": "statement-16-exact",
        "name": "Statement 16 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "16. True"
      },
      {
        "id": "statement-17-exact",
        "name": "Statement 17 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "17. False"
      },
      {
        "id": "statement-18-exact",
        "name": "Statement 18 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "18. False"
      },
      {
        "id": "statement-19-exact",
        "name": "Statement 19 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "19. False"
      },
      {
        "id": "statement-20-exact",
        "name": "Statement 20 ExactMatch",
        "type": "semantic",
        "description": "",
        "weight": 3,
        "grader_type": "ExactMatch",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "20. False"
      },
      {
        "id": "statement-01-llm-judge",
        "name": "Statement 1 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 1: If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for at least one pair $x_0,y_0$, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for all $x_0,y_0$.\nExpected classification: False.\nReference rationale: For example, consider two SRWs in $\\mathbb{Z}$, starting at sites of same or different parity.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-02-llm-judge",
        "name": "Statement 2 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 2: If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$ and $X$ is aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\nExpected classification: True.\nReference rationale: Standard argument using the irreducibility of the process $(X,Y)$.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-03-llm-judge",
        "name": "Statement 3 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 3: If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$ and both $X$ and $Y$ are aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\nExpected classification: True.\nReference rationale: Same as in the previous item.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-04-llm-judge",
        "name": "Statement 4 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 4: If $X$ is recurrent and $Y$ is transient, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for all $x_0,y_0$.\nExpected classification: False.\nReference rationale: As a counterexample, one can take $X$ to be the two-dimensional SRW, and $Y$ to be the two-dimensional SRW conditioned on never hitting the origin.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-05-llm-judge",
        "name": "Statement 5 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 5: If $X$ is recurrent and $Y$ is transient, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ for at least one pair $x_0,y_0$.\nExpected classification: False.\nReference rationale: Same as in the previous item (one has to modify the counterexample taking e.g. a lazy SRW, so that it is aperiodic).\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-06-llm-judge",
        "name": "Statement 6 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 6: If both $X$ and $Y$ are recurrent, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for at least one pair $x_0,y_0$.\nExpected classification: False.\nReference rationale: Consider two SRWs on the comb lattice.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-07-llm-judge",
        "name": "Statement 7 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 7: For any Markov chains $X$, $Y$ and all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often).\nExpected classification: False.\nReference rationale: Think of one-dimensional transient chains that “choose the direction” (i.e., can go to $+\\infty$ or $-\\infty$ with positive probabilities).\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-08-llm-judge",
        "name": "Statement 8 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 8: If $\\mathcal{X}=\\mathbb{Z}^d$ and $X$ and $Y$ are spatially homogeneous random walks, then for all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often).\nExpected classification: True.\nReference rationale: Follows from the fact that $X-Y$ is a Markov chain in the homogeneous case.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-09-llm-judge",
        "name": "Statement 9 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 9: If $X$ and $Y$ are simple random walks on an infinite graph of uniformly bounded degree, then for all $x_0,y_0$ it holds that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=0$ or $1$ (i.e., we have a 0-1 law for the event $X$ and $Y$ meet infinitely often).\nExpected classification: True.\nReference rationale: See Remark 1.2 and Proposition 2.1 of \"Collisions of random walks\" by Barlow, Peres, and Sousi.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-10-llm-judge",
        "name": "Statement 10 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 10: For any $\\alpha\\in(0,1)$ it is possible to construct an example of two Markov chains $X$, $Y$ with state space $\\mathbb{Z}$ and only nearest-neighbor jumps, such that $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=\\alpha$ for all $x_0,y_0$.\nExpected classification: True.\nReference rationale: Again, think of one-dimensional transient chains that “choose the direction” (i.e., can go to $+\\infty$ or $-\\infty$ with positive probabilities). Then manipulate their near-the-origin transition probabilities in such a way that they choose the same direction with probability $\\alpha$.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-11-llm-judge",
        "name": "Statement 11 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 11: If both $X$ and $Y$ are positive recurrent, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\nExpected classification: False.\nReference rationale: Think about the case when both chains are periodic, with the same period; then for some initial conditions they will never meet.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-12-llm-judge",
        "name": "Statement 12 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 12: If both $X$ and $Y$ are positive recurrent and $Y$ is aperiodic, then $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ for all $x_0,y_0$.\nExpected classification: True.\nReference rationale: Here $(X,Y)$ will be positive recurrent and aperiodic (so there will be infinite collisions even in any fixed site).\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-13-llm-judge",
        "name": "Statement 13 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 13: If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=1$ and $\\mathbb{P}_{x_0,z_0}[M^{X,Z}]=1$, than we have $\\mathbb{P}_{y_0,z_0}[M^{Y,Z}]=1$.\nExpected classification: False.\nReference rationale: Think of a one-dimensional example where $Y$ goes to the right, $Z$ to the left, and $X$ oscillates.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-14-llm-judge",
        "name": "Statement 14 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 14: If $\\mathbb{P}_{x_0,y_0}[M^{X,Y}]=\\mathbb{P}_{x_0,z_0}[M^{X,Z}]=\\mathbb{P}_{y_0,z_0}[M^{Y,Z}]=1$ for all $x_0,y_0,z_0$, then $\\mathbb{P}_{x_1,y_1,z_1}[\\text{there exists }n>0 \\text{ such that }X_n=Y_n=Z_n]=1$ for at least one triple $(x_1,y_1,z_1)$.\nExpected classification: False.\nReference rationale: For example, think about lazy SRWs in two dimensions.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-15-llm-judge",
        "name": "Statement 15 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 15: Let $X,Y$ be one-dimensional null-recurrent Markov chains with only nearest-neighbour jumps, and let $U$ be a positive recurrent Markov chain (also in $\\mathcal{X}=\\mathbb{Z}$). It is possible to construct an example such that $\\mathbb{P}_{0,0,0}[U_n=X_n=Y_n\\text{ infinitely often}]=1$.\nExpected classification: True.\nReference rationale: Consider a product of Sinai’s random walks.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-16-llm-judge",
        "name": "Statement 16 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 16: Let $X^{(1)},\\ldots,X^{(k)}$ be one-dimensional null-recurrent Markov chains with only nearest-neighbour jumps, and let $U$ be a positive recurrent Markov chain (also in $\\mathcal{X}=\\mathbb{Z}$). For any $k$ it is possible to construct an example such that $\\mathbb{P}_{0,\\ldots,0}[U_n=X^{(1)}_n=\\ldots=X^{(k)}_n\\text{ infinitely often}]=1$.\nExpected classification: True.\nReference rationale: Again, consider a product of Sinai’s random walks.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-17-llm-judge",
        "name": "Statement 17 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 17: Let $X,Y$ be one-dimensional Markov chains with only nearest-neighbour jumps, such that $X$ is positive recurrent and $Y$ is transient. Then $\\mathbb{P}_{0,0}[M^{X,Y}]=0$.\nExpected classification: False.\nReference rationale: Think e.g. of a case when $Y$ goes to infinity very slowly.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-18-llm-judge",
        "name": "Statement 18 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 18: Assume that the state space $\\mathcal{X} = \\mathbb{Z}^3$. It is possible to construct an example of spatially homogeneous independent random walks $X$ and $Y$ with bounded jumps such that $\\mathbb{P}_{0,0}[M^{X,Y}]=1$.\nExpected classification: False.\nReference rationale: In this case $X-Y$ is a transient Markov chain.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-19-llm-judge",
        "name": "Statement 19 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 19: Assume that the state space $\\mathcal{X} = \\mathbb{Z}^3$. We say that a Markov chain is \\emph{uniformly elliptic} if $\\mathbb{P}_{xy}>\\varepsilon$ for all neighbours $x,y\\in \\mathbb{Z}^3$, where $\\varepsilon>0$ is some constant. Let $X$ be a uniformly elliptic zero-mean Markov chain with uniformly bounded jumps, and $Y$ be a positive-recurrent Markov chain. Then $\\mathbb{P}_{0,0}[M^{X,Y}]=0$.\nExpected classification: False.\nReference rationale: $X$ can still be recurrent; see \"On recurrence and transience of self-interacting random walks\" by Peres, Popov, and Sousi.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      },
      {
        "id": "statement-20-llm-judge",
        "name": "Statement 20 Comparable Rationale",
        "type": "semantic",
        "description": "",
        "weight": 2,
        "grader_type": "llm-judge",
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Statement 20: Assume that the state space $\\mathcal{X} = \\mathbb{Z}^2$ and $X$ is a simple random walk. Then, for any uniformly elliptic zero-mean Markov chain $Y$ with uniformly bounded jumps we have $\\mathbb{P}_{0,0}[M^{X,Y}]=1$.\nExpected classification: False.\nReference rationale: Think about a difference Markov chain $X-Y$ which is zero-mean but transient.\n\nEvaluate whether the model's response provides a rationale that is comparable in substance to the reference rationale for the same True/False classification. Accept paraphrases, equivalent proof ideas, equivalent counterexamples, or a different rigorous argument/counterexample that comparably supports the classification. Reject if the response gives no real rationale, gives reasoning that is incompatible with the classification, or gives reasoning that is materially weaker or unrelated."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "ISZkG7",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "99593f0b-e3da-4186-96bb-096a6f219844",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "llm-judge",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "$(-1)^{k + 1} (d - 2 k) ((d - 3) d + 4 k (k - d) + 2)$"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-001",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "f7df93d8-1cf4-48f2-a09a-54d72314c190",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "Thiocyanatobenzene"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-002",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "35d38f25-1085-4a25-9b74-11b9e7926db8",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "BaCl2"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-003",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "1f0a19e7-e07e-4207-ac0c-c7d6d1ae6cc2",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "C3h"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-007",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "b6adea97-4ea5-4351-81e2-4efc521762f8",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "llm-judge",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "Tribrominated product"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-010",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "cc9493fc-514e-4b5b-b572-b67aee6b98a4",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "B"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-013",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "f23b4b6e-c357-4435-bcc8-50032ba37361",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "B"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LLI-016",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "be8864f8-3f75-4742-8b62-292670f1c4fc",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "A"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "LbE4sy",
    "reference_file": "Compound_10.png",
    "tools": [],
    "criteria": [
      {
        "id": "12955066-5373-46de-8a72-af7b51893758",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "C84H56O36"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "Mxadld",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "78a6b471-8410-4793-9997-76259fba3711",
        "name": "Split-resonance shape",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that the S21 trace should show a split or double resonance rather than a single resonance."
      },
      {
        "id": "ff99e1b8-38fd-56d9-8ae1-3f58000276e8",
        "name": "Cat-with-ears profile",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 35.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission describes the two resonant peaks as lying on either side of the coil's operating frequency with a smaller flat or depressed region in the center, i.e. a 'cat with ears' shape."
      },
      {
        "id": "4660adaf-4ee8-5f9a-ac84-ee17d75672fd",
        "name": "Preamplifier filter cause",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that the resonance splitting is caused by the preamplifier's tuned input network or LC band-pass filter interacting with the coil resonance."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "N8iZko",
    "reference_file": "Compound_2.png",
    "tools": [],
    "criteria": [
      {
        "id": "1686b2ad-f935-4a02-9783-4f224e8b84fa",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "C23H38O4"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "WmZccO",
    "reference_file": "main.pdf",
    "tools": [],
    "criteria": [
      {
        "id": "9b2e083a-7442-4127-bc4c-7cc44434ab9c",
        "name": "Answer to T1 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 10,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Answer to T1 is 0."
      },
      {
        "id": "f7e0028e-6518-457c-82eb-12df7c9bb6d1",
        "name": "Answer to T2 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 18,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Answer to T2 is 1."
      },
      {
        "id": "73f107c5-f430-454b-9939-c04a665039f4",
        "name": "Answer to T3 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 18,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Answer to T3 is 3."
      },
      {
        "id": "02b65ceb-070a-4017-bd19-dcae477f4ff0",
        "name": "Answer to T4 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 18,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Answer to T4 is 8."
      },
      {
        "id": "5ea7641c-d829-4a54-92a5-bcb7e6b46d4c",
        "name": "Answer to T5 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 18,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "The model correctly answers subtask T5 as  1 - m + C(m,2) - C(m,3) + C(m,4) where C(x,y) denote the binomial coefficient of x and y. Every simple arithmetic expression equal to the provided answer above counts as correct. Specifically, answers that provide an indexed sum, i.e., by just inserting the definition of the reduced Euler characteristic, should not be counted as correct."
      },
      {
        "id": "caa0648e-a275-4c1f-9607-5ea1be63d5ec",
        "name": "Answer to T6 is correct.",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 18,
        "rationale": "",
        "examples": [],
        "semanticPrompt": "Answer to T6 is 1."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "XE4oUN",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "5295c172-399d-491a-b230-a7e8bc45a8f1",
        "name": "Coefficient c_1",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission gives c_1 = 1/8."
      },
      {
        "id": "9c8c5204-9f2f-5d10-8641-fb273becfe78",
        "name": "Coefficient c_2",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission gives c_2 = ±1/8, i.e. c_2 depends on the parity choice and is plus-or-minus one eighth."
      },
      {
        "id": "a81e3b49-c834-532d-948b-152b886c330c",
        "name": "Coefficient c_3",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission gives c_3 = 1/8."
      },
      {
        "id": "cb1c7f9e-148f-574d-9e5a-65b01682223f",
        "name": "Coefficient c_4",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission gives c_4 = ±1/8, i.e. c_4 depends on the parity choice and is plus-or-minus one eighth."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "eXEvuO",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
        "id": "212c3d00-5097-4398-8b8a-649ad5465b08",
        "name": "Answer correctness",
        "type": "binary",
        "grader_type": "ExactMatch",
        "description": "Answer matches ground truth.",
        "weight": 100.0,
        "rationale": "Single-criterion fallback",
        "examples": [],
        "semanticPrompt": "116.25 nanotesla"
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "electron-microscopy_0b02fbe1",
    "reference_file": "619e2c2c.png",
    "tools": [],
    "criteria": [
      {
        "id": "d28c0f50-678f-4e06-b51b-c96af99071ab",
        "name": "Final structure selection",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 40.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately identifies the main flagellar structure in the image as the basal body."
      },
      {
        "id": "a1287738-ec71-56eb-a880-b1b1c9ad1789",
        "name": "Ninefold symmetry evidence",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that the image shows nine-fold rotational symmetry, which is a key structural clue."
      },
      {
        "id": "1dc509c1-b51c-59b0-8720-83be7d2b1c54",
        "name": "Triplet microtubule evidence",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission explains that triplet microtubules are visible and that this feature is characteristic of a basal body or centriole rather than an axoneme or transition zone."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "electron-microscopy_27300012",
    "reference_file": "0bde932a.png",
    "tools": [],
    "criteria": [
      {
        "id": "aea7388c-6496-4b36-9556-3fe03f53c0f2",
        "name": "Final organelle identification",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 40.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately identifies the main feature as an actin filament."
      },
      {
        "id": "7d3dbcdf-7fa7-56f2-bfa3-86d30ea81799",
        "name": "Central filament observation",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission points out that the main visible feature is a filament running through the center of the image."
      },
      {
        "id": "1a036f91-f452-5582-8041-c588e9e17da3",
        "name": "Scale-based actin reasoning",
        "type": "semantic",
        "grader_type": "llm-judge",
        "description": "",
        "weight": 30.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission justifies the actin identification by comparing the filament's apparent width or scale with nearby structures such as membranes and ribosomes."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "electron-microscopy_3cb48fbe",
    "reference_file": "0a63453b.png",
    "tools": [],
    "criteria": [
      {
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        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission justifies the direction using axonemal chirality, specifically the dynein-arm and/or A-tubule-versus-B-tubule orientation rule used to distinguish tipward from baseward views."
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    "passThreshold": 100
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  {
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        "semanticPrompt": "The submission ultimately selects option B, meaning the micrograph is looking up out of the feeding groove."
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        "semanticPrompt": "The submission states that the observed A tubule being clockwise of the B tubule indicates a view towards the ciliary tip."
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        "semanticPrompt": "The submission connects the tipward ciliary view and groove geometry to the conclusion that the view is from inside the cell looking up out of the feeding groove."
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        "name": "Final dynein-set choice",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission ultimately selects option D, meaning the over-active dyneins are on doublets 6, 7, 8, and 9."
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        "id": "5cb5ce7a-4967-57f7-8812-c9175f4d2dc9",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission explains that the paraflagellar rod lies adjacent to doublets 4 to 6 and appears on the outside of the curve, so the flagellum is bending away from the PFR and towards doublet 1."
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        "semanticPrompt": "The submission narrows the dynein candidates to the two bend-compatible groups {1,2,3,4} and {6,7,8,9}, or an equivalent pair of alternatives, based on axoneme geometry."
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      {
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        "semanticPrompt": "The submission uses microtubule-sliding or minus-end-directed dynein reasoning to conclude that doublets 6, 7, 8, and 9 are the over-active set."
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    "reference_file": "Compound_6.png",
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        "examples": [],
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        "semanticPrompt": "$16 k^4-32 d k^3 + 8 (3 (d - 1) d + 4) k^2 - \n  8 d ((d - 3) d + 4) k + (d - 3) (d - 2) (d - 1) d$"
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        "semanticPrompt": "C219H376N60O72S"
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        "description": "Answer matches ground truth.",
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        "rationale": "",
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        "semanticPrompt": "446"
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    ],
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        "semanticPrompt": "$p=c$"
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        "semanticPrompt": "1.142 microtesla"
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        "semanticPrompt": "C18H17N3O2"
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        "semanticPrompt": "The submission identifies that the boundary onsite factors include a vanishing product such as (1 + sigma_N^z)(1 - sigma_N^z) = 0, or an equivalent annihilating factor."
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        "name": "Operator D is zero",
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        "semanticPrompt": "The submission concludes that the non-local operator D itself vanishes before taking its expectation value."
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    "passThreshold": 100
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        "semanticPrompt": "The submission ultimately concludes that the bond dimension of the matrix product operator O is 0."
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        "semanticPrompt": "The submission shows that the onsite factor involving (1 + H_N) and (sqrt(2) - X_N - Z_N) vanishes, for example by rewriting it as a multiple of (1 + H_N)(1 - H_N)."
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission concludes that the whole operator O is the zero operator before inferring the bond dimension."
      }
    ],
    "passThreshold": 100
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  {
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        "semanticPrompt": "The submission uses the sum rule for A(omega) to conclude that alpha_1 + alpha_2 = 0.5."
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        "name": "Substitution of alpha_1",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission substitutes alpha_1 = 0.15 into the sum-rule relation to solve for alpha_2."
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        "name": "Final value of alpha_2",
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        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately concludes that alpha_2 = 0.35."
      }
    ],
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  },
  {
    "task_id": "prob_120",
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        "name": "Reduced state rho_AB after tracing out C",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission computes the pre-channel AB state as rho_AB = 1/2 |00><00| + 1/4 |11><11| + 1/4 |22><22|, or an equivalent diagonal form in the |ii> basis."
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        "name": "Post-channel AB spectrum or Renyi entropy",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission correctly obtains the post-channel rho'_AB, or its nonzero eigenvalues, in a way that leads to S^{1/2}(rho'_AB) ≈ 1.690."
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      {
        "id": "cb7276ca-2658-5469-9dfd-88354c7c3782",
        "name": "State of subsystem B",
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        "description": "",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission correctly computes rho_B as diag(1/2, 1/4, 1/4), or an equivalent statement leading to S^{1/2}(rho_B) ≈ 1.064."
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission ultimately concludes that S^{1/2}(A|B) is approximately 0.624 (to three decimal places)."
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    "passThreshold": 100
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  {
    "task_id": "prob_138",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission uses the FWHM data to conclude that Gamma_a + gamma_a = 15 meV."
      },
      {
        "id": "753ccf27-a61d-50f2-b393-c1a3ec5271c6",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission uses the minimum reflectivity to obtain Gamma_a * gamma_a = 45 meV^2 and solves for Gamma_a ≈ 10.85 meV and gamma_a ≈ 4.15 meV, or the same pair in reverse order with the larger root chosen as the radiative rate."
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        "name": "Triple-resonance absorption formula",
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        "description": "",
        "weight": 25.0,
        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission writes or uses the triple-resonance expression eta_ISB = 4 gamma_P Gamma_a Omega_R^2 / [gamma_P (gamma_a + Gamma_a) + Omega_R^2]^2, or an equivalent simplified resonance formula."
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        "name": "Final absorption efficiency",
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        "description": "",
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        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately concludes that eta_ISB is approximately 0.25, i.e. about 25%."
      }
    ],
    "passThreshold": 100
  },
  {
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    "reference_file": "",
    "tools": [],
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        "name": "Normal-ordering setup",
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        "semanticPrompt": "The submission sets up the calculation by normal ordering (a^dagger^3 a^2)^4, for example using generalized Stirling numbers or an equivalent combinatorial expansion."
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        "semanticPrompt": "The submission obtains the correct coefficients in the normal-ordered polynomial, namely 1440, 5760, 6120, 2520, 456, 36, and 1 for the successive powers."
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      {
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        "name": "Final coherent-state expectation value",
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        "description": "",
        "weight": 45.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately concludes that <z|(a^dagger^2 a^3)^4|z> = z^4(|z|^16 + 36|z|^14 + 456|z|^12 + 2520|z|^10 + 6120|z|^8 + 5760|z|^6 + 1440|z|^4), or an equivalent algebraic form."
      }
    ],
    "passThreshold": 100
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  {
    "task_id": "prob_148",
    "reference_file": "",
    "tools": [],
    "criteria": [
      {
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        "name": "SU(4) to SU(3) decomposition",
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission decomposes the SU(4) quartic scalar term into SU(3)-invariant pieces involving (varphi varphi)(bar varphi bar varphi) and (varphi bar varphi)(bar varphi varphi), or an equivalent matching setup."
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      {
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        "name": "Matching equations from D-term or superpotential",
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        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission matches the decomposed terms to the N=1 SYM plus three WZ multiplet interactions and obtains the coefficient relation 8 k_{D+F} = -1/2, or an equivalent set of equations leading to the same result."
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      {
        "id": "56d4338c-2326-5d3d-9358-eae11de2584c",
        "name": "Final value of k_{D+F}",
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        "description": "",
        "weight": 40.0,
        "rationale": "Separated from the original single-criterion rubric.",
        "examples": [],
        "semanticPrompt": "The submission ultimately concludes that k_{D+F} = -1/16."
      }
    ],
    "passThreshold": 100
  },
  {
    "task_id": "prob_150",
    "reference_file": "",
    "tools": [],
    "criteria": [
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        "semanticPrompt": "The submission states the relevant three-dimensional identity relating e gamma^{mu nu rho} to epsilon^{mu nu rho}, specifically with the sign that implies e gamma^{mu nu rho} = - epsilon^{mu nu rho} I, or an equivalent statement."
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        "semanticPrompt": "The submission checks the sign using an explicit gamma-matrix convention, for example by obtaining gamma^{012} = -I while epsilon^{012} = +1."
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        "rationale": "Separated from the original single-criterion rubric.",
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        "semanticPrompt": "The submission ultimately concludes that B = -1."
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