{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0100",
  "pilot_number": "R0002-01550",
  "number": 1550,
  "family_id": "qf-app20-dynamics-invariant-subset-audit",
  "title": {
    "zh-Hant": "第 1550 題：有限離散動力核對",
    "en": "Question 1550: Forward Invariance versus Pointwise Fixing"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0100",
    "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-discrete-dynamical-system"
    ],
    "title_zh": "有限離散動力核對",
    "stem_zh": "明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":25,\"b\":21,\"subset\":[0,1,3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], 否)",
      "explanation_zh": "應用基礎核對=(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限離散動力核對"
      ]
    },
    "variant": {
      "family_id": "qf-app20-dynamics-invariant-subset-audit",
      "seed": 202610201100,
      "parameters": {
        "modulus": 9,
        "a": 25,
        "b": 21,
        "subset": [
          0,
          1,
          3,
          4
        ],
        "domain": "Fin_discrete_dynamics",
        "operation": "dynamics-invariant-subset-audit",
        "semantic_template": "qf-app20-dynamics-invariant-subset-audit/{\"a\":25,\"b\":21,\"modulus\":9,\"subset\":[0,1,3,4]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0100",
    "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "587c1f4e9bb0c0add33857acc71f1932df7797ed16c4787f3811a3068e5163d0",
      "family_id": "qf-app20-dynamics-invariant-subset-audit",
      "seed": 202610201100,
      "semantic_template": "qf-app20-dynamics-invariant-subset-audit/{\"a\":25,\"b\":21,\"modulus\":9,\"subset\":[0,1,3,4]}"
    },
    "fields": {
      "title": "Forward Invariance versus Pointwise Fixing",
      "stem": "Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).",
      "answer": {
        "kind": "expression",
        "value": "(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No)",
        "explanation": "Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No)."
      },
      "skills": [
        "Forward Invariance versus Pointwise Fixing"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":25,\"b\":21,\"subset\":[0,1,3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。",
        "en": "Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "X={0 through 8}; T(x)=(25x+(21)) mod 9",
      "after": "remainder representative always lies in X",
      "relation": "implies",
      "rule_id": "finite_total_map",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定完整有限狀態域與同一更新規則；負整數也取0到n−1的標準餘數。",
        "en": "Fix the full finite state carrier and the same update rule;negative integers also use the standard remainder from0 through n−1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "listed subset=[0, 1, 3, 4]",
      "after": "set S=[0, 1, 3, 4]",
      "relation": "implies",
      "rule_id": "subset_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "集合刪除重複後排序只為一致呈現，沒有增加新狀態。",
        "en": "Deduplication and sorting give a consistent set presentation without adding states."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "member0: a·x := mul(25, 0)",
      "after": "member0: a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把原係數乘當前狀態，不先把負數截成0。",
        "en": "Multiply the original coefficient by the current state;do not clip negative values to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member0: a·x，不以小數近似替代。",
          "en": "Recompute member0: a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "0"
        ],
        "output": "0",
        "label": "member0: a·x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "member0: a·x+b := sum(0, 21)",
      "after": "member0: a·x+b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加上同一常數平移後才取模。",
        "en": "Add the same offset before reducing modulo n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member0: a·x+b，不以小數近似替代。",
          "en": "Recompute member0: a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "21"
        ],
        "output": "21",
        "label": "member0: a·x+b"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "reduce 21 modulo 9",
      "after": "21=2·9+3",
      "relation": "implies",
      "rule_id": "euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此整數等式及0≤餘數<n一起確定下一個狀態。",
        "en": "The integer identity together with0≤remainder<n determines the next state."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "21",
          "9",
          "2"
        ],
        "output": "3",
        "label": "remainder"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "member1: a·x := mul(25, 1)",
      "after": "member1: a·x = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把原係數乘當前狀態，不先把負數截成0。",
        "en": "Multiply the original coefficient by the current state;do not clip negative values to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member1: a·x，不以小數近似替代。",
          "en": "Recompute member1: a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "1"
        ],
        "output": "25",
        "label": "member1: a·x"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "member1: a·x+b := sum(25, 21)",
      "after": "member1: a·x+b = 46",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加上同一常數平移後才取模。",
        "en": "Add the same offset before reducing modulo n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member1: a·x+b，不以小數近似替代。",
          "en": "Recompute member1: a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25",
          "21"
        ],
        "output": "46",
        "label": "member1: a·x+b"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "reduce 46 modulo 9",
      "after": "46=5·9+1",
      "relation": "implies",
      "rule_id": "euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此整數等式及0≤餘數<n一起確定下一個狀態。",
        "en": "The integer identity together with0≤remainder<n determines the next state."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "46",
          "9",
          "5"
        ],
        "output": "1",
        "label": "remainder"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "member3: a·x := mul(25, 3)",
      "after": "member3: a·x = 75",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把原係數乘當前狀態，不先把負數截成0。",
        "en": "Multiply the original coefficient by the current state;do not clip negative values to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member3: a·x，不以小數近似替代。",
          "en": "Recompute member3: a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "3"
        ],
        "output": "75",
        "label": "member3: a·x"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "member3: a·x+b := sum(75, 21)",
      "after": "member3: a·x+b = 96",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加上同一常數平移後才取模。",
        "en": "Add the same offset before reducing modulo n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member3: a·x+b，不以小數近似替代。",
          "en": "Recompute member3: a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "75",
          "21"
        ],
        "output": "96",
        "label": "member3: a·x+b"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "reduce 96 modulo 9",
      "after": "96=10·9+6",
      "relation": "implies",
      "rule_id": "euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此整數等式及0≤餘數<n一起確定下一個狀態。",
        "en": "The integer identity together with0≤remainder<n determines the next state."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "96",
          "9",
          "10"
        ],
        "output": "6",
        "label": "remainder"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "member4: a·x := mul(25, 4)",
      "after": "member4: a·x = 100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把原係數乘當前狀態，不先把負數截成0。",
        "en": "Multiply the original coefficient by the current state;do not clip negative values to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member4: a·x，不以小數近似替代。",
          "en": "Recompute member4: a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "4"
        ],
        "output": "100",
        "label": "member4: a·x"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "member4: a·x+b := sum(100, 21)",
      "after": "member4: a·x+b = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加上同一常數平移後才取模。",
        "en": "Add the same offset before reducing modulo n."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 member4: a·x+b，不以小數近似替代。",
          "en": "Recompute member4: a·x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "100",
          "21"
        ],
        "output": "121",
        "label": "member4: a·x+b"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "reduce 121 modulo 9",
      "after": "121=13·9+4",
      "relation": "implies",
      "rule_id": "euclidean_remainder",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此整數等式及0≤餘數<n一起確定下一個狀態。",
        "en": "The integer identity together with0≤remainder<n determines the next state."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "9",
          "13"
        ],
        "output": "4",
        "label": "remainder"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "T(S)=[1, 3, 4, 6]",
      "after": "T(S)⊆S: 否; exiting witnesses=[[3, 6]]",
      "relation": "implies",
      "rule_id": "forward_invariance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正例要求所有像都留在S；一個明示的離開狀態就足以否定。空S則真值為空量詞成立。",
        "en": "Every image must remain in S;one explicit exit refutes invariance.The empty S satisfies the condition vacuously."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "x₀∈S and T(S)⊆S",
      "after": "induction keeps every later iterate in S",
      "relation": "implies",
      "rule_id": "invariance_induction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若條件成立，起始在S且每一步留在S，歸納得到所有後續狀態留在S；不只檢查一條任意短軌道。",
        "en": "When the condition holds,the initial state and every update stay in S,so induction gives all later iterates,not merely one arbitrary short orbit."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      9,
      25,
      21,
      [
        0,
        1,
        3,
        4
      ],
      [
        1,
        3,
        4,
        6
      ],
      false
    ],
    "evaluations": [
      [
        0,
        3
      ],
      [
        1,
        1
      ],
      [
        3,
        6
      ],
      [
        4,
        4
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/questions.jsonl",
      "batch_sha256": "5f5f7ffde34f79b3a160c843b058e6c24e8dd5fda6cb219b2588cfe96e5c846e",
      "original_line": 100,
      "original_line_sha256": "8ea6451156026b3c5136dd149ed9a7bce0a0a7796906333a1570d997983e76f0",
      "question_compact_sha256": "587c1f4e9bb0c0add33857acc71f1932df7797ed16c4787f3811a3068e5163d0",
      "english_row_compact_sha256": "d11ca1c006ee71220e364cebcd7e3b08f0502cc9aa247fa5135ef073d3b026df",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-discrete-dynamical-system": {
        "sha256": "83db6a4ce40522862935c10fd58632bc890dbcf7e2d2277368dad51ee0b72837",
        "english_sha256": "8513c6a6ed28df30295d27ab05a1d1c2a008427aceaf56ba9bfa33819c320e29",
        "original_path": "public/data/mko/discrete-dynamical-system.json",
        "english_path": "public/data/i18n/editions/mko-discrete-dynamical-system.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e476360038ede38e6f70fbe8c039b14a4f4bd45937ee88050de8370944911f19",
    "original_candidate_sha256": "7dab1279a17730bdbea43f26533822ac883e6fc8e83c982f60e85b37f447baeb",
    "original_candidate_line_sha256": "7dab1279a17730bdbea43f26533822ac883e6fc8e83c982f60e85b37f447baeb"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p3",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861",
      "question_first_witness_sha256": "9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 16,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e63259d12771380e0655b94a8d666dbbcdfbcf7f13fd20ed9fff45a07599463c",
      "arithmetic_steps": 12
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}