{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r04-proof-0058",
  "pilot_number": "R0011-06208",
  "number": 6208,
  "family_id": "qf-density13-proof-horn-candidate-trace",
  "title": {
    "zh-Hant": "第 6208 題：候選MP序列的逐行稽核",
    "en": "Question 6208: Sequential Candidate Trace and First Rejected Row"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r04-proof-0058",
    "batch_id": "batch-20261009-advance-20261013-r04-proof-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-proof"
    ],
    "title_zh": "候選MP序列的逐行稽核",
    "stem_zh": "給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。有序trace=[2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。",
    "answer": {
      "kind": "expression",
      "value": "([否, 是, 是], 0)",
      "explanation_zh": "候選逐行資格與首錯=([否, 是, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "候選MP序列的逐行稽核"
      ]
    },
    "variant": {
      "family_id": "qf-density13-proof-horn-candidate-trace",
      "seed": 202610134058,
      "parameters": {
        "n": 3,
        "facts": [
          0
        ],
        "rules": [
          [
            0,
            1
          ],
          [
            1,
            2
          ]
        ],
        "trace": [
          2,
          1,
          2
        ],
        "operation": "density13_proof_horn_candidate_trace",
        "domain": "finite_given_MP_protocol",
        "semantic_template": "qf-density13-proof-horn-candidate-trace/[false,true,true]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r04-proof-0058",
    "batch_id": "batch-20261009-advance-20261013-r04-proof-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "bd163d239e756b5a5003f55872eb9d5bc530934e3ea59a86220daa023a6c6936",
      "family_id": "qf-density13-proof-horn-candidate-trace",
      "seed": 202610134058,
      "semantic_template": "qf-density13-proof-horn-candidate-trace/[false,true,true]"
    },
    "fields": {
      "title": "Sequential Candidate Trace and First Rejected Row",
      "stem": "Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.",
      "answer": {
        "kind": "expression",
        "value": "([No, Yes, Yes], 0)",
        "explanation": "Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], 0)."
      },
      "skills": [
        "Sequential Candidate Trace and First Rejected Row"
      ],
      "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": "給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。有序trace=[2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。",
        "en": "Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=3;given facts=[0];original indexed rules=[[0, 0, 1], [1, 1, 2]]",
      "after": "use only given facts and single-premise MP;not-derived does not mean False",
      "relation": "implies",
      "rule_id": "given_unary_MP_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原子以0起算索引表示，規則i→j只在i已得時可用。沒有事實的循環不能自行產生結論；未得不等於假，有限核對不是一般邏輯完備性。",
        "en": "Atoms use0-based indices and a rule i→j requires an already derived i.A cycle without facts cannot generate conclusions.Not-derived is not False,and this finite check is not general logical completeness."
      },
      "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": "given facts=[0]",
      "after": "initial accepted atoms=[0]",
      "relation": "implies",
      "rule_id": "candidate_trace_given_facts",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選列只能使用給定事實與此前合法列，不預先把全圖可達結論當成已列出的前提。",
        "en": "Candidate lines may use only the given facts and preceding legal lines;globally reachable conclusions are not automatically accepted prefix premises."
      },
      "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": "line=0;candidate=2;accepted before=[0]",
      "after": "line=0;valid=否;accepted=[0];rule=rejected",
      "relation": "implies",
      "rule_id": "horn_candidate_prefix_line",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用加入本列之前的已接受集合驗證：已接受原子可重列，或依一條給定MP推出；無效列不加入，所以不能替後列補造前提。",
        "en": "Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "line=1;candidate=1;accepted before=[0]",
      "after": "line=1;valid=是;accepted=[0,1];rule=0",
      "relation": "implies",
      "rule_id": "horn_candidate_prefix_line",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用加入本列之前的已接受集合驗證：已接受原子可重列，或依一條給定MP推出；無效列不加入，所以不能替後列補造前提。",
        "en": "Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "line=2;candidate=2;accepted before=[0,1]",
      "after": "line=2;valid=是;accepted=[0,1,2];rule=1",
      "relation": "implies",
      "rule_id": "horn_candidate_prefix_line",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用加入本列之前的已接受集合驗證：已接受原子可重列，或依一條給定MP推出；無效列不加入，所以不能替後列補造前提。",
        "en": "Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "line validity=[否, 是, 是];final accepted=[0, 1, 2]",
      "after": "first invalid line=0",
      "relation": "implies",
      "rule_id": "horn_candidate_trace_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "首錯用候選列的0起位置；錯後繼續稽核但不接受錯列。空序列沒有錯列，仍不因此證明某個目標。",
        "en": "The first error is a0-based trace position.Continue auditing after an error without accepting invalid lines.An empty sequence has no bad line but does not thereby prove a target."
      },
      "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": [
      [
        false,
        true,
        true
      ],
      "0"
    ],
    "accepted_after": [
      0,
      1,
      2
    ]
  },
  "conclusion": {
    "canonical_value": "([否, 是, 是], 0)",
    "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-20261009-advance-20261013-r04-proof-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r04-proof-foundations/questions.jsonl",
      "batch_sha256": "9d2b0a978c354c8317efdeb07f00669c8dabe08c14a9392d3101a946b73b4ac0",
      "original_line": 58,
      "original_line_sha256": "0c6cd128ea4d5b403f3710ed39ab5ae5944b064aab6746864a7a744b811d684c",
      "question_compact_sha256": "bd163d239e756b5a5003f55872eb9d5bc530934e3ea59a86220daa023a6c6936",
      "english_row_compact_sha256": "ebcec93112cf87daef2cfa57198e86a421c817824333e6b7ff17e5ced6dc45c9",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r04-proof-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r04-proof-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-proof": {
        "sha256": "72f6009b912caa3c05a2948a97ca422367574a19ad5fc665964a540875417b91",
        "english_sha256": "1c8ec335b48f7d913d3057b2d632c371b092ca869f799b056b72464d948504fe",
        "original_path": "public/data/mko/proof.json",
        "english_path": "public/data/i18n/editions/mko-proof.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "6f407a1b0ebfa85476086323f10fed347cdd9b450ac6fc4036cbfc997e391d6f",
    "original_candidate_line_sha256": "6f407a1b0ebfa85476086323f10fed347cdd9b450ac6fc4036cbfc997e391d6f"
  },
  "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-20261026-005-p12",
    "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": "b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80",
      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6002ee2c0e9f1559ad51acbc0a4e3f9a480e8393d590465a24d40ff6b477262f",
      "arithmetic_steps": 0
    },
    "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."
  }
}