{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261117-r06-path-connected-space-0012",
  "pilot_number": "R0052-26687",
  "number": 26687,
  "family_id": "qf-mod48-path-interval-linear",
  "title": {
    "zh-Hant": "第 26687 題：道路連通空間",
    "en": "Question 26687: An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261117-r06-path-connected-space-0012",
    "batch_id": "batch-20261015-advance-20261117-r06-path-connected-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-path-connected-space"
    ],
    "title_zh": "道路連通空間",
    "stem_zh": "unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-13,\"right\":14,\"start\":-11,\"end\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)",
      "explanation_zh": "Wave6第十一層核對=(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "道路連通空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod48-path-interval-linear",
      "seed": 202611176012,
      "parameters": {
        "left": -13,
        "right": 14,
        "start": -11,
        "end": 12,
        "domain": "Exact_explicit_path_connectedness",
        "operation": "path-interval-linear",
        "semantic_template": "path-interval-linear|{end:[12,1],left:[-13,1],right:[14,1],start:[-11,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261117-r06-path-connected-space-0012",
    "batch_id": "batch-20261015-advance-20261117-r06-path-connected-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ab86194866fed14bf83417ed2150944e0c9b1f1d16435f7883b676a05d7618f8",
      "family_id": "qf-mod48-path-interval-linear",
      "seed": 202611176012,
      "semantic_template": "path-interval-linear|{end:[12,1],left:[-13,1],right:[14,1],start:[-11,1]}"
    },
    "fields": {
      "title": "An Affine Path Has Exact Endpoints and Stays in Its Convex Interval",
      "stem": "Use closed real interval [-13,14], start=-11,end=12, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)",
        "explanation": "The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."
      },
      "skills": [
        "An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"
      ],
      "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": "unit-interval path的continuity、兩端點與全range containment責任。操作=path-interval-linear；資料={\"left\":-13,\"right\":14,\"start\":-11,\"end\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use closed real interval [-13,14], start=-11,end=12, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [path-interval-linear]",
      "after": "derived = [[left, right, start, end, formula, start_endpoint, end_endpoint, continuous_path, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列出此操作及mode的完整來源欄序，不需JavaScript；布林、字串與null角色保持區別，不補欄或以答案形狀猜介面。",
        "en": "List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Path ORIGINAL end minus start := sub(12, -11)",
      "after": "Path ORIGINAL end minus start = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Path ORIGINAL end minus start，不以小數近似替代。",
          "en": "Recompute Path ORIGINAL end minus start with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "12",
          "-11"
        ],
        "output": "23",
        "label": "Path ORIGINAL end minus start"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Path endpoint at zero := sum(-11, 0)",
      "after": "Path endpoint at zero = -11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Path endpoint at zero，不以小數近似替代。",
          "en": "Recompute Path endpoint at zero with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-11",
          "0"
        ],
        "output": "-11",
        "label": "Path endpoint at zero"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Path endpoint at one := sum(-11, 23)",
      "after": "Path endpoint at one = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。",
        "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Path endpoint at one，不以小數近似替代。",
          "en": "Recompute Path endpoint at one with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-11",
          "23"
        ],
        "output": "12",
        "label": "Path endpoint at one"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [-13, 14, -11, 12, 23]",
      "after": "derived = [gamma(t)=(1-t)*start+t*end, 0<=t<=1, continuous affine function, 是]",
      "relation": "implies",
      "rule_id": "Path_WHOLE_affine_convex_range_not_endpoint_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將a+(b-a)t展成(1-t)a+tb；所有t的權重非負且和1，故全range留在閉interval，affine連續並有指定endpoint。left=right也可，額外endpoint算式不施加來源未要求的數值界。",
        "en": "Expand a+(b-a)t as(1-t)a+tb.For every t the weights are nonnegative with sum1,so the whole range remains in the closed interval.The affine function is continuous with the stated endpoints.Degenerate intervals are allowed;extra endpoint identities add no source numeric gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。",
          "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-13",
      "14",
      "-11",
      "12",
      "gamma(t)=-11+(23)t",
      "gamma(0)=start",
      "gamma(1)=end",
      true,
      "convex_combination_stays_in_interval"
    ]
  },
  "conclusion": {
    "canonical_value": "(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, 是, convex_combination_stays_in_interval)",
    "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-20261015-advance-20261117-r06-path-connected-space-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261117-r06-path-connected-space-foundations/questions.jsonl",
      "batch_sha256": "2455669d9f59c769c41b1462899255c75387a7fdce752cd36e1fad7467bb3f32",
      "original_line": 12,
      "original_line_sha256": "df4b7bca680f10a986a2393b08cce6af6512e4c0fc3dbf08d19bb8b196ea5808",
      "question_compact_sha256": "ab86194866fed14bf83417ed2150944e0c9b1f1d16435f7883b676a05d7618f8",
      "english_row_compact_sha256": "e622283f8007a1f76bfe64420bc9919aea4c7b59c3d484640256d82ea1c76c4a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r06-path-connected-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r06-path-connected-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-path-connected-space": {
        "sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17",
        "english_sha256": "bb59736c31f281af474355507a3b5172c4fd2b948847c2332ec737e88490c538",
        "original_path": "public/data/mko/path-connected-space.json",
        "english_path": "public/data/i18n/editions/mko-path-connected-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a3def7368821a4d3dc7757fefd3c4a099713f8de8fb8e6f3434025e9d7b5a6d0",
    "original_candidate_sha256": "433d331249359bc21d0b9e74d07db89865f04f1000d5303987faea08e0839118",
    "original_candidate_line_sha256": "433d331249359bc21d0b9e74d07db89865f04f1000d5303987faea08e0839118",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-path-connected-space": "b17c14e650f00c01de1e7a49d95f7fbc2a4d9dcdac715c039f9e61fb4f9de607"
      }
    }
  },
  "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-20261026-010-p53",
    "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": "729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252",
      "question_first_witness_sha256": "d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "60bb4b49858158be3a9a50184e06640d7a63c9d37eb0c3d63fa808d9a863bccd",
      "arithmetic_steps": 3
    },
    "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."
  }
}