{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r04-0059",
  "pilot_number": "R0054-27984",
  "number": 27984,
  "family_id": "qf-mod50-lagrange-single-linear-constraint",
  "title": {
    "zh-Hant": "第 27984 題：Lagrange乘子條件",
    "en": "Question 27984: Squared Distance with One Regular Linear Constraint Has an Exact Solution"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r04-0059",
    "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-lagrange-multiplier-condition"
    ],
    "title_zh": "Lagrange乘子條件",
    "stem_zh": "feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,1],\"constraint_gradient\":[1,9],\"constraint_rhs\":11}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)",
      "explanation_zh": "Wave6第十三層核對=([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Lagrange乘子條件"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-lagrange-single-linear-constraint",
      "seed": 202611194059,
      "parameters": {
        "center": [
          3,
          1
        ],
        "constraint_gradient": [
          1,
          9
        ],
        "constraint_rhs": 11,
        "domain": "Exact_equality_constraint_Lagrange_condition",
        "operation": "lagrange-single-linear-constraint",
        "semantic_template": "lagrange-single-linear-constraint|{\"center\":[[3,1],[1,1]],\"constraint_gradient\":[[1,1],[9,1]],\"constraint_rhs\":[11,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_contract_recomputation",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "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-pulse-20261119-r04-0059",
    "batch_id": "batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2f04af6845e43f5d05d770a27bde2843b3d6033798979ad417dd79caf83ddcc2",
      "family_id": "qf-mod50-lagrange-single-linear-constraint",
      "seed": 202611194059,
      "semantic_template": "lagrange-single-linear-constraint|{\"center\":[[3,1],[1,1]],\"constraint_gradient\":[[1,1],[9,1]],\"constraint_rhs\":[11,1]}"
    },
    "fields": {
      "title": "Squared Distance with One Regular Linear Constraint Has an Exact Solution",
      "stem": "Minimize squared Euclidean distance to center=[3, 1] subject to normal a=[1, 9], a dot x=11; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).",
      "answer": {
        "kind": "expression",
        "value": "([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)",
        "explanation": "Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."
      },
      "skills": [
        "Squared Distance with One Regular Linear Constraint Has an Exact Solution"
      ],
      "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": "feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,1],\"constraint_gradient\":[1,9],\"constraint_rhs\":11}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Minimize squared Euclidean distance to center=[3, 1] subject to normal a=[1, 9], a dot x=11; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [lagrange-single-linear-constraint]",
      "after": "derived = [[center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。",
        "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Lagrange single ORIGINAL normal squared norm ORIGINAL product0 := mul(1, 1)",
      "after": "Lagrange single ORIGINAL normal squared norm ORIGINAL product0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal squared norm ORIGINAL product0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Lagrange single ORIGINAL normal squared norm ORIGINAL product0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 := sum(0, 1)",
      "after": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Lagrange single ORIGINAL normal squared norm ORIGINAL product1 := mul(9, 9)",
      "after": "Lagrange single ORIGINAL normal squared norm ORIGINAL product1 = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal squared norm ORIGINAL product1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "9"
        ],
        "output": "81",
        "label": "Lagrange single ORIGINAL normal squared norm ORIGINAL product1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 := sum(1, 81)",
      "after": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 = 82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "81"
        ],
        "output": "82",
        "label": "Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Lagrange single ORIGINAL normal dot center ORIGINAL product0 := mul(1, 3)",
      "after": "Lagrange single ORIGINAL normal dot center ORIGINAL product0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal dot center ORIGINAL product0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "Lagrange single ORIGINAL normal dot center ORIGINAL product0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 := sum(0, 3)",
      "after": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal dot center ORIGINAL running sum0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "3"
        ],
        "output": "3",
        "label": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Lagrange single ORIGINAL normal dot center ORIGINAL product1 := mul(9, 1)",
      "after": "Lagrange single ORIGINAL normal dot center ORIGINAL product1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal dot center ORIGINAL product1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "Lagrange single ORIGINAL normal dot center ORIGINAL product1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 := sum(3, 9)",
      "after": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal dot center ORIGINAL running sum1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "9"
        ],
        "output": "12",
        "label": "Lagrange single ORIGINAL normal dot center ORIGINAL running sum1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Lagrange single ORIGINAL rhs minus center value := sub(11, 12)",
      "after": "Lagrange single ORIGINAL rhs minus center value = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL rhs minus center value，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL rhs minus center value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "12"
        ],
        "output": "-1",
        "label": "Lagrange single ORIGINAL rhs minus center value"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Lagrange single ORIGINAL normal displacement := div(-1, 82)",
      "after": "Lagrange single ORIGINAL normal displacement = -1/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal displacement，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal displacement with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-1",
          "82"
        ],
        "output": "-1/82",
        "label": "Lagrange single ORIGINAL normal displacement"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Lagrange single ORIGINAL normal shift0 := mul(-1/82, 1)",
      "after": "Lagrange single ORIGINAL normal shift0 = -1/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal shift0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal shift0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/82",
          "1"
        ],
        "output": "-1/82",
        "label": "Lagrange single ORIGINAL normal shift0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Lagrange single ORIGINAL solution coordinate0 := sum(3, -1/82)",
      "after": "Lagrange single ORIGINAL solution coordinate0 = 245/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL solution coordinate0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL solution coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "-1/82"
        ],
        "output": "245/82",
        "label": "Lagrange single ORIGINAL solution coordinate0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Lagrange single ORIGINAL normal shift1 := mul(-1/82, 9)",
      "after": "Lagrange single ORIGINAL normal shift1 = -9/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL normal shift1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL normal shift1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/82",
          "9"
        ],
        "output": "-9/82",
        "label": "Lagrange single ORIGINAL normal shift1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Lagrange single ORIGINAL solution coordinate1 := sum(1, -9/82)",
      "after": "Lagrange single ORIGINAL solution coordinate1 = 73/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL solution coordinate1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL solution coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "-9/82"
        ],
        "output": "73/82",
        "label": "Lagrange single ORIGINAL solution coordinate1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Lagrange single ORIGINAL multiplier2tau := mul(2, -1/82)",
      "after": "Lagrange single ORIGINAL multiplier2tau = -1/41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL multiplier2tau，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL multiplier2tau with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1/82"
        ],
        "output": "-1/41",
        "label": "Lagrange single ORIGINAL multiplier2tau"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Lagrange single ORIGINAL displacement coordinate0 := sub(245/82, 3)",
      "after": "Lagrange single ORIGINAL displacement coordinate0 = -1/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL displacement coordinate0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL displacement coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "245/82",
          "3"
        ],
        "output": "-1/82",
        "label": "Lagrange single ORIGINAL displacement coordinate0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Lagrange single ORIGINAL objective gradient0 := mul(2, -1/82)",
      "after": "Lagrange single ORIGINAL objective gradient0 = -1/41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL objective gradient0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL objective gradient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1/82"
        ],
        "output": "-1/41",
        "label": "Lagrange single ORIGINAL objective gradient0"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Lagrange single ORIGINAL displacement coordinate1 := sub(73/82, 1)",
      "after": "Lagrange single ORIGINAL displacement coordinate1 = -9/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL displacement coordinate1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL displacement coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "73/82",
          "1"
        ],
        "output": "-9/82",
        "label": "Lagrange single ORIGINAL displacement coordinate1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Lagrange single ORIGINAL objective gradient1 := mul(2, -9/82)",
      "after": "Lagrange single ORIGINAL objective gradient1 = -9/41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL objective gradient1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL objective gradient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-9/82"
        ],
        "output": "-9/41",
        "label": "Lagrange single ORIGINAL objective gradient1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Lagrange single ORIGINAL full feasibility ORIGINAL product0 := mul(1, 245/82)",
      "after": "Lagrange single ORIGINAL full feasibility ORIGINAL product0 = 245/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL full feasibility ORIGINAL product0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "245/82"
        ],
        "output": "245/82",
        "label": "Lagrange single ORIGINAL full feasibility ORIGINAL product0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 := sum(0, 245/82)",
      "after": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 = 245/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL full feasibility ORIGINAL running sum0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "245/82"
        ],
        "output": "245/82",
        "label": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Lagrange single ORIGINAL full feasibility ORIGINAL product1 := mul(9, 73/82)",
      "after": "Lagrange single ORIGINAL full feasibility ORIGINAL product1 = 657/82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL full feasibility ORIGINAL product1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "73/82"
        ],
        "output": "657/82",
        "label": "Lagrange single ORIGINAL full feasibility ORIGINAL product1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 := sum(245/82, 657/82)",
      "after": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL full feasibility ORIGINAL running sum1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "245/82",
          "657/82"
        ],
        "output": "11",
        "label": "Lagrange single ORIGINAL full feasibility ORIGINAL running sum1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Lagrange single ORIGINAL lambda times normal0 := mul(-1/41, 1)",
      "after": "Lagrange single ORIGINAL lambda times normal0 = -1/41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL lambda times normal0，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL lambda times normal0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/41",
          "1"
        ],
        "output": "-1/41",
        "label": "Lagrange single ORIGINAL lambda times normal0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Lagrange single ORIGINAL lambda times normal1 := mul(-1/41, 9)",
      "after": "Lagrange single ORIGINAL lambda times normal1 = -9/41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以原始角色與已證中間量精確代值，不用近似或自洽但換過的操作數；額外教學代數不另縮來源域。",
        "en": "Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lagrange single ORIGINAL lambda times normal1，不以小數近似替代。",
          "en": "Recompute Lagrange single ORIGINAL lambda times normal1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/41",
          "9"
        ],
        "output": "-9/41",
        "label": "Lagrange single ORIGINAL lambda times normal1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [[3, 1], [1, 9], 11, 82, 12, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41]]",
      "after": "derived = [是, 是, ANY feasible y=x+v has a dot v=0;f(y)=f(x)+norm(v)^2]",
      "relation": "implies",
      "rule_id": "Lagrange_specified_squared_distance_complete_orthogonality_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "normal非0使norm²>0，tau解a·(c+tau a)=rhs；objective是平方距離無1/2，所以gradient與lambda含2。任意可行y=x+v，cross term=2tau(a·v)=0，差為||v||²>=0，故本指定問題有唯一minimum；不外推任意multiplier方程充分。",
        "en": "A nonzero normal gives norm²>0 and tau solves a dot(c+tau a)=rhs.The objective is squared distance WITHOUT1/2,so gradient and lambda contain2.For any feasible y=x+v,the cross term2tau(a dot v)=0 and the objective difference is||v||²>=0.This specified problem has a unique minimum,not general sufficiency of multiplier equations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "3",
        "1"
      ],
      [
        "1",
        "9"
      ],
      "11",
      "-1/82",
      [
        "245/82",
        "73/82"
      ],
      "-1/41",
      [
        "-1/41",
        "-9/41"
      ],
      true,
      true,
      "squared_distance_with_one_regular_linear_constraint_solved_exactly"
    ]
  },
  "conclusion": {
    "canonical_value": "([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)",
    "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-20261119-r04-lagrange-multiplier-condition-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations/questions.jsonl",
      "batch_sha256": "d71d7d0bcecf04d4e0a1a6c42431e568d00d31857240898ab7af52725d5943da",
      "original_line": 59,
      "original_line_sha256": "6006dbc4300b48edcf9d1c0235d2a5de858aa20584557462f60636673c6d5796",
      "question_compact_sha256": "2f04af6845e43f5d05d770a27bde2843b3d6033798979ad417dd79caf83ddcc2",
      "english_row_compact_sha256": "af683f01597511b9421462d7bc57042783f8de823f6898b592cc077c15bae1f5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations.en.jsonl"
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    "mko_bindings": {
      "mko-lagrange-multiplier-condition": {
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        "english_sha256": "51b359b63416bacca8d5d5350982535ac76896b00ad3397742174d696a57ddbb",
        "original_path": "public/data/mko/lagrange-multiplier-condition.json",
        "english_path": "public/data/i18n/editions/mko-lagrange-multiplier-condition.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8",
    "original_candidate_sha256": "1d52f82532d20b4f154095fe2c06c52d1c36b172bda0ebbeffadf139b032f6f8",
    "original_candidate_line_sha256": "1d52f82532d20b4f154095fe2c06c52d1c36b172bda0ebbeffadf139b032f6f8"
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  "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-p55",
    "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": "0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54",
      "question_first_witness_sha256": "09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 27,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2fe3bfac6a1497ff90ce7834a7751eb05f643ee7d524d0788c6f8a6a244c061e",
      "arithmetic_steps": 25
    },
    "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."
  }
}