{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r03-0032",
  "pilot_number": "R0054-27832",
  "number": 27832,
  "family_id": "qf-mod50-gd-one-step",
  "title": {
    "zh-Hant": "第 27832 題：梯度下降",
    "en": "Question 27832: Positive Step Size Alone Does Not Guarantee Objective Descent"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r03-0032",
    "batch_id": "batch-20261015-advance-20261119-r03-gradient-descent-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-gradient-descent"
    ],
    "title_zh": "梯度下降",
    "stem_zh": "standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,4],\"linear\":[0,-3],\"point\":[-1,3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)",
      "explanation_zh": "Wave6第十三層核對=([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "梯度下降"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-gd-one-step",
      "seed": 202611193032,
      "parameters": {
        "diagonal": [
          1,
          4
        ],
        "linear": [
          0,
          -3
        ],
        "point": [
          -1,
          3
        ],
        "step_size": [
          1,
          4
        ],
        "domain": "Exact_diagonal_quadratic_gradient_descent",
        "operation": "gd-one-step",
        "semantic_template": "gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4]}"
      }
    },
    "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-r03-0032",
    "batch_id": "batch-20261015-advance-20261119-r03-gradient-descent-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9c9a80b7af03b6c087e1ff44408e14f38059967e72f950257e494636210330d5",
      "family_id": "qf-mod50-gd-one-step",
      "seed": 202611193032,
      "semantic_template": "gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4]}"
    },
    "fields": {
      "title": "Positive Step Size Alone Does Not Guarantee Objective Descent",
      "stem": "Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).",
      "answer": {
        "kind": "expression",
        "value": "([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent)",
        "explanation": "Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent)."
      },
      "skills": [
        "Positive Step Size Alone Does Not Guarantee Objective Descent"
      ],
      "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": "standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,4],\"linear\":[0,-3],\"point\":[-1,3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [gd-one-step]",
      "after": "derived = [[point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_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": "GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, -1)",
      "after": "GD ORIGINAL ORIGINAL gradient d*x coordinate0 = -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 GD ORIGINAL ORIGINAL gradient d*x coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "GD ORIGINAL ORIGINAL gradient d*x coordinate0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(-1, 0)",
      "after": "GD ORIGINAL ORIGINAL gradient plus b coordinate0 = -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 GD ORIGINAL ORIGINAL gradient plus b coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "-1",
        "label": "GD ORIGINAL ORIGINAL gradient plus b coordinate0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(4, 3)",
      "after": "GD ORIGINAL ORIGINAL gradient d*x coordinate1 = 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 GD ORIGINAL ORIGINAL gradient d*x coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "3"
        ],
        "output": "12",
        "label": "GD ORIGINAL ORIGINAL gradient d*x coordinate1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(12, -3)",
      "after": "GD ORIGINAL ORIGINAL gradient plus b coordinate1 = 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 GD ORIGINAL ORIGINAL gradient plus b coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "-3"
        ],
        "output": "9",
        "label": "GD ORIGINAL ORIGINAL gradient plus b coordinate1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1, -1)",
      "after": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 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 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 1)",
      "after": "GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 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 GD ORIGINAL ORIGINAL diagonal times square coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "GD ORIGINAL ORIGINAL diagonal times square coordinate0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(1, 2)",
      "after": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 1/2",
      "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 GD ORIGINAL ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "GD ORIGINAL ORIGINAL linear product coordinate0 := mul(0, -1)",
      "after": "GD ORIGINAL ORIGINAL linear product coordinate0 = 0",
      "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 GD ORIGINAL ORIGINAL linear product coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "GD ORIGINAL ORIGINAL linear product coordinate0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(1/2, 0)",
      "after": "GD ORIGINAL ORIGINAL coordinate objective sum0 = 1/2",
      "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 GD ORIGINAL ORIGINAL coordinate objective sum0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1/2",
        "label": "GD ORIGINAL ORIGINAL coordinate objective sum0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 1/2)",
      "after": "GD ORIGINAL ORIGINAL objective running sum0 = 1/2",
      "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 GD ORIGINAL ORIGINAL objective running sum0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "GD ORIGINAL ORIGINAL objective running sum0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(3, 3)",
      "after": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 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 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(4, 9)",
      "after": "GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 36",
      "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 GD ORIGINAL ORIGINAL diagonal times square coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "9"
        ],
        "output": "36",
        "label": "GD ORIGINAL ORIGINAL diagonal times square coordinate1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(36, 2)",
      "after": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 18",
      "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 GD ORIGINAL ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "36",
          "2"
        ],
        "output": "18",
        "label": "GD ORIGINAL ORIGINAL quadratic divide2 coordinate1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "GD ORIGINAL ORIGINAL linear product coordinate1 := mul(-3, 3)",
      "after": "GD ORIGINAL ORIGINAL linear product coordinate1 = -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 GD ORIGINAL ORIGINAL linear product coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "3"
        ],
        "output": "-9",
        "label": "GD ORIGINAL ORIGINAL linear product coordinate1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(18, -9)",
      "after": "GD ORIGINAL ORIGINAL coordinate objective sum1 = 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 GD ORIGINAL ORIGINAL coordinate objective sum1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "-9"
        ],
        "output": "9",
        "label": "GD ORIGINAL ORIGINAL coordinate objective sum1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "GD ORIGINAL ORIGINAL objective running sum1 := sum(1/2, 9)",
      "after": "GD ORIGINAL ORIGINAL objective running sum1 = 19/2",
      "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 GD ORIGINAL ORIGINAL objective running sum1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "9"
        ],
        "output": "19/2",
        "label": "GD ORIGINAL ORIGINAL objective running sum1"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [GD ORIGINAL, [1, 4], [0, -3], [-1, 3]]",
      "after": "derived = [[1, 4], [0, -3], [-1, 3], [-1, 9], 19/2]",
      "relation": "implies",
      "rule_id": "GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective",
      "conditions": [],
      "explanation": {
        "zh-Hant": "給定完整f(x)=1/2 Σd_i x_i²+Σb_i x_i，對每座標微分得d_i x_i+b_i（不是2d_i x_i）。完整gradient先算，再按原順序算objective，square先於乘di；正di與維度、座標次序均核對。",
        "en": "For the WHOLE specified f(x)=1/2 sum d_i x_i²+sum b_i x_i,differentiate each coordinate to d_i x_i+b_i,not2d_i x_i.Compute ALL gradient entries first,then the objective in original order,squaring before multiplying d_i.Qualify positive diagonals,dimensions and coordinate order."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "GD ORIGINAL one step ORIGINAL alpha times gradient0 := mul(1/4, -1)",
      "after": "GD ORIGINAL one step ORIGINAL alpha times gradient0 = -1/4",
      "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 GD ORIGINAL one step ORIGINAL alpha times gradient0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step ORIGINAL alpha times gradient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "-1"
        ],
        "output": "-1/4",
        "label": "GD ORIGINAL one step ORIGINAL alpha times gradient0"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "GD ORIGINAL one step ORIGINAL next coordinate0 := sub(-1, -1/4)",
      "after": "GD ORIGINAL one step ORIGINAL next coordinate0 = -3/4",
      "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 GD ORIGINAL one step ORIGINAL next coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step ORIGINAL next coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-1/4"
        ],
        "output": "-3/4",
        "label": "GD ORIGINAL one step ORIGINAL next coordinate0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "GD ORIGINAL one step ORIGINAL alpha times gradient1 := mul(1/4, 9)",
      "after": "GD ORIGINAL one step ORIGINAL alpha times gradient1 = 9/4",
      "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 GD ORIGINAL one step ORIGINAL alpha times gradient1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step ORIGINAL alpha times gradient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/4",
          "9"
        ],
        "output": "9/4",
        "label": "GD ORIGINAL one step ORIGINAL alpha times gradient1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "GD ORIGINAL one step ORIGINAL next coordinate1 := sub(3, 9/4)",
      "after": "GD ORIGINAL one step ORIGINAL next coordinate1 = 3/4",
      "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 GD ORIGINAL one step ORIGINAL next coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step ORIGINAL next coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "9/4"
        ],
        "output": "3/4",
        "label": "GD ORIGINAL one step ORIGINAL next coordinate1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, -3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 = -3/4",
      "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 GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-3/4"
        ],
        "output": "-3/4",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 := sum(-3/4, 0)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 = -3/4",
      "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 GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3/4",
          "0"
        ],
        "output": "-3/4",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 := mul(4, 3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 = 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 GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "3/4"
        ],
        "output": "3",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 := sum(3, -3)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 = 0",
      "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 GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "-3"
        ],
        "output": "0",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-3/4, -3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 9/16",
      "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 GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3/4",
          "-3/4"
        ],
        "output": "9/16",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 9/16)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 = 9/16",
      "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 GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "9/16"
        ],
        "output": "9/16",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 := div(9/16, 2)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 = 9/32",
      "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 GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9/16",
          "2"
        ],
        "output": "9/32",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 := mul(0, -3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 = 0",
      "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 GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-3/4"
        ],
        "output": "0",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 := sum(9/32, 0)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 = 9/32",
      "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 GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9/32",
          "0"
        ],
        "output": "9/32",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 := sum(0, 9/32)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 = 9/32",
      "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 GD ORIGINAL one step next recomputation ORIGINAL objective running sum0，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "9/32"
        ],
        "output": "9/32",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum0"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(3/4, 3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 9/16",
      "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 GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/4",
          "3/4"
        ],
        "output": "9/16",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 := mul(4, 9/16)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 = 9/4",
      "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 GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "9/16"
        ],
        "output": "9/4",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1 := div(9/4, 2)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1 = 9/8",
      "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 GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9/4",
          "2"
        ],
        "output": "9/8",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 := mul(-3, 3/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 = -9/4",
      "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 GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "3/4"
        ],
        "output": "-9/4",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 := sum(9/8, -9/4)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 = -9/8",
      "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 GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9/8",
          "-9/4"
        ],
        "output": "-9/8",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 := sum(9/32, -9/8)",
      "after": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 = -27/32",
      "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 GD ORIGINAL one step next recomputation ORIGINAL objective running sum1，不以小數近似替代。",
          "en": "Recompute GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9/32",
          "-9/8"
        ],
        "output": "-27/32",
        "label": "GD ORIGINAL one step next recomputation ORIGINAL objective running sum1"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "source roles = [GD ORIGINAL one step next recomputation, [[1, 1], [4, 1]], [[0, 1], [-3, 1]], [[-3, 4], [3, 4]]]",
      "after": "derived = [[1, 4], [0, -3], [-3/4, 3/4], [-3/4, 0], -27/32]",
      "relation": "implies",
      "rule_id": "GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective",
      "conditions": [],
      "explanation": {
        "zh-Hant": "給定完整f(x)=1/2 Σd_i x_i²+Σb_i x_i，對每座標微分得d_i x_i+b_i（不是2d_i x_i）。完整gradient先算，再按原順序算objective，square先於乘di；正di與維度、座標次序均核對。",
        "en": "For the WHOLE specified f(x)=1/2 sum d_i x_i²+sum b_i x_i,differentiate each coordinate to d_i x_i+b_i,not2d_i x_i.Compute ALL gradient entries first,then the objective in original order,squaring before multiplying d_i.Qualify positive diagonals,dimensions and coordinate order."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [[-1, 3], [-1, 9], 1/4]",
      "after": "derived = [[-3/4, 3/4], 19/2, -27/32, 是]",
      "relation": "implies",
      "rule_id": "GD_one_positive_step_ACTUAL_objective_comparison_not_safe_assumption",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此題只要求正步長，逐真正前後objective比較。flag可False；不能因positive_step字串就假設下降，也不能私加safe-step條件來排除合法例子。",
        "en": "This family requires only a positive step and compares the ACTUAL objectives.The flag can be False.A positive_step description does not imply descent,and no stronger safe-step guard is added."
      },
      "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": [
      [
        "-1",
        "3"
      ],
      [
        "-1",
        "9"
      ],
      "1/4",
      [
        "-3/4",
        "3/4"
      ],
      "19/2",
      "-27/32",
      true,
      "positive_step_alone_does_not_guarantee_descent"
    ]
  },
  "conclusion": {
    "canonical_value": "([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)",
    "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-r03-gradient-descent-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r03-gradient-descent-foundations/questions.jsonl",
      "batch_sha256": "ab1c07c8672115b16a6477b3ab604a2116f08a2db6bf163d8cbf723885ab4189",
      "original_line": 32,
      "original_line_sha256": "f053767b1e9edfc69a4631a56723ed9c51a53324785af60b5ca827cbbde9403a",
      "question_compact_sha256": "9c9a80b7af03b6c087e1ff44408e14f38059967e72f950257e494636210330d5",
      "english_row_compact_sha256": "d110d5f0e069e8768d264bd57651747632fffb4352a719eb7563adc91a184146",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r03-gradient-descent-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r03-gradient-descent-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-gradient-descent": {
        "sha256": "b886e6f39050667744f8b88cc6e5d065b39d33d9bbcaf87612a4129ae035973b",
        "english_sha256": "0e010c35c5db93d7d0e7c33c6218d6f9ae774248a498ad97fc98c2be454a1fef",
        "original_path": "public/data/mko/gradient-descent.json",
        "english_path": "public/data/i18n/editions/mko-gradient-descent.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8",
    "original_candidate_sha256": "96e2fa9cc879fe10cbc61750ff0825f223057a2be0b0cf4d81e3861296075270",
    "original_candidate_line_sha256": "96e2fa9cc879fe10cbc61750ff0825f223057a2be0b0cf4d81e3861296075270"
  },
  "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": 40,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "106363f16d27a232d1bfee613f0ff0bca7efa4546a9573ef25862cafcca30440",
      "arithmetic_steps": 36
    },
    "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."
  }
}