{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r06-0079",
  "pilot_number": "R0066-33829",
  "number": 33829,
  "family_id": "qf-mod50-graddeep-boundary",
  "title": {
    "zh-Hant": "第 33829 題：梯度向量",
    "en": "Question 33829: Stationarity, Metric Identification and Finite Differences Have Different Responsibilities"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r06-0079",
    "batch_id": "batch-20261015-advance-20261119-r06-gradient-vector-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-gradient-vector"
    ],
    "title_zh": "梯度向量",
    "stem_zh": "fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":4,\"step\":[1,5]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)",
      "explanation_zh": "Wave6第十三層核對=(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "梯度向量"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-graddeep-boundary",
      "seed": 202611196079,
      "parameters": {
        "mode": "finite_difference_confusion",
        "model": {
          "difference_quotient": 4,
          "step": [
            1,
            5
          ]
        },
        "domain": "Exact_gradient_vector_density",
        "operation": "graddeep-boundary",
        "semantic_template": "graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[4,1],\"step\":[1,5]}}"
      }
    },
    "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-r06-0079",
    "batch_id": "batch-20261015-advance-20261119-r06-gradient-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "562d01c975a1832c20fc720dc1ddda30fea508b58b54006c2f97391656766147",
      "family_id": "qf-mod50-graddeep-boundary",
      "seed": 202611196079,
      "semantic_template": "graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[4,1],\"step\":[1,5]}}"
    },
    "fields": {
      "title": "Stationarity, Metric Identification and Finite Differences Have Different Responsibilities",
      "stem": "Use mode=finite_difference_confusion, complete model={\"difference_quotient\":4,\"step\":[1,5]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(finite_difference_confusion, 4, 1/5, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)",
        "explanation": "A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 4, 1/5, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)."
      },
      "skills": [
        "Stationarity, Metric Identification and Finite Differences Have Different Responsibilities"
      ],
      "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": "fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":4,\"step\":[1,5]}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use mode=finite_difference_confusion, complete model={\"difference_quotient\":4,\"step\":[1,5]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-graddeep-boundary, [domain, mode, model, operation]]",
      "after": "derived = [graddeep-boundary, Exact_gradient_vector_density]",
      "relation": "implies",
      "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。",
        "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [finite difference ORIGINAL quotient, 100000000, 4, 1]",
      "after": "derived = [[4, 1], 4]",
      "relation": "implies",
      "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。",
        "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "finite difference ORIGINAL quotient ORIGINAL numerator over denominator := div(4, 1)",
      "after": "finite difference ORIGINAL quotient ORIGINAL numerator over denominator = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite difference ORIGINAL quotient ORIGINAL numerator over denominator，不以小數近似替代。",
          "en": "Recompute finite difference ORIGINAL quotient ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "finite difference ORIGINAL quotient ORIGINAL numerator over denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [finite difference ORIGINAL step, 100000000, 1, 5]",
      "after": "derived = [[1, 5], 1/5]",
      "relation": "implies",
      "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。",
        "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "finite difference ORIGINAL step ORIGINAL numerator over denominator := div(1, 5)",
      "after": "finite difference ORIGINAL step ORIGINAL numerator over denominator = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。",
        "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 finite difference ORIGINAL step ORIGINAL numerator over denominator，不以小數近似替代。",
          "en": "Recompute finite difference ORIGINAL step ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "5"
        ],
        "output": "1/5",
        "label": "finite difference ORIGINAL step ORIGINAL numerator over denominator"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [4, 1/5]",
      "after": "derived = [否]",
      "relation": "implies",
      "rule_id": "Gradient_ONE_finite_difference_not_derivative_limit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "單一非零step的差商未提供h→0整個極限責任，所以不認證derivative或gradient。此No是證書不足，不是宣稱指定真導數數值一定不同。",
        "en": "ONE quotient at a nonzero step does not supply the full h→0 limit obligation,so it does not certify a derivative or gradient.No withholds certification;it does not assert the true derivative must have a different numerical value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。",
          "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "finite_difference_confusion",
      "4",
      "1/5",
      false,
      "one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)",
    "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": {
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    "original_candidate_sha256": "6b63f4b246df69db0aed88dc20404a58b25122ee1437857acf8f0056847422c2",
    "original_candidate_line_sha256": "6b63f4b246df69db0aed88dc20404a58b25122ee1437857acf8f0056847422c2"
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  "historical_source_state": {
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      "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-20261026-012-p67",
    "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": {
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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}