{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r06-0076",
  "pilot_number": "R0066-33826",
  "number": 33826,
  "family_id": "qf-mod50-graddeep-boundary",
  "title": {
    "zh-Hant": "第 33826 題：梯度向量",
    "en": "Question 33826: Stationarity, Metric Identification and Finite Differences Have Different Responsibilities"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r06-0076",
    "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\":\"zero_gradient\",\"model\":{\"dimension\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)",
      "explanation_zh": "Wave6第十三層核對=(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)；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": 202611196076,
      "parameters": {
        "mode": "zero_gradient",
        "model": {
          "dimension": 1
        },
        "domain": "Exact_gradient_vector_density",
        "operation": "graddeep-boundary",
        "semantic_template": "graddeep-boundary|{\"mode\":\"zero_gradient\",\"model\":{\"dimension\":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-r06-0076",
    "batch_id": "batch-20261015-advance-20261119-r06-gradient-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d0e12d9102a389dda4e36c9d53d978eeaff698770061a3d6042c7a1b5ea20249",
      "family_id": "qf-mod50-graddeep-boundary",
      "seed": 202611196076,
      "semantic_template": "graddeep-boundary|{\"mode\":\"zero_gradient\",\"model\":{\"dimension\":1}}"
    },
    "fields": {
      "title": "Stationarity, Metric Identification and Finite Differences Have Different Responsibilities",
      "stem": "Use mode=zero_gradient, complete model={\"dimension\":1}. Give (mode, dimension, zero_gradient, stationary, minimum_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(zero_gradient, 1, [0], Yes, No, zero_gradient_only_certifies_stationarity)",
        "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 (zero_gradient, 1, [0], Yes, No, zero_gradient_only_certifies_stationarity)."
      },
      "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\":\"zero_gradient\",\"model\":{\"dimension\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use mode=zero_gradient, complete model={\"dimension\":1}. Give (mode, dimension, zero_gradient, stationary, minimum_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 = [zero_gradient, 1]",
      "after": "derived = [[0], 是, 否]",
      "relation": "implies",
      "rule_id": "Gradient_ZERO_stationary_not_minimum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零gradient只定義stationary；例如−sum x_i²在0的gradient0卻是極大，不能以zero gradient認證minimum。",
        "en": "A zero gradient certifies stationarity ONLY.For example−sum x_i² has gradient0 at0 but a maximum,so zero gradient cannot certify a minimum."
      },
      "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": [
      "zero_gradient",
      1,
      [
        "0"
      ],
      true,
      false,
      "zero_gradient_only_certifies_stationarity"
    ]
  },
  "conclusion": {
    "canonical_value": "(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)",
    "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-r06-gradient-vector-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r06-gradient-vector-foundations/questions.jsonl",
      "batch_sha256": "7aeac0f5d0efe49c24d6c45348a1a0f24e40ba206983593e56a4cf547072e1ff",
      "original_line": 76,
      "original_line_sha256": "fbc835823ba37878df9cf1b9dddd5dc796ad996f6595ddf1998ae649b3ed4703",
      "question_compact_sha256": "d0e12d9102a389dda4e36c9d53d978eeaff698770061a3d6042c7a1b5ea20249",
      "english_row_compact_sha256": "21c831826b922b93e6698a4dca3793f8aeb98e0a4641f8b08cd4eda0f39d6750",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r06-gradient-vector-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r06-gradient-vector-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-gradient-vector": {
        "sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa",
        "english_sha256": "6ba53da50d59e50b734006c70a0c1e5cb7e4986a4a57e6d0b1f27a8ae84bd363",
        "original_path": "public/data/mko/gradient-vector.json",
        "english_path": "public/data/i18n/editions/mko-gradient-vector.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310",
    "original_candidate_sha256": "a606fe7e17f3e22728623b21d875142d85e1b88c9cf9cc2421c1cd66bdaf0e91",
    "original_candidate_line_sha256": "a606fe7e17f3e22728623b21d875142d85e1b88c9cf9cc2421c1cd66bdaf0e91"
  },
  "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-20261026-012-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": {
      "semantic_review_sha256": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267",
      "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d00e3f5cf93866e7eebe3ebee97a59a1c64b5cb1d5ee3936bc957db8ff9d199f",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}