{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r07-lp-convergence-0064",
  "pilot_number": "R0035-18289",
  "number": 18289,
  "family_id": "qf-mod35-lp-qualification",
  "title": {
    "zh-Hant": "第 18289 題：Lp收斂",
    "en": "Question 18289: Keep the Fixed p, Common Law and Whole Symbolic Tail Together"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r07-lp-convergence-0064",
    "batch_id": "batch-20261014-advance-20261104-r07-lp-convergence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-lp-convergence"
    ],
    "title_zh": "Lp收斂",
    "stem_zh": "同一有限原子probability space、固定整數p大於等於1與symbolic moment tail。操作=lp-qualification；資料={\"masses\":[[1,2],[1,2]],\"coefficients\":[14,-15],\"shift\":2,\"p\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, [1/2, 1/2], [14, -15], 2, 是, common_probability_space_and_finite_integer_p_moments)",
      "explanation_zh": "Wave5第四層核對=(2, [1/2, 1/2], [14, -15], 2, 是, common_probability_space_and_finite_integer_p_moments)；依完整明示資料重算；核E absolute error power p趨零且moments有限；finite prefix不足，a.s. probability或其他Lq implications另需假設。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Lp收斂"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-lp-qualification",
      "seed": 202611047064,
      "parameters": {
        "masses": [
          [
            1,
            2
          ],
          [
            1,
            2
          ]
        ],
        "coefficients": [
          14,
          -15
        ],
        "shift": 2,
        "p": 2,
        "domain": "Finite_atomic_symbolic_Lp_tail",
        "operation": "lp-qualification",
        "semantic_template": "qf-mod35-lp-qualification/{\"coefficients\":[14,-15],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261104-r07-lp-convergence-0064",
    "batch_id": "batch-20261014-advance-20261104-r07-lp-convergence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "af46efed1dba35ca378cf1de702f878d62c1e4e5c5ca17a756f1178d009d56c3",
      "family_id": "qf-mod35-lp-qualification",
      "seed": 202611047064,
      "semantic_template": "qf-mod35-lp-qualification/{\"coefficients\":[14,-15],\"masses\":[[1,2],[1,2]],\"p\":2,\"shift\":2}"
    },
    "fields": {
      "title": "Keep the Fixed p, Common Law and Whole Symbolic Tail Together",
      "stem": "Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments)",
        "explanation": "The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments)."
      },
      "skills": [
        "Keep the Fixed p, Common Law and Whole Symbolic Tail Together"
      ],
      "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": "同一有限原子probability space、固定整數p大於等於1與symbolic moment tail。操作=lp-qualification；資料={\"masses\":[[1,2],[1,2]],\"coefficients\":[14,-15],\"shift\":2,\"p\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=lp-qualification;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[p, masses, coefficients, shift, qualified, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題的歷史介面標記保留在快照；本頁直接列原始欄位順序，不需JavaScript或外部表。欄位描述不自動成為數學證據，No標記須依本題的聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Lp complete atomic probability mass := sum(1/2, 1/2)",
      "after": "Lp complete atomic probability mass = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp complete atomic probability mass，不以小數近似替代。",
          "en": "Recompute Lp complete atomic probability mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "Lp complete atomic probability mass"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Lp coefficient atom0 absolute power product stage1 := mul(1, 14)",
      "after": "Lp coefficient atom0 absolute power product stage1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp coefficient atom0 absolute power product stage1，不以小數近似替代。",
          "en": "Recompute Lp coefficient atom0 absolute power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "14"
        ],
        "output": "14",
        "label": "Lp coefficient atom0 absolute power product stage1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Lp coefficient atom0 absolute power product stage2 := mul(14, 14)",
      "after": "Lp coefficient atom0 absolute power product stage2 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp coefficient atom0 absolute power product stage2，不以小數近似替代。",
          "en": "Recompute Lp coefficient atom0 absolute power product stage2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "14"
        ],
        "output": "196",
        "label": "Lp coefficient atom0 absolute power product stage2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Lp constant weighted coefficient atom0 := mul(1/2, 196)",
      "after": "Lp constant weighted coefficient atom0 = 98",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp constant weighted coefficient atom0，不以小數近似替代。",
          "en": "Recompute Lp constant weighted coefficient atom0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "196"
        ],
        "output": "98",
        "label": "Lp constant weighted coefficient atom0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original atom=0;mass=1/2;SIGNED coefficient=14;fixed integer p=2",
      "after": "absolute coefficient power=196;weighted coefficient power=98",
      "relation": "implies",
      "rule_id": "lp_atomic_absolute_power_weight_not_signed_or_unweighted",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取每個signed係數絕對值再作p次方，最後乘原原子mass；不能用signed奇次方或無權平均，零質量原子仍保留原資料。",
        "en": "Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Lp coefficient atom1 absolute power product stage1 := mul(1, 15)",
      "after": "Lp coefficient atom1 absolute power product stage1 = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp coefficient atom1 absolute power product stage1，不以小數近似替代。",
          "en": "Recompute Lp coefficient atom1 absolute power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "15"
        ],
        "output": "15",
        "label": "Lp coefficient atom1 absolute power product stage1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Lp coefficient atom1 absolute power product stage2 := mul(15, 15)",
      "after": "Lp coefficient atom1 absolute power product stage2 = 225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp coefficient atom1 absolute power product stage2，不以小數近似替代。",
          "en": "Recompute Lp coefficient atom1 absolute power product stage2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "15"
        ],
        "output": "225",
        "label": "Lp coefficient atom1 absolute power product stage2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Lp constant weighted coefficient atom1 := mul(1/2, 225)",
      "after": "Lp constant weighted coefficient atom1 = 225/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp constant weighted coefficient atom1，不以小數近似替代。",
          "en": "Recompute Lp constant weighted coefficient atom1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "225"
        ],
        "output": "225/2",
        "label": "Lp constant weighted coefficient atom1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original atom=1;mass=1/2;SIGNED coefficient=-15;fixed integer p=2",
      "after": "absolute coefficient power=225;weighted coefficient power=225/2",
      "relation": "implies",
      "rule_id": "lp_atomic_absolute_power_weight_not_signed_or_unweighted",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取每個signed係數絕對值再作p次方，最後乘原原子mass；不能用signed奇次方或無權平均，零質量原子仍保留原資料。",
        "en": "Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Lp full weighted absolute coefficient power sum := sum(98, 225/2)",
      "after": "Lp full weighted absolute coefficient power sum = 421/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Lp full weighted absolute coefficient power sum，不以小數近似替代。",
          "en": "Recompute Lp full weighted absolute coefficient power sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "98",
          "225/2"
        ],
        "output": "421/2",
        "label": "Lp full weighted absolute coefficient power sum"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[14, -15];shift=2;fixed p=2;weighted constant=421/2",
      "after": "for EVERY n>=0,p-moment E|error_n|^p=(421/2)/(n+2)^2;all moments finite and tend0;this is NOT the p-th-root norm value",
      "relation": "implies",
      "rule_id": "lp_common_space_full_moment_formula_not_norm_or_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "共同空間、固定p與全尾振幅是明示模型。有限多有限係數給moment有限，正分母隨n增大而p≥1，故moment趨零。此欄是moment，不是取p次根後的norm，也不從有限觀察推出。",
        "en": "The common space,fixed p and whole-tail amplitudes are declared.Finitely many finite coefficients give finite moments;the positive denominator grows with n and p≥1,so moments tend0.The field is the moment,not its p-th-root norm,and is not inferred from finite observations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      2,
      [
        "1/2",
        "1/2"
      ],
      [
        14,
        -15
      ],
      2,
      true,
      "common_probability_space_and_finite_integer_p_moments"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, [1/2, 1/2], [14, -15], 2, 是, common_probability_space_and_finite_integer_p_moments)",
    "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-20261014-advance-20261104-r07-lp-convergence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r07-lp-convergence-foundations/questions.jsonl",
      "batch_sha256": "cdf9b0146c4bab5661f3de96c43077fa29c0500df8be10d7335c7859f5d2b750",
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      "question_compact_sha256": "af46efed1dba35ca378cf1de702f878d62c1e4e5c5ca17a756f1178d009d56c3",
      "english_row_compact_sha256": "0ada216300de8364681ef894d4fab5be5315efa79f31e68efe2ac8f5a31f7146",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r07-lp-convergence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r07-lp-convergence-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/lp-convergence.json",
        "english_path": "public/data/i18n/editions/mko-lp-convergence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9adff0bbee39d9c3dec346304161ea9633576ccd5ca954f6b64ad482d2533a8a",
    "original_candidate_sha256": "52be0af9468ce12441e931bd16fda36cece8702a0ac5cc32846131bfb5e763cd",
    "original_candidate_line_sha256": "52be0af9468ce12441e931bd16fda36cece8702a0ac5cc32846131bfb5e763cd"
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    "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-20261026-008-p36",
    "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": "713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc",
      "question_first_witness_sha256": "0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2"
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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1ca2a979fa88e2542e45f432e5c3d2893a401c0600be95abcf09b17c44cbad11",
      "arithmetic_steps": 8
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    "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."
  }
}