{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r07-lp-convergence-0035",
  "pilot_number": "R0035-18210",
  "number": 18210,
  "family_id": "qf-mod35-lp-epsilon-witness",
  "title": {
    "zh-Hant": "第 18210 題：Lp收斂",
    "en": "Question 18210: Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r07-lp-convergence-0035",
    "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-epsilon-witness；資料={\"masses\":[[1,2],[1,2]],\"coefficients\":[10,-11],\"shift\":2,\"p\":1,\"epsilon\":[1,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 1/19, 198, 21/400, 是, forall_n_ge_N_symbolic_monotone_tail)",
      "explanation_zh": "Wave5第四層核對=(1, 1/19, 198, 21/400, 是, forall_n_ge_N_symbolic_monotone_tail)；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-epsilon-witness",
      "seed": 202611047035,
      "parameters": {
        "masses": [
          [
            1,
            2
          ],
          [
            1,
            2
          ]
        ],
        "coefficients": [
          10,
          -11
        ],
        "shift": 2,
        "p": 1,
        "epsilon": [
          1,
          19
        ],
        "domain": "Finite_atomic_symbolic_Lp_tail",
        "operation": "lp-epsilon-witness",
        "semantic_template": "qf-mod35-lp-epsilon-witness/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"masses\":[[1,2],[1,2]],\"p\":1,\"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-0035",
    "batch_id": "batch-20261014-advance-20261104-r07-lp-convergence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "73517ae049993ca4a129791668bcfd3a2dea3678062fa1a6dba3ff98d0a41bd6",
      "family_id": "qf-mod35-lp-epsilon-witness",
      "seed": 202611047035,
      "semantic_template": "qf-mod35-lp-epsilon-witness/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"masses\":[[1,2],[1,2]],\"p\":1,\"shift\":2}"
    },
    "fields": {
      "title": "Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number",
      "stem": "Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 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. Positive input epsilon=1/19 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail)",
        "explanation": "Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail)."
      },
      "skills": [
        "Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number"
      ],
      "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-epsilon-witness；資料={\"masses\":[[1,2],[1,2]],\"coefficients\":[10,-11],\"shift\":2,\"p\":1,\"epsilon\":[1,19]}。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=[10, -11], positive shift=2, and fixed integer p=1 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. Positive input epsilon=1/19 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=lp-epsilon-witness;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[p, moment_tolerance, N, p_moment_at_N, strict_tail, 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, 10)",
      "after": "Lp coefficient atom0 absolute power product stage1 = 10",
      "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",
          "10"
        ],
        "output": "10",
        "label": "Lp coefficient atom0 absolute power product stage1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Lp constant weighted coefficient atom0 := mul(1/2, 10)",
      "after": "Lp constant weighted coefficient atom0 = 5",
      "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",
          "10"
        ],
        "output": "5",
        "label": "Lp constant weighted coefficient atom0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original atom=0;mass=1/2;SIGNED coefficient=10;fixed integer p=1",
      "after": "absolute coefficient power=10;weighted coefficient power=5",
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Lp coefficient atom1 absolute power product stage1 := mul(1, 11)",
      "after": "Lp coefficient atom1 absolute power product stage1 = 11",
      "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",
          "11"
        ],
        "output": "11",
        "label": "Lp coefficient atom1 absolute power product stage1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Lp constant weighted coefficient atom1 := mul(1/2, 11)",
      "after": "Lp constant weighted coefficient atom1 = 11/2",
      "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",
          "11"
        ],
        "output": "11/2",
        "label": "Lp constant weighted coefficient atom1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original atom=1;mass=1/2;SIGNED coefficient=-11;fixed integer p=1",
      "after": "absolute coefficient power=11;weighted coefficient power=11/2",
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Lp full weighted absolute coefficient power sum := sum(5, 11/2)",
      "after": "Lp full weighted absolute coefficient power sum = 21/2",
      "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": [
          "5",
          "11/2"
        ],
        "output": "21/2",
        "label": "Lp full weighted absolute coefficient power sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[10, -11];shift=2;fixed p=1;weighted constant=21/2",
      "after": "for EVERY n>=0,p-moment E|error_n|^p=(21/2)/(n+2)^1;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
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Lp strict moment comparison numerator := mul(21, 19)",
      "after": "Lp strict moment comparison numerator = 399",
      "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 strict moment comparison numerator，不以小數近似替代。",
          "en": "Recompute Lp strict moment comparison numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "19"
        ],
        "output": "399",
        "label": "Lp strict moment comparison numerator"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Lp strict moment comparison coefficient := mul(2, 1)",
      "after": "Lp strict moment comparison coefficient = 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 strict moment comparison coefficient，不以小數近似替代。",
          "en": "Recompute Lp strict moment comparison coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Lp strict moment comparison coefficient"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Lp least cutoff positive denominator := sum(198, 2)",
      "after": "Lp least cutoff positive denominator = 200",
      "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 least cutoff positive denominator，不以小數近似替代。",
          "en": "Recompute Lp least cutoff positive denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "198",
          "2"
        ],
        "output": "200",
        "label": "Lp least cutoff positive denominator"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Lp cutoff denominator power product stage1 := mul(1, 200)",
      "after": "Lp cutoff denominator power product stage1 = 200",
      "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 cutoff denominator power product stage1，不以小數近似替代。",
          "en": "Recompute Lp cutoff denominator power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "200"
        ],
        "output": "200",
        "label": "Lp cutoff denominator power product stage1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Lp strict cutoff integer right side := mul(2, 200)",
      "after": "Lp strict cutoff integer right side = 400",
      "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 strict cutoff integer right side，不以小數近似替代。",
          "en": "Recompute Lp strict cutoff integer right side with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "200"
        ],
        "output": "400",
        "label": "Lp strict cutoff integer right side"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Lp previous denominator := sub(200, 1)",
      "after": "Lp previous denominator = 199",
      "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 previous denominator，不以小數近似替代。",
          "en": "Recompute Lp previous denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "200",
          "1"
        ],
        "output": "199",
        "label": "Lp previous denominator"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Lp previous denominator power product stage1 := mul(1, 199)",
      "after": "Lp previous denominator power product stage1 = 199",
      "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 previous denominator power product stage1，不以小數近似替代。",
          "en": "Recompute Lp previous denominator power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "199"
        ],
        "output": "199",
        "label": "Lp previous denominator power product stage1"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Lp previous integer right side := mul(2, 199)",
      "after": "Lp previous integer right side = 398",
      "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 previous integer right side，不以小數近似替代。",
          "en": "Recompute Lp previous integer right side with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "199"
        ],
        "output": "398",
        "label": "Lp previous integer right side"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "cutoff N=198;denominator=200;integer left=399;current right=400;previous right=398",
      "after": "strict comparison holds at N and fails at N-1;all smaller allowed indices fail",
      "relation": "implies",
      "rule_id": "lp_least_moment_cutoff_integer_power_not_float_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交叉乘正數後比較整數p次方，前一項失敗且右邊隨n單調，證最小N；不以浮點p次根四捨五入。",
        "en": "Cross-multiply positive factors and compare exact integer p-th powers.Failure at the previous index and monotonicity prove the least N,without rounding a floating-point root."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Lp cutoff evaluation common error denominator := sum(198, 2)",
      "after": "Lp cutoff evaluation common error denominator = 200",
      "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 cutoff evaluation common error denominator，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation common error denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "198",
          "2"
        ],
        "output": "200",
        "label": "Lp cutoff evaluation common error denominator"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Lp cutoff evaluation atom0 absolute error amplitude := div(10, 200)",
      "after": "Lp cutoff evaluation atom0 absolute error amplitude = 1/20",
      "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 cutoff evaluation atom0 absolute error amplitude，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom0 absolute error amplitude with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "200"
        ],
        "output": "1/20",
        "label": "Lp cutoff evaluation atom0 absolute error amplitude"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Lp cutoff evaluation atom0 error power product stage1 := mul(1, 1/20)",
      "after": "Lp cutoff evaluation atom0 error power product stage1 = 1/20",
      "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 cutoff evaluation atom0 error power product stage1，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom0 error power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/20"
        ],
        "output": "1/20",
        "label": "Lp cutoff evaluation atom0 error power product stage1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 1/20)",
      "after": "Lp cutoff evaluation atom0 weighted moment term = 1/40",
      "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 cutoff evaluation atom0 weighted moment term，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom0 weighted moment term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/20"
        ],
        "output": "1/40",
        "label": "Lp cutoff evaluation atom0 weighted moment term"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Lp cutoff evaluation atom1 absolute error amplitude := div(11, 200)",
      "after": "Lp cutoff evaluation atom1 absolute error amplitude = 11/200",
      "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 cutoff evaluation atom1 absolute error amplitude，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom1 absolute error amplitude with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "11",
          "200"
        ],
        "output": "11/200",
        "label": "Lp cutoff evaluation atom1 absolute error amplitude"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Lp cutoff evaluation atom1 error power product stage1 := mul(1, 11/200)",
      "after": "Lp cutoff evaluation atom1 error power product stage1 = 11/200",
      "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 cutoff evaluation atom1 error power product stage1，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom1 error power product stage1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "11/200"
        ],
        "output": "11/200",
        "label": "Lp cutoff evaluation atom1 error power product stage1"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 11/200)",
      "after": "Lp cutoff evaluation atom1 weighted moment term = 11/400",
      "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 cutoff evaluation atom1 weighted moment term，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation atom1 weighted moment term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "11/200"
        ],
        "output": "11/400",
        "label": "Lp cutoff evaluation atom1 weighted moment term"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Lp cutoff evaluation full p-moment := sum(1/40, 11/400)",
      "after": "Lp cutoff evaluation full p-moment = 21/400",
      "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 cutoff evaluation full p-moment，不以小數近似替代。",
          "en": "Recompute Lp cutoff evaluation full p-moment with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/40",
          "11/400"
        ],
        "output": "21/400",
        "label": "Lp cutoff evaluation full p-moment"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Lp cutoff evaluation:original n=198;fixed p=1;all weighted absolute-power terms=[1/40, 11/400]",
      "after": "exact p-moment=21/400;neither its p-th root nor unweighted atom mean",
      "relation": "implies",
      "rule_id": "lp_selected_exact_moment_not_root_norm",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把每個原子振幅的絕對p次方乘mass後加全部，沒有再取p次根或除原子數。",
        "en": "Sum all mass-weighted absolute p-th-power amplitudes,without taking a p-th root or dividing by the atom count."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "moment_tolerance epsilon=1/19;fixed p=1;least N=198;EXACT moment at N=21/400",
      "after": "for EVERY later n,moment=K/(n+shift)^p<=moment_at_N<epsilon;norm bound with SAME epsilon would instead require moment<epsilon^1",
      "relation": "implies",
      "rule_id": "lp_moment_tolerance_not_same_numeric_norm_tolerance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原epsilon直接界定E|error|^p，不能偷換成norm<同一epsilon。固定p時兩者趨零等價，但單個數值門檻不同；本見證依來源保moment criterion與所有後續n。",
        "en": "The source epsilon directly bounds E|error|^p,not the norm by the same number.For fixed p both tend-to-zero notions agree,but their numerical thresholds differ.Keep the source moment criterion and its all-later guarantee."
      },
      "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": [
      1,
      "1/19",
      198,
      "21/400",
      true,
      "forall_n_ge_N_symbolic_monotone_tail"
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 1/19, 198, 21/400, 是, forall_n_ge_N_symbolic_monotone_tail)",
    "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",
      "original_line": 35,
      "original_line_sha256": "b4cce535b614ce20945bbaee679db251ca55400e56fa971bf6796dcf8ff9c370",
      "question_compact_sha256": "73517ae049993ca4a129791668bcfd3a2dea3678062fa1a6dba3ff98d0a41bd6",
      "english_row_compact_sha256": "f280849ef2f1ec6d685e576ee06754f887890a91327ddfd3ec502ac036190118",
      "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"
    },
    "mko_bindings": {
      "mko-lp-convergence": {
        "sha256": "aec69290c870938d15b40e7ffb6849ffa5fa53265e0fa29d59021aa1f01ce47f",
        "english_sha256": "c406a88686066c71f8dd62d25eb6931b8411e40497c57a3e0826bbddfc8655cb",
        "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": "6f7ebdcc9c0a0fb971fcb9a131cad7474252ff0047b85c0497d5f17e1051585d",
    "original_candidate_line_sha256": "6f7ebdcc9c0a0fb971fcb9a131cad7474252ff0047b85c0497d5f17e1051585d"
  },
  "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-20261026-008-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"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 29,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "52dfb6ae63c659831751b5b610758aada7cbded53faeda477bbf2109a43f27ad",
      "arithmetic_steps": 22
    },
    "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."
  }
}