{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r07-subsequence-0057",
  "pilot_number": "R0039-20432",
  "number": 20432,
  "family_id": "qf-mod38-subseq-composed-index",
  "title": {
    "zh-Hant": "第 20432 題：子數列",
    "en": "Question 20432: Compose the Inner Index into the Outer Index in That Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r07-subsequence-0057",
    "batch_id": "batch-20261014-advance-20261107-r07-subsequence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-subsequence"
    ],
    "title_zh": "子數列",
    "stem_zh": "自然數索引與symbolic嚴格遞增證書。操作=subseq-composed-index；資料={\"outer_start\":0,\"outer_step\":2,\"inner_start\":2,\"inner_step\":1,\"j\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, 9, 18, 2, 是, composition_remains_strictly_increasing)",
      "explanation_zh": "Wave6根層核對=(7, 9, 18, 2, 是, composition_remains_strictly_increasing)；依完整明示資料獨立重算；subsequence索引必須嚴格遞增並保留原數列次序，不是重排或任意子集合；有限索引列表只證finite prefix；無限subsequence需要通用increasing-index certificate。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "子數列"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-subseq-composed-index",
      "seed": 202611077057,
      "parameters": {
        "outer_start": 0,
        "outer_step": 2,
        "inner_start": 2,
        "inner_step": 1,
        "j": 7,
        "domain": "Symbolic_affine_subsequence_index",
        "operation": "subseq-composed-index",
        "semantic_template": "qf-mod38-subseq-composed-index/{\"inner_start\":2,\"inner_step\":1,\"j\":7,\"outer_start\":0,\"outer_step\":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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r07-subsequence-0057",
    "batch_id": "batch-20261014-advance-20261107-r07-subsequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "69012b74ca715c54434b854be74291adce1c0fa34e28cb65794475339cc91c97",
      "family_id": "qf-mod38-subseq-composed-index",
      "seed": 202611077057,
      "semantic_template": "qf-mod38-subseq-composed-index/{\"inner_start\":2,\"inner_step\":1,\"j\":7,\"outer_start\":0,\"outer_step\":2}"
    },
    "fields": {
      "title": "Compose the Inner Index into the Outer Index in That Order",
      "stem": "Use outer index k(i)=0+2i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=7. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(7, 9, 18, 2, Yes, composition_remains_strictly_increasing)",
        "explanation": "Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (7, 9, 18, 2, Yes, composition_remains_strictly_increasing)."
      },
      "skills": [
        "Compose the Inner Index into the Outer Index in That Order"
      ],
      "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": "自然數索引與symbolic嚴格遞增證書。操作=subseq-composed-index；資料={\"outer_start\":0,\"outer_step\":2,\"inner_start\":2,\"inner_step\":1,\"j\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use outer index k(i)=0+2i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=7. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=subseq-composed-index;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[j, inner_index, source_index, composed_step, strictly_increasing, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。",
        "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source 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": "outer index k:ORIGINAL nonnegative start=0;positive integer step=2;F(q)=start+step*q for EVERY natural q>=0",
      "after": "F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(2)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix",
      "relation": "implies",
      "rule_id": "subseq_affine_all_natural_index_order_not_value_monotonicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正整數step讓任意兩個自然索引的差都保持正，故保原順序，不只是單射或任意重排。此全稱數學law不以某個數值輸出安全範圍截斷，也不宣稱原數列的值遞增。",
        "en": "A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "inner index l:ORIGINAL nonnegative start=2;positive integer step=1;F(q)=start+step*q for EVERY natural q>=0",
      "after": "F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(1)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix",
      "relation": "implies",
      "rule_id": "subseq_affine_all_natural_index_order_not_value_monotonicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正整數step讓任意兩個自然索引的差都保持正，故保原順序，不只是單射或任意重排。此全稱數學law不以某個數值輸出安全範圍截斷，也不宣稱原數列的值遞增。",
        "en": "A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "subsequence inner step times ORIGINAL j := mul(1, 7)",
      "after": "subsequence inner step times ORIGINAL j = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 subsequence inner step times ORIGINAL j，不以小數近似替代。",
          "en": "Recompute subsequence inner step times ORIGINAL j with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "7"
        ],
        "output": "7",
        "label": "subsequence inner step times ORIGINAL j"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "subsequence ORIGINAL inner index l(j) := sum(2, 7)",
      "after": "subsequence ORIGINAL inner index l(j) = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 subsequence ORIGINAL inner index l(j)，不以小數近似替代。",
          "en": "Recompute subsequence ORIGINAL inner index l(j) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "7"
        ],
        "output": "9",
        "label": "subsequence ORIGINAL inner index l(j)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "subsequence outer step times INNER result := mul(2, 9)",
      "after": "subsequence outer step times INNER result = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 subsequence outer step times INNER result，不以小數近似替代。",
          "en": "Recompute subsequence outer step times INNER result with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "9"
        ],
        "output": "18",
        "label": "subsequence outer step times INNER result"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "subsequence ORIGINAL composed index k(l(j)) := sum(0, 18)",
      "after": "subsequence ORIGINAL composed index k(l(j)) = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 subsequence ORIGINAL composed index k(l(j))，不以小數近似替代。",
          "en": "Recompute subsequence ORIGINAL composed index k(l(j)) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "18"
        ],
        "output": "18",
        "label": "subsequence ORIGINAL composed index k(l(j))"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "subsequence composed positive step := mul(2, 1)",
      "after": "subsequence composed positive step = 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代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 subsequence composed positive step，不以小數近似替代。",
          "en": "Recompute subsequence composed positive step with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "subsequence composed positive step"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ORIGINAL outer k(i)=0+2i;inner l(j)=2+1j;actual j=7;inner value=9",
      "after": "outer(inner(j))=18;for EVERY j,composed consecutive difference=2>0;exact evaluated fields within SAFE integer range",
      "relation": "implies",
      "rule_id": "subseq_composition_inner_then_outer_all_j_positive_step",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把j送進inner，再把inner索引送進outer，不能倒置；所有自然j的合成差是兩正step的乘積。SAFE限制只處理本次明列的數字投影，並非限制無限索引law的數學定義域。",
        "en": "Apply inner to j first,then feed that index into outer,not the reverse.The composed difference is the product of positive steps for every natural j.The SAFE check concerns explicitly evaluated numeric fields,not the mathematical domain of the infinite index law."
      },
      "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": [
      7,
      9,
      18,
      2,
      true,
      "composition_remains_strictly_increasing"
    ]
  },
  "conclusion": {
    "canonical_value": "(7, 9, 18, 2, 是, composition_remains_strictly_increasing)",
    "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-20261107-r07-subsequence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r07-subsequence-foundations/questions.jsonl",
      "batch_sha256": "9ba8fa0ce8414b0f93d4e18a0af4fe49e8c59529895aa0a7d8920db4ef56b98f",
      "original_line": 57,
      "original_line_sha256": "ddffe6dc9b70cbd586c9c95dcde2b029e6e62d57ce37d58ca3703cefa1e7f4c8",
      "question_compact_sha256": "69012b74ca715c54434b854be74291adce1c0fa34e28cb65794475339cc91c97",
      "english_row_compact_sha256": "d0d3e0749305e0b70b36f0107808345feb1d055ad03892304e73993301b5ff63",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r07-subsequence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r07-subsequence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-subsequence": {
        "sha256": "f4791c70590c6e099961027b14a591894bd691565f6e1bc8cd191792d0e2a44f",
        "english_sha256": "e54379f1fd9f6ccb9ef4be2d69a165c97d4e793ec8b2ecf051c61d8e2163b155",
        "original_path": "public/data/mko/source13-20261018-subsequence.json",
        "english_path": "public/data/i18n/editions/mko-subsequence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "521929dfe9b0919224c38312ad78e5fa56457fc886faaa12b137edd83962251e",
    "original_candidate_sha256": "df884af618c1552b1f0ee6f2b512fdb8cb575349fb7241ae608980e71addf929",
    "original_candidate_line_sha256": "df884af618c1552b1f0ee6f2b512fdb8cb575349fb7241ae608980e71addf929"
  },
  "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-p40",
    "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": "6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034",
      "question_first_witness_sha256": "cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7a410bd9db99c3eb1ab91d08417aaa42faae09f0bfed6835b852bc403826b11e",
      "arithmetic_steps": 5
    },
    "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."
  }
}