{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261004-advance-20261005-r04-inverse-function-0098",
  "pilot_number": "R0064-32723",
  "number": 32723,
  "family_id": "qf-inverse-affine-coefficients",
  "title": {
    "zh-Hant": "第 32723 題：非零斜率仿射反函數的係數",
    "en": "Question 32723: Inverse Coefficients of a Nonzero-Slope Affine Function"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261004-advance-20261005-r04-inverse-function-0098",
    "batch_id": "batch-20261004-advance-20261005-r04-inverse-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-inverse-function"
    ],
    "title_zh": "非零斜率仿射反函數的係數",
    "stem_zh": "在實數論域明示 f:R→R，f(x)=23x+(10)，m=23≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。",
    "answer": {
      "kind": "expression",
      "value": "(1/23, -10/23)",
      "explanation_zh": "g係數=(1/23, -10/23)。由 y=mx+b 解得 x=(y−b)/m，因此 u=1/m、v=−b/m；非零斜率確保此實仿射函數可逆。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "非零斜率仿射反函數的係數"
      ]
    },
    "variant": {
      "family_id": "qf-inverse-affine-coefficients",
      "seed": 202610054098,
      "parameters": {
        "m": 23,
        "b": 10,
        "domain": "reals",
        "operation": "affine_inverse_coefficients",
        "semantic_template": "qf-inverse-affine-coefficients/{\"b\":10,\"domain\":\"reals\",\"m\":23,\"operation\":\"affine_inverse_coefficients\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent producer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-04",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261004-advance-20261005-r04-inverse-function-0098",
    "batch_id": "batch-20261004-advance-20261005-r04-inverse-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8956c491c8561fc354b5336e415e9b89db5aab242a74d8783fc633ca82455c52",
      "family_id": "qf-inverse-affine-coefficients",
      "seed": 202610054098,
      "semantic_template": "qf-inverse-affine-coefficients/{\"b\":10,\"domain\":\"reals\",\"m\":23,\"operation\":\"affine_inverse_coefficients\"}"
    },
    "fields": {
      "title": "Inverse Coefficients of a Nonzero-Slope Affine Function",
      "stem": "Over R, let f:R→R be f(x)=(23)x+(10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(1/23, -10/23)",
        "explanation": "Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/23, -10/23), with nonzero slope ensuring invertibility in this real affine setting."
      },
      "skills": [
        "Inverse Coefficients of a Nonzero-Slope Affine Function"
      ],
      "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": "在實數論域明示 f:R→R，f(x)=23x+(10)，m=23≠0。反函數寫成 g(y)=uy+v，求 (u,v)，每個分量採最簡正分母 p/q，整數也保留 /1。",
        "en": "Over R, let f:R→R be f(x)=(23)x+(10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-inverse-affine-coefficients, affine_inverse_coefficients, reals, []]",
      "after": "derived = [[SLOPE_FIRST_pair_fractions_RETAIN1]]",
      "relation": "implies",
      "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。",
        "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Affine inverse ORIGINAL reciprocal nonzero slope := div(1, 23)",
      "after": "Affine inverse ORIGINAL reciprocal nonzero slope = 1/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse ORIGINAL reciprocal nonzero slope，不以小數近似替代。",
          "en": "Recompute Affine inverse ORIGINAL reciprocal nonzero slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "23"
        ],
        "output": "1/23",
        "label": "Affine inverse ORIGINAL reciprocal nonzero slope"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Affine inverse ORIGINAL negative constant := sub(0, 10)",
      "after": "Affine inverse ORIGINAL negative constant = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse ORIGINAL negative constant，不以小數近似替代。",
          "en": "Recompute Affine inverse ORIGINAL negative constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "10"
        ],
        "output": "-10",
        "label": "Affine inverse ORIGINAL negative constant"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Affine inverse ORIGINAL negative b over m := div(-10, 23)",
      "after": "Affine inverse ORIGINAL negative b over m = -10/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse ORIGINAL negative b over m，不以小數近似替代。",
          "en": "Recompute Affine inverse ORIGINAL negative b over m with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-10",
          "23"
        ],
        "output": "-10/23",
        "label": "Affine inverse ORIGINAL negative b over m"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Affine inverse reverse g after f slope := mul(1/23, 23)",
      "after": "Affine inverse reverse g after f slope = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse g after f slope，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse g after f slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/23",
          "23"
        ],
        "output": "1",
        "label": "Affine inverse reverse g after f slope"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Affine inverse reverse u times original b := mul(1/23, 10)",
      "after": "Affine inverse reverse u times original b = 10/23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse u times original b，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse u times original b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/23",
          "10"
        ],
        "output": "10/23",
        "label": "Affine inverse reverse u times original b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Affine inverse reverse g after f constant := sum(10/23, -10/23)",
      "after": "Affine inverse reverse g after f constant = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse g after f constant，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse g after f constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10/23",
          "-10/23"
        ],
        "output": "0",
        "label": "Affine inverse reverse g after f constant"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Affine inverse reverse f after g slope := mul(23, 1/23)",
      "after": "Affine inverse reverse f after g slope = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse f after g slope，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse f after g slope with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "1/23"
        ],
        "output": "1",
        "label": "Affine inverse reverse f after g slope"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Affine inverse reverse m times inverse constant := mul(23, -10/23)",
      "after": "Affine inverse reverse m times inverse constant = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse m times inverse constant，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse m times inverse constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "-10/23"
        ],
        "output": "-10",
        "label": "Affine inverse reverse m times inverse constant"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Affine inverse reverse f after g constant := sum(-10, 10)",
      "after": "Affine inverse reverse f after g constant = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Affine inverse reverse f after g constant，不以小數近似替代。",
          "en": "Recompute Affine inverse reverse f after g constant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "10"
        ],
        "output": "0",
        "label": "Affine inverse reverse f after g constant"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [23, 10]",
      "after": "derived = [1/23, -10/23, 1, 0, 1, 0]",
      "relation": "implies",
      "rule_id": "Inverse_AFFINE_real_whole_law_SLOPE_FIRST_BOTH_compositions",
      "conditions": [],
      "explanation": {
        "zh-Hant": "m非0才由y=mx+b解x=(y−b)/m，g(y)=uy+v的pair為[slope,constant]。列g(f(x))與f(g(y))的係數／常數都還原，對全部R成立；負m合法，fraction保/1。",
        "en": "ONLY m≠0 permits x=(y−b)/m.The pair for g(y)=uy+v is[SLOPE,constant].BOTH whole compositions restore slope1/constant0 on ALL R.Negative m is legal;fractions retain/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "1/23",
      "-10/23"
    ]
  },
  "conclusion": {
    "canonical_value": "(1/23, -10/23)",
    "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-20261004-advance-20261005-r04-inverse-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r04-inverse-function-foundations/questions.jsonl",
      "batch_sha256": "ff8d56ee8ee70775611adcab47240ff9097e9da25da577e3cd139cf9b146853d",
      "original_line": 98,
      "original_line_sha256": "3680f221d4bf63731b9b98f40ededdb1f4499847dd0c23314ec95a7c4ae3f06f",
      "question_compact_sha256": "8956c491c8561fc354b5336e415e9b89db5aab242a74d8783fc633ca82455c52",
      "english_row_compact_sha256": "323abfdf57ab9b7c592a41646bcf5f7b52636d1abd8b2b5506987ef8e2a3d0ef",
      "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r04-inverse-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r04-inverse-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-inverse-function": {
        "sha256": "bad163b9a218429518937fb4e86abe0e8707ebda0fc06c7e27f625184bc05c4f",
        "english_sha256": "c8aec9f36446191c0529002aef5c9e758f57dd5f6a6f73d01d62310e907d6bea",
        "original_path": "public/data/mko/inverse-function.json",
        "english_path": "public/data/i18n/editions/mko-inverse-function.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470",
    "original_candidate_sha256": "51d70f37ae2f6ef6ef7032a6bc7079e6cd05a462f2abcb46fd522c1e68a0db5f",
    "original_candidate_line_sha256": "51d70f37ae2f6ef6ef7032a6bc7079e6cd05a462f2abcb46fd522c1e68a0db5f"
  },
  "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-20261026-011-p65",
    "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": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b",
      "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "b5f5991b746fd56e101991b9257e569b8121c00fd60b6c7b48b40bb2540ae388",
      "arithmetic_steps": 9
    },
    "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."
  }
}