{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261031-r01-at-most-countable-set-0048",
  "pilot_number": "R0030-15823",
  "number": 15823,
  "family_id": "qf-mod31-countable-affine-injection",
  "title": {
    "zh-Hant": "第 15823 題：至多可數集",
    "en": "Question 15823: A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261031-r01-at-most-countable-set-0048",
    "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-at-most-countable-set"
    ],
    "title_zh": "至多可數集",
    "stem_zh": "有限完整清單或明示全稱注入證書。操作=countable-affine-injection；資料={\"slope\":3,\"intercept\":10,\"left\":-23,\"right\":24}。請依 [slope,intercept,left,right,image_left,image_right,injective_check,universal_certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3, 10, -23, 24, -59, 82, 是, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)",
      "explanation_zh": "Wave4收束核對=(3, 10, -23, 24, -59, 82, 是, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)；依完整明示資料重算；有限與可數無限不可混同；單一失敗encoding不證不可數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "至多可數集"
      ]
    },
    "variant": {
      "family_id": "qf-mod31-countable-affine-injection",
      "seed": 202610311048,
      "parameters": {
        "slope": 3,
        "intercept": 10,
        "left": -23,
        "right": 24,
        "domain": "Explicit_at_most_countable_certificate",
        "operation": "countable-affine-injection",
        "semantic_template": "qf-mod31-countable-affine-injection/{\"intercept\":10,\"left\":-23,\"right\":24,\"slope\":3}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r01-at-most-countable-set-0048",
    "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5c0ee4681afa9c43351bfeb771a84d39e1f5d2b43ec25def360930e2eb345c1c",
      "family_id": "qf-mod31-countable-affine-injection",
      "seed": 202610311048,
      "semantic_template": "qf-mod31-countable-affine-injection/{\"intercept\":10,\"left\":-23,\"right\":24,\"slope\":3}"
    },
    "fields": {
      "title": "A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess",
      "stem": "On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).",
      "answer": {
        "kind": "expression",
        "value": "(3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)",
        "explanation": "A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."
      },
      "skills": [
        "A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"
      ],
      "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": "有限完整清單或明示全稱注入證書。操作=countable-affine-injection；資料={\"slope\":3,\"intercept\":10,\"left\":-23,\"right\":24}。請依 [slope,intercept,left,right,image_left,image_right,injective_check,universal_certificate] 欄序回答。",
        "en": "On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "affine left nonzero-slope product := mul(3, -23)",
      "after": "affine left nonzero-slope product = -69",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 affine left nonzero-slope product，不以小數近似替代。",
          "en": "Recompute affine left nonzero-slope product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "-23"
        ],
        "output": "-69",
        "label": "affine left nonzero-slope product"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "affine original left image := sum(-69, 10)",
      "after": "affine original left image = -59",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 affine original left image，不以小數近似替代。",
          "en": "Recompute affine original left image with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-69",
          "10"
        ],
        "output": "-59",
        "label": "affine original left image"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "affine right nonzero-slope product := mul(3, 24)",
      "after": "affine right nonzero-slope product = 72",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 affine right nonzero-slope product，不以小數近似替代。",
          "en": "Recompute affine right nonzero-slope product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "24"
        ],
        "output": "72",
        "label": "affine right nonzero-slope product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "affine original right image := sum(72, 10)",
      "after": "affine original right image = 82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 affine original right image，不以小數近似替代。",
          "en": "Recompute affine original right image with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "72",
          "10"
        ],
        "output": "82",
        "label": "affine original right image"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "fixed integer map f(x)=(3)*x+(10);nonzero slope=3",
      "after": "for ANY integers x,y:f(x)=f(y) implies(3)*(x-y)=0,hence x=y",
      "relation": "implies",
      "rule_id": "affine_universal_injection_by_integer_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先同減截距，得到非零整數斜率乘(x−y)為0；整數無零因子才消去斜率，對所有x,y成立。選定左右兩點的核算不是全稱注入性的證明來源。",
        "en": "Subtract the same intercept;the nonzero integer slope times(x−y) is zero.Cancellation in the integers gives x=y for every pair.The displayed left/right check is not the source of universal injectivity."
      },
      "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": "f maps Z injectively to Z;selected images=[-59, 82];selected-pair implication=是",
      "after": "encode integer z into N_0 by E(z)=2*z if z>=0,else -2*z-1;even/odd branches disjoint and each injective;E composed with f is injective",
      "relation": "implies",
      "rule_id": "integer_to_natural_encoding_parity_and_universal_composition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正負分支分別輸出非負偶數與非負奇數，異分支不撞，同分支可消去±2。因此合成到自然數的注入，才是至多可數證書；負斜率與相同選定輸入皆合法。",
        "en": "The two branches yield nonnegative even and odd values,so cross-branch collisions are impossible;within each branch cancel±2.Composition gives an injection into the naturals.Negative slopes and equal selected inputs are allowed."
      },
      "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": [
      3,
      10,
      -23,
      24,
      -59,
      82,
      true,
      "universal_nonzero_slope_injection_Z_to_Z_then_encode_N"
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 10, -23, 24, -59, 82, 是, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)",
    "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-20261031-r01-at-most-countable-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/questions.jsonl",
      "batch_sha256": "7b382f15a68b0f6e0c1e44dc793ede056d8bd39aff2ca4e878a95654740ee01a",
      "original_line": 48,
      "original_line_sha256": "39b40fac4503161520c28fd575f1af8461f4192b1a80423685601327c49ccf68",
      "question_compact_sha256": "5c0ee4681afa9c43351bfeb771a84d39e1f5d2b43ec25def360930e2eb345c1c",
      "english_row_compact_sha256": "902082796daff1b235402413e85fc5eedee56de97eb863200466600e86faa581",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-at-most-countable-set": {
        "sha256": "9c7748686b7974341920a59d07628f20017ef6d65476d751b2feb7424f7b3b57",
        "english_sha256": "a73faf7f1c04553efad774692eab18a2de59165f0fedaac3ba03ee75c7276def",
        "original_path": "public/data/mko/at-most-countable-set.json",
        "english_path": "public/data/i18n/editions/mko-at-most-countable-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3",
    "original_candidate_sha256": "6407416881abacef2b1a804e8fec62b034ac9a4192f9e76e4156bfc8dada5331",
    "original_candidate_line_sha256": "6407416881abacef2b1a804e8fec62b034ac9a4192f9e76e4156bfc8dada5331"
  },
  "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-20261026-007-p31",
    "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": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc",
      "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f74d855d0ec6d9f8798c293f9999ef79b5f22f09ccd1adc5bab0d0e65f1a6ffc",
      "arithmetic_steps": 4
    },
    "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."
  }
}