{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261102-r03-linear-map-image-0095",
  "pilot_number": "R0033-17045",
  "number": 17045,
  "family_id": "qf-mod33-image-bounded-search-boundary",
  "title": {
    "zh-Hant": "第 17045 題：線性映射的像",
    "en": "Question 17045: Failure to Find an Integer Preimage Does Not Establish Real Nonmembership"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261102-r03-linear-map-image-0095",
    "batch_id": "batch-20261014-advance-20261102-r03-linear-map-image-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-map-image"
    ],
    "title_zh": "線性映射的像",
    "stem_zh": "同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[39,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([39, 1], 5, [], 是, bounded_failure_not_global_nonmembership)",
      "explanation_zh": "Wave5第二層核對=([39, 1], 5, [], 是, bounded_failure_not_global_nonmembership)；依完整明示資料重算；image是full-real column span且不等同codomain；bounded fiber search失敗不證全域nonmembership。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "線性映射的像"
      ]
    },
    "variant": {
      "family_id": "qf-mod33-image-bounded-search-boundary",
      "seed": 202611023095,
      "parameters": {
        "matrix": [
          [
            2,
            0
          ],
          [
            0,
            2
          ]
        ],
        "target": [
          39,
          1
        ],
        "bound": 5,
        "domain": "Exact_R2_matrix_image",
        "operation": "image-bounded-search-boundary",
        "semantic_template": "qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,39]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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-20261102-r03-linear-map-image-0095",
    "batch_id": "batch-20261014-advance-20261102-r03-linear-map-image-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b76e0686466ac3a333a6958aa85938e0bd2a7d7a87621f5b68ad872765f0b512",
      "family_id": "qf-mod33-image-bounded-search-boundary",
      "seed": 202611023095,
      "semantic_template": "qf-mod33-image-bounded-search-boundary/{\"bound\":5,\"matrix\":[[2,0],[0,2]],\"target\":[1,39]}"
    },
    "fields": {
      "title": "Failure to Find an Integer Preimage Does Not Establish Real Nonmembership",
      "stem": "Use this same exact integer matrix A=[[2, 0], [0, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Target=[39, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).",
      "answer": {
        "kind": "expression",
        "value": "([39, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)",
        "explanation": "List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([39, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)."
      },
      "skills": [
        "Failure to Find an Integer Preimage Does Not Establish Real Nonmembership"
      ],
      "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": "同一精確2×2線性映射與column-span證書。操作=image-bounded-search-boundary；資料={\"matrix\":[[2,0],[0,2]],\"target\":[39,1],\"bound\":5}。請依 [target,bound,integer_witnesses,global_membership,boundary] 欄序回答。",
        "en": "Use this same exact integer matrix A=[[2, 0], [0, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Target=[39, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "real map determinant diagonal product := mul(2, 2)",
      "after": "real map determinant diagonal product = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 real map determinant diagonal product，不以小數近似替代。",
          "en": "Recompute real map determinant diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "real map determinant diagonal product"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "real map determinant off-diagonal product := mul(0, 0)",
      "after": "real map determinant off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 real map determinant off-diagonal product，不以小數近似替代。",
          "en": "Recompute real map determinant off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "real map determinant off-diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "real map exact determinant := sub(4, 0)",
      "after": "real map exact determinant = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 real map exact determinant，不以小數近似替代。",
          "en": "Recompute real map exact determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "real map exact determinant"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "same ordered integer matrix=[[2, 0], [0, 2]];usual real map T:R2 to R2;original columns=[[2, 0], [0, 2]]",
      "after": "exact determinant=4;rank=2",
      "relation": "implies",
      "rule_id": "real_map_same_matrix_det_and_nonzero_rank_cases",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零det給秩2；det=0時還要看是否全零，才分秩0或1。固定同一矩陣與通常實向量空間，不把列當欄或自動把像當整個codomain。",
        "en": "A nonzero determinant gives rank2.With determinant0,distinguish the zero matrix from a nonzero singular matrix to get rank0 or1.Keep the same matrix and real vector spaces;do not replace columns by rows or assume the image is the entire codomain."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "first adjugate column row0 first term := mul(2, 2)",
      "after": "first adjugate column row0 first term = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column row0 first term，不以小數近似替代。",
          "en": "Recompute first adjugate column row0 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "first adjugate column row0 first term"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "first adjugate column row0 second term := mul(0, 0)",
      "after": "first adjugate column row0 second term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column row0 second term，不以小數近似替代。",
          "en": "Recompute first adjugate column row0 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "first adjugate column row0 second term"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "first adjugate column image coordinate0 := sum(4, 0)",
      "after": "first adjugate column image coordinate0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column image coordinate0，不以小數近似替代。",
          "en": "Recompute first adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "first adjugate column image coordinate0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "first adjugate column row1 first term := mul(0, 2)",
      "after": "first adjugate column row1 first term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column row1 first term，不以小數近似替代。",
          "en": "Recompute first adjugate column row1 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "first adjugate column row1 first term"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "first adjugate column row1 second term := mul(2, 0)",
      "after": "first adjugate column row1 second term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column row1 second term，不以小數近似替代。",
          "en": "Recompute first adjugate column row1 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "first adjugate column row1 second term"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "first adjugate column image coordinate1 := sum(0, 0)",
      "after": "first adjugate column image coordinate1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first adjugate column image coordinate1，不以小數近似替代。",
          "en": "Recompute first adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "first adjugate column image coordinate1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "second adjugate column row0 first term := mul(2, 0)",
      "after": "second adjugate column row0 first term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column row0 first term，不以小數近似替代。",
          "en": "Recompute second adjugate column row0 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "second adjugate column row0 first term"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "second adjugate column row0 second term := mul(0, 2)",
      "after": "second adjugate column row0 second term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column row0 second term，不以小數近似替代。",
          "en": "Recompute second adjugate column row0 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "second adjugate column row0 second term"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "second adjugate column image coordinate0 := sum(0, 0)",
      "after": "second adjugate column image coordinate0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column image coordinate0，不以小數近似替代。",
          "en": "Recompute second adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "second adjugate column image coordinate0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "second adjugate column row1 first term := mul(0, 0)",
      "after": "second adjugate column row1 first term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column row1 first term，不以小數近似替代。",
          "en": "Recompute second adjugate column row1 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "second adjugate column row1 first term"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "second adjugate column row1 second term := mul(2, 2)",
      "after": "second adjugate column row1 second term = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column row1 second term，不以小數近似替代。",
          "en": "Recompute second adjugate column row1 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "second adjugate column row1 second term"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "second adjugate column image coordinate1 := sum(0, 4)",
      "after": "second adjugate column image coordinate1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second adjugate column image coordinate1，不以小數近似替代。",
          "en": "Recompute second adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "4"
        ],
        "output": "4",
        "label": "second adjugate column image coordinate1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "same matrix=[[2, 0], [0, 2]];adjugate columns=[[2, 0], [0, 2]];verified products=[[4, 0], [0, 4]]",
      "after": "M*adjugate=det*I;det=4 nonzero;inverse=adjugate/det;kernel basis=[];image basis ORIGINAL columns=[[2, 0], [0, 2]]",
      "relation": "implies",
      "rule_id": "real_map_full_rank_inverse_both_kernel_and_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已逐座標核M乘兩個adjugate欄等於det倍標準基；除非零det得逆。因此Mx=0僅零解，且每個實目標有原像，兩個原欄形成像的basis；空kernel basis表示零空間，不是kernel空集合。",
        "en": "The two products give det times the standard basis.Dividing by nonzero det gives an inverse.Hence Mx=0 has only the zero solution,and every real target has a preimage.The original columns form an image basis.An empty kernel basis spans the zero space;the kernel itself is not empty."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "real kernel source dimension minus rank := sub(2, 2)",
      "after": "real kernel source dimension minus rank = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 real kernel source dimension minus rank，不以小數近似替代。",
          "en": "Recompute real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "real kernel source dimension minus rank"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "same real map rank=2;full kernel basis=[];full image basis=[[2, 0], [0, 2]]",
      "after": "nullity=0;image basis cardinality=2;rank plus nullity=2",
      "relation": "implies",
      "rule_id": "real_map_full_space_basis_counts_not_grid_counts",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這些是已證完整實空間的basis個數，不是搜尋格點數；rank、nullity與像basis數量分開保留，同一來源維度為2。",
        "en": "Count bases of the established full real spaces,not searched grid points.Keep rank,nullity and image-basis count distinct;the same source dimension is2."
      },
      "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": "image Cramer first diagonal target term := mul(2, 39)",
      "after": "image Cramer first diagonal target term = 78",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer first diagonal target term，不以小數近似替代。",
          "en": "Recompute image Cramer first diagonal target term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "39"
        ],
        "output": "78",
        "label": "image Cramer first diagonal target term"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "image Cramer first off-diagonal target term := mul(0, 1)",
      "after": "image Cramer first off-diagonal target term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer first off-diagonal target term，不以小數近似替代。",
          "en": "Recompute image Cramer first off-diagonal target term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "image Cramer first off-diagonal target term"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "image Cramer first numerator := sub(78, 0)",
      "after": "image Cramer first numerator = 78",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer first numerator，不以小數近似替代。",
          "en": "Recompute image Cramer first numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "78",
          "0"
        ],
        "output": "78",
        "label": "image Cramer first numerator"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "image Cramer first real coordinate := div(78, 4)",
      "after": "image Cramer first real coordinate = 39/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer first real coordinate，不以小數近似替代。",
          "en": "Recompute image Cramer first real coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "78",
          "4"
        ],
        "output": "39/2",
        "label": "image Cramer first real coordinate"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "image Cramer second signed target term := mul(0, 39)",
      "after": "image Cramer second signed target term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer second signed target term，不以小數近似替代。",
          "en": "Recompute image Cramer second signed target term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "39"
        ],
        "output": "0",
        "label": "image Cramer second signed target term"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "image Cramer second diagonal target term := mul(2, 1)",
      "after": "image Cramer second diagonal target term = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer second diagonal target term，不以小數近似替代。",
          "en": "Recompute image Cramer second diagonal target term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "image Cramer second diagonal target term"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "image Cramer second numerator := sum(0, 2)",
      "after": "image Cramer second numerator = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer second numerator，不以小數近似替代。",
          "en": "Recompute image Cramer second numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "image Cramer second numerator"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "image Cramer second real coordinate := div(2, 4)",
      "after": "image Cramer second real coordinate = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer second real coordinate，不以小數近似替代。",
          "en": "Recompute image Cramer second real coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "4"
        ],
        "output": "1/2",
        "label": "image Cramer second real coordinate"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "image Cramer substitution row0 first term := mul(2, 39/2)",
      "after": "image Cramer substitution row0 first term = 39",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution row0 first term，不以小數近似替代。",
          "en": "Recompute image Cramer substitution row0 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "39/2"
        ],
        "output": "39",
        "label": "image Cramer substitution row0 first term"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "image Cramer substitution row0 second term := mul(0, 1/2)",
      "after": "image Cramer substitution row0 second term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution row0 second term，不以小數近似替代。",
          "en": "Recompute image Cramer substitution row0 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "0",
        "label": "image Cramer substitution row0 second term"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "image Cramer substitution image coordinate0 := sum(39, 0)",
      "after": "image Cramer substitution image coordinate0 = 39",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution image coordinate0，不以小數近似替代。",
          "en": "Recompute image Cramer substitution image coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "39",
          "0"
        ],
        "output": "39",
        "label": "image Cramer substitution image coordinate0"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "image Cramer substitution row1 first term := mul(0, 39/2)",
      "after": "image Cramer substitution row1 first term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution row1 first term，不以小數近似替代。",
          "en": "Recompute image Cramer substitution row1 first term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "39/2"
        ],
        "output": "0",
        "label": "image Cramer substitution row1 first term"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "image Cramer substitution row1 second term := mul(2, 1/2)",
      "after": "image Cramer substitution row1 second term = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution row1 second term，不以小數近似替代。",
          "en": "Recompute image Cramer substitution row1 second term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1/2"
        ],
        "output": "1",
        "label": "image Cramer substitution row1 second term"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "image Cramer substitution image coordinate1 := sum(0, 1)",
      "after": "image Cramer substitution image coordinate1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 image Cramer substitution image coordinate1，不以小數近似替代。",
          "en": "Recompute image Cramer substitution image coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "image Cramer substitution image coordinate1"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "original target=[39, 1];same invertible real matrix=[[2, 0], [0, 2]];exact solved coordinates=[39/2, 1/2]",
      "after": "substitution equals target;unique real preimage;integer coordinates NOT required",
      "relation": "implies",
      "rule_id": "image_full_rank_real_solution_substitution_and_uniqueness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用同一det的Cramer分式求解，逐座標代回原M；任何兩解之差在零kernel，所以唯一。分數座標亦是合法實原像，不能加整數條件。",
        "en": "Solve with Cramer fractions using the same determinant and substitute in the original matrix.The difference of two solutions lies in the zero kernel,hence uniqueness.Rational coordinates are legal real preimages;do not add an integer-only condition."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-5, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-10, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-4, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-8, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-3, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-6, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-2, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-4, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[-1, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[-2, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[0, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[0, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[1, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[2, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[2, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[4, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[3, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[6, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[4, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[8, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, -5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, -10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, -4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, -8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, -3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, -6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, -2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, -4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, -1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, -2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 0];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 0];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 1];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 2];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 2];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 4];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 3];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 6];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 4];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 8];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "image bounded search:integer point in x-outer/y-inner order=[5, 5];same matrix=[[2, 0], [0, 2]];target=[39, 1]",
      "after": "exact row-dot image=[10, 10];equals target=否",
      "relation": "implies",
      "rule_id": "real_map_every_bounded_integer_point_exact_image",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。",
        "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs."
      },
      "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": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "image bounded search:bound=5;complete integer box pair count=121",
      "after": "all hits in original scan order=[]",
      "relation": "implies",
      "rule_id": "real_map_complete_bounded_list_not_unbounded_search",
      "conditions": [],
      "explanation": {
        "zh-Hant": "上下界含端點，兩座標共有(2b+1)²組；每組已核，清單只對這個盒完整。",
        "en": "Both endpoints are included,giving(2b+1)² pairs.Every pair was checked;completeness of the list is relative to this box only."
      },
      "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": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "bounded integer preimages in original scan order=[];SEPARATE full-real membership=是;global certificate=[unique_solution, [39/2, 1/2]]",
      "after": "boundary=bounded_failure_not_global_nonmembership",
      "relation": "implies",
      "rule_id": "image_bounded_failure_not_global_real_nonmembership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "盒內無整數原像與存在實原像可同時成立，分數解甚至所有整數盒都找不到。若全域不屬像，也必須靠前面完整實空間證書，不能只用這次搜尋失敗。",
        "en": "No integer preimage in the box can coexist with a real preimage;noninteger solutions may be missed by every integer box.Global nonmembership must likewise come from the separate full-real certificate,not this search failure alone."
      },
      "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": [
      [
        39,
        1
      ],
      5,
      [],
      true,
      "bounded_failure_not_global_nonmembership"
    ]
  },
  "conclusion": {
    "canonical_value": "([39, 1], 5, [], 是, bounded_failure_not_global_nonmembership)",
    "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-20261102-r03-linear-map-image-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r03-linear-map-image-foundations/questions.jsonl",
      "batch_sha256": "0af6e99557f03113cce20f1da2577d4a65149b5da726e6ca6c66a50f0ba07eaf",
      "original_line": 95,
      "original_line_sha256": "aedeaec26d0bcd57253c40ce15ce4639ea5d7cff0f3e16aad763af9a95f7bd37",
      "question_compact_sha256": "b76e0686466ac3a333a6958aa85938e0bd2a7d7a87621f5b68ad872765f0b512",
      "english_row_compact_sha256": "89d0d9b12e0fafa9b1815077850db74646214d0bd223302563eb52e4fd51465a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r03-linear-map-image-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r03-linear-map-image-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-map-image": {
        "sha256": "9901554d389b316f90187d99a94cc8c17a03fc69cde338962d359b8d5e95eb8d",
        "english_sha256": "b431ff922c8e97a289c8ba4ddab5f8dc659cce34249fd397c52e03398f0f34c2",
        "original_path": "public/data/mko/linear-map-image.json",
        "english_path": "public/data/i18n/editions/mko-linear-map-image.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5",
    "original_candidate_sha256": "b4339fb6e13f0744dcfa8036e7719e2dfcd8b7eb42a177824b680f43b09d3dbe",
    "original_candidate_line_sha256": "b4339fb6e13f0744dcfa8036e7719e2dfcd8b7eb42a177824b680f43b09d3dbe"
  },
  "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-p34",
    "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": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3",
      "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 157,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "73404b831e2d6fb35984f43b9cf13ba0617b6eb569a45064ebe66bf2a0898aff",
      "arithmetic_steps": 30
    },
    "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."
  }
}