{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261025-r06-finite-markov-chain-0067",
  "pilot_number": "R0065-33342",
  "number": 33342,
  "family_id": "qf-mod25-markovdeep-endpoint-joint-probability",
  "title": {
    "zh-Hant": "第 33342 題：有限Markov密度",
    "en": "Question 33342: Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261025-r06-finite-markov-chain-0067",
    "batch_id": "batch-20261013-advance-20261025-r06-finite-markov-chain-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-markov-chain"
    ],
    "title_zh": "有限Markov密度",
    "stem_zh": "精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)",
      "explanation_zh": "收束波核對=([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限Markov密度"
      ]
    },
    "variant": {
      "family_id": "qf-mod25-markovdeep-endpoint-joint-probability",
      "seed": 202610256067,
      "parameters": {
        "initial": [
          [
            1,
            2
          ],
          [
            1,
            2
          ]
        ],
        "transition": [
          [
            [
              1,
              5
            ],
            [
              4,
              5
            ]
          ],
          [
            [
              1,
              2
            ],
            [
              1,
              2
            ]
          ]
        ],
        "from": 0,
        "to": 1,
        "steps": 2,
        "domain": "Finite_exact_Markov_density",
        "operation": "markovdeep-endpoint-joint-probability",
        "semantic_template": "qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261025-r06-finite-markov-chain-0067",
    "batch_id": "batch-20261013-advance-20261025-r06-finite-markov-chain-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9b934336be6839b8533463828ad31aa4f133d3dd4a865375dacb4680e0cece93",
      "family_id": "qf-mod25-markovdeep-endpoint-joint-probability",
      "seed": 202610256067,
      "semantic_template": "qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"
    },
    "fields": {
      "title": "Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability",
      "stem": "Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).",
      "answer": {
        "kind": "expression",
        "value": "([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)",
        "explanation": "This is the joint probability of the initial and later endpoint, not the conditional transition entry alone or the later-time marginal. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)."
      },
      "skills": [
        "Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability"
      ],
      "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": "精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。",
        "en": "Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/2, 1/2]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[1/2, 1/2], from=0, to=1, steps=2. Give (initial,from,to,steps,initial[from],P^steps[from][to],product)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod25-markovdeep-endpoint-joint-probability, markovdeep-endpoint-joint-probability, Finite_exact_Markov_density, []]",
      "after": "derived = [[initial, from, to, steps, initial_mass, power_entry, joint]]",
      "relation": "implies",
      "rule_id": "R65_SOURCE_bank_keys_operation_domain_ordered_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "MOD25用LOCAL hyphen及具名domain，矩陣legacy用underscore/reals，BASElogic不補domain/op。輸出tuple/矩陣/數字欄與fraction文字各依來源，不一律保/1；原中英/AST/status不變。",
        "en": "MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [Markov RAW P row0 column0, RAW-pair, [1, 5], [], 1, 5]",
      "after": "derived = [1, 5, 1/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 5)",
      "after": "Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "5"
        ],
        "output": "1/5",
        "label": "Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Markov RAW P row0 column1, RAW-pair, [4, 5], [], 4, 5]",
      "after": "derived = [4, 5, 4/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator := div(4, 5)",
      "after": "Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "5"
        ],
        "output": "4/5",
        "label": "Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Markov RAW row0 normalization through column0, radd, [[0, 1], [1, 5]], [1], 1, 5]",
      "after": "derived = [1, 5, 1/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Markov RAW row0 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/5)",
      "after": "Markov RAW row0 normalization through column0 ORIGINAL exact rational sum = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW row0 normalization through column0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov RAW row0 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/5"
        ],
        "output": "1/5",
        "label": "Markov RAW row0 normalization through column0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [Markov RAW row0 normalization through column1, radd, [[1, 5], [4, 5]], [5], 5, 5]",
      "after": "derived = [1, 1, 1]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Markov RAW row0 normalization through column1 ORIGINAL exact rational sum := sum(1/5, 4/5)",
      "after": "Markov RAW row0 normalization through column1 ORIGINAL exact rational sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW row0 normalization through column1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov RAW row0 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/5",
          "4/5"
        ],
        "output": "1",
        "label": "Markov RAW row0 normalization through column1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [Markov RAW P row1 column0, RAW-pair, [1, 2], [], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 2)",
      "after": "Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [Markov RAW P row1 column1, RAW-pair, [1, 2], [], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator := div(1, 2)",
      "after": "Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [Markov RAW row1 normalization through column0, radd, [[0, 1], [1, 2]], [1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Markov RAW row1 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/2)",
      "after": "Markov RAW row1 normalization through column0 ORIGINAL exact rational sum = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW row1 normalization through column0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov RAW row1 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov RAW row1 normalization through column0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [Markov RAW row1 normalization through column1, radd, [[1, 2], [1, 2]], [2], 2, 2]",
      "after": "derived = [1, 1, 1]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Markov RAW row1 normalization through column1 ORIGINAL exact rational sum := sum(1/2, 1/2)",
      "after": "Markov RAW row1 normalization through column1 ORIGINAL exact rational sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW row1 normalization through column1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov RAW row1 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "Markov RAW row1 normalization through column1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [[[[1, 5], [4, 5]], [[1, 2], [1, 2]]]]",
      "after": "derived = [2, [[1/5, 4/5], [1/2, 1/2]]]",
      "relation": "implies",
      "rule_id": "Markov_FULL_RAW_kernel_ALL_cells_row_stochastic_fixed_state_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部raw rational先過界、sign/gcd後非負，各row精確sum1，row=current/column=next固定。steps0或路徑prob0也不能略過unused cells。",
        "en": "Validate EVERY raw rational/envelope,then nonnegativity after sign/gcd normalization and exact row sum1.Keep row=current/column=next order.Even steps0/path probability0 cannot waive unused cells."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [2, 2]",
      "after": "derived = [[[1, 0], [0, 1]]]",
      "relation": "implies",
      "rule_id": "Markov_ZERO_step_identity_then_fixed_RIGHT_multiplication",
      "conditions": [],
      "explanation": {
        "zh-Hant": "P0是identity，之後按原state順序逐次右乘P；保source每個dot的middle順序與rational運算界，不是平穩或極限證明。",
        "en": "P0 is the identity,then multiply RIGHT by P in original state order.Preserve every dot-product middle order and SOURCE rational envelope.This is not a stationarity or limit proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [Markov power1 entry0,0 middle0 product, rmul, [[1, 1], [1, 5]], [1, 1], 1, 5]",
      "after": "derived = [1, 5, 1/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Markov power1 entry0,0 middle0 product ORIGINAL exact rational product := mul(1, 1/5)",
      "after": "Markov power1 entry0,0 middle0 product ORIGINAL exact rational product = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/5"
        ],
        "output": "1/5",
        "label": "Markov power1 entry0,0 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [Markov power1 entry0,0 through middle0, radd, [[0, 1], [1, 5]], [1], 1, 5]",
      "after": "derived = [1, 5, 1/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/5)",
      "after": "Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/5"
        ],
        "output": "1/5",
        "label": "Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [Markov power1 entry0,0 middle1 product, rmul, [[0, 1], [1, 2]], [2, 1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Markov power1 entry0,0 middle1 product ORIGINAL exact rational product := mul(0, 1/2)",
      "after": "Markov power1 entry0,0 middle1 product ORIGINAL exact rational product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "0",
        "label": "Markov power1 entry0,0 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [Markov power1 entry0,0 through middle1, radd, [[1, 5], [0, 1]], [1], 1, 5]",
      "after": "derived = [1, 5, 1/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/5, 0)",
      "after": "Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum = 1/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/5",
          "0"
        ],
        "output": "1/5",
        "label": "Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [Markov power1 entry0,1 middle0 product, rmul, [[1, 1], [4, 5]], [1, 1], 4, 5]",
      "after": "derived = [4, 5, 4/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Markov power1 entry0,1 middle0 product ORIGINAL exact rational product := mul(1, 4/5)",
      "after": "Markov power1 entry0,1 middle0 product ORIGINAL exact rational product = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "4/5"
        ],
        "output": "4/5",
        "label": "Markov power1 entry0,1 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [Markov power1 entry0,1 through middle0, radd, [[0, 1], [4, 5]], [1], 4, 5]",
      "after": "derived = [4, 5, 4/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 4/5)",
      "after": "Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "4/5"
        ],
        "output": "4/5",
        "label": "Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [Markov power1 entry0,1 middle1 product, rmul, [[0, 1], [1, 2]], [2, 1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Markov power1 entry0,1 middle1 product ORIGINAL exact rational product := mul(0, 1/2)",
      "after": "Markov power1 entry0,1 middle1 product ORIGINAL exact rational product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "0",
        "label": "Markov power1 entry0,1 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [Markov power1 entry0,1 through middle1, radd, [[4, 5], [0, 1]], [1], 4, 5]",
      "after": "derived = [4, 5, 4/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum := sum(4/5, 0)",
      "after": "Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum = 4/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4/5",
          "0"
        ],
        "output": "4/5",
        "label": "Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "source roles = [Markov power1 entry1,0 middle0 product, rmul, [[0, 1], [1, 5]], [5, 1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Markov power1 entry1,0 middle0 product ORIGINAL exact rational product := mul(0, 1/5)",
      "after": "Markov power1 entry1,0 middle0 product ORIGINAL exact rational product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1/5"
        ],
        "output": "0",
        "label": "Markov power1 entry1,0 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [Markov power1 entry1,0 through middle0, radd, [[0, 1], [0, 1]], [1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 0)",
      "after": "Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [Markov power1 entry1,0 middle1 product, rmul, [[1, 1], [1, 2]], [1, 1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Markov power1 entry1,0 middle1 product ORIGINAL exact rational product := mul(1, 1/2)",
      "after": "Markov power1 entry1,0 middle1 product ORIGINAL exact rational product = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov power1 entry1,0 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "source roles = [Markov power1 entry1,0 through middle1, radd, [[0, 1], [1, 2]], [1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum := sum(0, 1/2)",
      "after": "Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "source roles = [Markov power1 entry1,1 middle0 product, rmul, [[0, 1], [4, 5]], [5, 1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Markov power1 entry1,1 middle0 product ORIGINAL exact rational product := mul(0, 4/5)",
      "after": "Markov power1 entry1,1 middle0 product ORIGINAL exact rational product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "4/5"
        ],
        "output": "0",
        "label": "Markov power1 entry1,1 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "source roles = [Markov power1 entry1,1 through middle0, radd, [[0, 1], [0, 1]], [1], 0, 1]",
      "after": "derived = [0, 1, 0]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 0)",
      "after": "Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "source roles = [Markov power1 entry1,1 middle1 product, rmul, [[1, 1], [1, 2]], [1, 1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Markov power1 entry1,1 middle1 product ORIGINAL exact rational product := mul(1, 1/2)",
      "after": "Markov power1 entry1,1 middle1 product ORIGINAL exact rational product = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov power1 entry1,1 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "source roles = [Markov power1 entry1,1 through middle1, radd, [[0, 1], [1, 2]], [1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum := sum(0, 1/2)",
      "after": "Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [1, [[1, 0], [0, 1]], [[1/5, 4/5], [1/2, 1/2]]]",
      "after": "derived = [[[1/5, 4/5], [1/2, 1/2]]]",
      "relation": "implies",
      "rule_id": "Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全矩陣所有entries都由完整dot products累加；不是只算詢問那一格。此source在path分支前也算power，雖不輸出仍核運算界。",
        "en": "Compute EVERY full-matrix entry by its COMPLETE dot product,not only a queried entry.The SOURCE also computes this power before a path branch;its envelope remains required even when not returned."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "source roles = [Markov power2 entry0,0 middle0 product, rmul, [[1, 5], [1, 5]], [1, 1], 1, 25]",
      "after": "derived = [1, 25, 1/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Markov power2 entry0,0 middle0 product ORIGINAL exact rational product := mul(1/5, 1/5)",
      "after": "Markov power2 entry0,0 middle0 product ORIGINAL exact rational product = 1/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/5",
          "1/5"
        ],
        "output": "1/25",
        "label": "Markov power2 entry0,0 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "source roles = [Markov power2 entry0,0 through middle0, radd, [[0, 1], [1, 25]], [1], 1, 25]",
      "after": "derived = [1, 25, 1/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/25)",
      "after": "Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum = 1/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/25"
        ],
        "output": "1/25",
        "label": "Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "source roles = [Markov power2 entry0,0 middle1 product, rmul, [[4, 5], [1, 2]], [2, 1], 2, 5]",
      "after": "derived = [2, 5, 2/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Markov power2 entry0,0 middle1 product ORIGINAL exact rational product := mul(4/5, 1/2)",
      "after": "Markov power2 entry0,0 middle1 product ORIGINAL exact rational product = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/5",
          "1/2"
        ],
        "output": "2/5",
        "label": "Markov power2 entry0,0 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [Markov power2 entry0,0 through middle1, radd, [[1, 25], [2, 5]], [5], 11, 25]",
      "after": "derived = [11, 25, 11/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/25, 2/5)",
      "after": "Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum = 11/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/25",
          "2/5"
        ],
        "output": "11/25",
        "label": "Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "source roles = [Markov power2 entry0,1 middle0 product, rmul, [[1, 5], [4, 5]], [1, 1], 4, 25]",
      "after": "derived = [4, 25, 4/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Markov power2 entry0,1 middle0 product ORIGINAL exact rational product := mul(1/5, 4/5)",
      "after": "Markov power2 entry0,1 middle0 product ORIGINAL exact rational product = 4/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/5",
          "4/5"
        ],
        "output": "4/25",
        "label": "Markov power2 entry0,1 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "source roles = [Markov power2 entry0,1 through middle0, radd, [[0, 1], [4, 25]], [1], 4, 25]",
      "after": "derived = [4, 25, 4/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 4/25)",
      "after": "Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum = 4/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "4/25"
        ],
        "output": "4/25",
        "label": "Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "source roles = [Markov power2 entry0,1 middle1 product, rmul, [[4, 5], [1, 2]], [2, 1], 2, 5]",
      "after": "derived = [2, 5, 2/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Markov power2 entry0,1 middle1 product ORIGINAL exact rational product := mul(4/5, 1/2)",
      "after": "Markov power2 entry0,1 middle1 product ORIGINAL exact rational product = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4/5",
          "1/2"
        ],
        "output": "2/5",
        "label": "Markov power2 entry0,1 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "source roles = [Markov power2 entry0,1 through middle1, radd, [[4, 25], [2, 5]], [5], 14, 25]",
      "after": "derived = [14, 25, 14/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum := sum(4/25, 2/5)",
      "after": "Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum = 14/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4/25",
          "2/5"
        ],
        "output": "14/25",
        "label": "Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "source roles = [Markov power2 entry1,0 middle0 product, rmul, [[1, 2], [1, 5]], [1, 1], 1, 10]",
      "after": "derived = [1, 10, 1/10]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "Markov power2 entry1,0 middle0 product ORIGINAL exact rational product := mul(1/2, 1/5)",
      "after": "Markov power2 entry1,0 middle0 product ORIGINAL exact rational product = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/5"
        ],
        "output": "1/10",
        "label": "Markov power2 entry1,0 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "source roles = [Markov power2 entry1,0 through middle0, radd, [[0, 1], [1, 10]], [1], 1, 10]",
      "after": "derived = [1, 10, 1/10]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/10)",
      "after": "Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/10"
        ],
        "output": "1/10",
        "label": "Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [Markov power2 entry1,0 middle1 product, rmul, [[1, 2], [1, 2]], [1, 1], 1, 4]",
      "after": "derived = [1, 4, 1/4]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Markov power2 entry1,0 middle1 product ORIGINAL exact rational product := mul(1/2, 1/2)",
      "after": "Markov power2 entry1,0 middle1 product ORIGINAL exact rational product = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "Markov power2 entry1,0 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "source roles = [Markov power2 entry1,0 through middle1, radd, [[1, 10], [1, 4]], [2], 7, 20]",
      "after": "derived = [7, 20, 7/20]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum := sum(1/10, 1/4)",
      "after": "Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum = 7/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/10",
          "1/4"
        ],
        "output": "7/20",
        "label": "Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "source roles = [Markov power2 entry1,1 middle0 product, rmul, [[1, 2], [4, 5]], [1, 2], 2, 5]",
      "after": "derived = [2, 5, 2/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Markov power2 entry1,1 middle0 product ORIGINAL exact rational product := mul(1/2, 4/5)",
      "after": "Markov power2 entry1,1 middle0 product ORIGINAL exact rational product = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "4/5"
        ],
        "output": "2/5",
        "label": "Markov power2 entry1,1 middle0 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "source roles = [Markov power2 entry1,1 through middle0, radd, [[0, 1], [2, 5]], [1], 2, 5]",
      "after": "derived = [2, 5, 2/5]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 2/5)",
      "after": "Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum = 2/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2/5"
        ],
        "output": "2/5",
        "label": "Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "source roles = [Markov power2 entry1,1 middle1 product, rmul, [[1, 2], [1, 2]], [1, 1], 1, 4]",
      "after": "derived = [1, 4, 1/4]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Markov power2 entry1,1 middle1 product ORIGINAL exact rational product := mul(1/2, 1/2)",
      "after": "Markov power2 entry1,1 middle1 product ORIGINAL exact rational product = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1/4",
        "label": "Markov power2 entry1,1 middle1 product ORIGINAL exact rational product"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "source roles = [Markov power2 entry1,1 through middle1, radd, [[2, 5], [1, 4]], [1], 13, 20]",
      "after": "derived = [13, 20, 13/20]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum := sum(2/5, 1/4)",
      "after": "Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum = 13/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2/5",
          "1/4"
        ],
        "output": "13/20",
        "label": "Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "source roles = [2, [[1/5, 4/5], [1/2, 1/2]], [[1/5, 4/5], [1/2, 1/2]]]",
      "after": "derived = [[[11/25, 14/25], [7/20, 13/20]]]",
      "relation": "implies",
      "rule_id": "Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全矩陣所有entries都由完整dot products累加；不是只算詢問那一格。此source在path分支前也算power，雖不輸出仍核運算界。",
        "en": "Compute EVERY full-matrix entry by its COMPLETE dot product,not only a queried entry.The SOURCE also computes this power before a path branch;its envelope remains required even when not returned."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "source roles = [Markov RAW initial state0, RAW-pair, [1, 2], [], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator := div(1, 2)",
      "after": "Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "source roles = [Markov RAW initial state1, RAW-pair, [1, 2], [], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator := div(1, 2)",
      "after": "Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "source roles = [Markov initial normalization through state0, radd, [[0, 1], [1, 2]], [1], 1, 2]",
      "after": "derived = [1, 2, 1/2]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Markov initial normalization through state0 ORIGINAL exact rational sum := sum(0, 1/2)",
      "after": "Markov initial normalization through state0 ORIGINAL exact rational sum = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov initial normalization through state0 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov initial normalization through state0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "1/2",
        "label": "Markov initial normalization through state0 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "source roles = [Markov initial normalization through state1, radd, [[1, 2], [1, 2]], [2], 2, 2]",
      "after": "derived = [1, 1, 1]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Markov initial normalization through state1 ORIGINAL exact rational sum := sum(1/2, 1/2)",
      "after": "Markov initial normalization through state1 ORIGINAL exact rational sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov initial normalization through state1 ORIGINAL exact rational sum，不以小數近似替代。",
          "en": "Recompute Markov initial normalization through state1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "Markov initial normalization through state1 ORIGINAL exact rational sum"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "source roles = [[[1, 2], [1, 2]]]",
      "after": "derived = [[1/2, 1/2]]",
      "relation": "implies",
      "rule_id": "Markov_RAW_initial_law_nonnegative_exact_sum1_ORDER_preserved",
      "conditions": [],
      "explanation": {
        "zh-Hant": "initial law按原state順序，全部非負且sum1；不換成uniform，也不把µ與matrix各自排序。",
        "en": "The initial law retains ORIGINAL state order,nonnegativity and exact sum1.Do not uniformize or independently sort mu and the matrix."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "source roles = [Markov endpoint joint ORIGINAL initial-from mass times conditional entry, rmul, [[1, 2], [14, 25]], [1, 2], 7, 25]",
      "after": "derived = [7, 25, 7/25]",
      "relation": "implies",
      "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。",
        "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product := mul(1/2, 14/25)",
      "after": "Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product = 7/25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。",
        "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product，不以小數近似替代。",
          "en": "Recompute Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "14/25"
        ],
        "output": "7/25",
        "label": "Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "source roles = [[1/2, 1/2], 0, 1, 2]",
      "after": "derived = [1/2, 14/25, 7/25]",
      "relation": "implies",
      "rule_id": "Markov_ENDPOINT_joint_initial_mass_times_entry_NOT_marginal",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個endpoint的joint是µ[from]*Ppower[from,to]，不是只取conditional entry或later marginal；µ[from]0時joint0合法。",
        "en": "The endpoint JOINT is mu[from]*Ppower[from,to],not the conditional entry alone or the later marginal.Zero initial mass gives a legal joint0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。",
          "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "1/2",
        "1/2"
      ],
      0,
      1,
      2,
      "1/2",
      "14/25",
      "7/25"
    ]
  },
  "conclusion": {
    "canonical_value": "([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)",
    "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-20261013-advance-20261025-r06-finite-markov-chain-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations/questions.jsonl",
      "batch_sha256": "78020bad9b4b61acc31e819b594a5dc1c0ab7a6248cf86fee77ffcaf6eebd849",
      "original_line": 67,
      "original_line_sha256": "85c1da51923964086b8a0a3be184c7ffdea314050bda437ee7c6c9842a6e1430",
      "question_compact_sha256": "9b934336be6839b8533463828ad31aa4f133d3dd4a865375dacb4680e0cece93",
      "english_row_compact_sha256": "2a1be775e5468750e2095d74e6494eebef0278a1ad5f69a689816c9e889b9eea",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-markov-chain": {
        "sha256": "713cef8fb3c1efc7627452592ab30fb69b9ebb7992c47aad78b9ae322ee7fd55",
        "english_sha256": "531d840a993901f295df6db2e167b57a16a1d3820e22c5212389593049a3ef38",
        "original_path": "public/data/mko/finite-markov-chain.json",
        "english_path": "public/data/i18n/editions/mko-finite-markov-chain.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "871e2c05026567a1deae741d6b90b0588100c6482ee167c7f9f45af5b72d8606",
    "original_candidate_sha256": "e213bb1551cd1e0bb697f202fc9785961ec8a06905cde4293e24bd86764fc17e",
    "original_candidate_line_sha256": "e213bb1551cd1e0bb697f202fc9785961ec8a06905cde4293e24bd86764fc17e"
  },
  "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-20261026-012-p66",
    "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": "92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6",
      "question_first_witness_sha256": "012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 97,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f344429ac918ee49bd7134764d0cfad499e58fe5679fd3007ba7a4a8165d0c97",
      "arithmetic_steps": 45
    },
    "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."
  }
}