{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261102-r07-regular-language-0009",
  "pilot_number": "R0033-17259",
  "number": 17259,
  "family_id": "qf-mod33-regular-mod-count-certificate",
  "title": {
    "zh-Hant": "第 17259 題：正規語言",
    "en": "Question 17259: A State Invariant Covers Every Word, Not Just Test Cases"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261102-r07-regular-language-0009",
    "batch_id": "batch-20261014-advance-20261102-r07-regular-language-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-regular-language"
    ],
    "title_zh": "正規語言",
    "stem_zh": "完整DFA總轉移與任意word invariant或明示有限語言構造。操作=regular-mod-count-certificate；資料={\"modulus\":10,\"residue\":0,\"target_symbol\":1,\"alphabet_count\":2}。請依 [states,transition,start,accepting,regular,universal_invariant] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]], 0, [0], 是, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)",
      "explanation_zh": "Wave5第二層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]], 0, [0], 是, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)；依完整明示資料重算；mod-count證書須涵蓋任意word length而不只是測試集；不引入regex、NFA、epsilon、minimization或pumping lemma；有限測試不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "正規語言"
      ]
    },
    "variant": {
      "family_id": "qf-mod33-regular-mod-count-certificate",
      "seed": 202611027009,
      "parameters": {
        "modulus": 10,
        "residue": 0,
        "target_symbol": 1,
        "alphabet_count": 2,
        "domain": "Exact_DFA_regular_language_certificate",
        "operation": "regular-mod-count-certificate",
        "semantic_template": "qf-mod33-regular-mod-count-certificate/{\"alphabet_count\":2,\"modulus\":10,\"residue\":0,\"target_symbol\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261102-r07-regular-language-0009",
    "batch_id": "batch-20261014-advance-20261102-r07-regular-language-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3c2327ff21e2cbbd4d23446a0c9ac68bc8bda4c5640dfaaefc9145125aafe97a",
      "family_id": "qf-mod33-regular-mod-count-certificate",
      "seed": 202611027009,
      "semantic_template": "qf-mod33-regular-mod-count-certificate/{\"alphabet_count\":2,\"modulus\":10,\"residue\":0,\"target_symbol\":1}"
    },
    "fields": {
      "title": "A State Invariant Covers Every Word, Not Just Test Cases",
      "stem": "Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=10. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order.",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]], 0, [0], Yes, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)",
        "explanation": "The invariant holds initially with count zero and is preserved by each symbol transition. Induction over arbitrary word length makes the accepting residue characterize the whole declared language. This explicit DFA/invariant is the certificate, not a finite suite or a new Lean proof. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]], 0, [0], Yes, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)."
      },
      "skills": [
        "A State Invariant Covers Every Word, Not Just Test Cases"
      ],
      "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": "完整DFA總轉移與任意word invariant或明示有限語言構造。操作=regular-mod-count-certificate；資料={\"modulus\":10,\"residue\":0,\"target_symbol\":1,\"alphabet_count\":2}。請依 [states,transition,start,accepting,regular,universal_invariant] 欄序回答。",
        "en": "Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=10. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(0+0) mod10=0",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(0+1) mod10=1",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(1+0) mod10=1",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(1+1) mod10=2",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(2+0) mod10=2",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(2+1) mod10=3",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "mod-count state=3;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(3+0) mod10=3",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "mod-count state=3;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(3+1) mod10=4",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "mod-count state=4;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(4+0) mod10=4",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "mod-count state=4;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(4+1) mod10=5",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "mod-count state=5;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(5+0) mod10=5",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "mod-count state=5;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(5+1) mod10=6",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "mod-count state=6;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(6+0) mod10=6",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "mod-count state=6;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(6+1) mod10=7",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "mod-count state=7;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(7+0) mod10=7",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "mod-count state=7;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(7+1) mod10=8",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "mod-count state=8;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(8+0) mod10=8",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "mod-count state=8;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(8+1) mod10=9",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "mod-count state=9;original symbol=0;target symbol=1;indicator=0;modulus=10",
      "after": "next state=(9+0) mod10=9",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "mod-count state=9;original symbol=1;target symbol=1;indicator=1;modulus=10",
      "after": "next state=(9+1) mod10=0",
      "relation": "implies",
      "rule_id": "regular_mod_count_every_state_symbol_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標符號才加1，其他加0，再取同一模數；每個狀態／符號格都核，特別保留最後一類加1回0的情況。",
        "en": "Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "states Fin(10);complete typed table=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]];start0;accepting residue=0",
      "after": "empty prefix count=0 equals start;append any symbol adds exactly its target indicator to BOTH count and state modulo the modulus",
      "relation": "implies",
      "rule_id": "regular_mod_count_all_prefix_induction_base_step",
      "conditions": [],
      "explanation": {
        "zh-Hant": "初始計數與state皆0；每追加一符號，計數與state依同一indicator更新，所以對任意字長的每個前綴都保state等於目標次數mod m。",
        "en": "Both count and state start at0.Appending any symbol updates both by the same indicator,so induction preserves state equal to target-symbol count modulo m for every prefix of every word."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "for EVERY finite word,state equals target-symbol count mod10;accepting set=[0]",
      "after": "accepted iff count congruent to0;whole declared language has this complete DFA certificate",
      "relation": "implies",
      "rule_id": "regular_mod_count_language_characterized_not_test_suite",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以最後state是否等於指定residue刻畫全部宣告語言；這是DFA加全稱不變量，不是有限測試，也不引入regex、epsilon、最小化或pumping lemma。",
        "en": "Final-state membership in the specified residue characterizes the entire declared language.This uses the complete DFA and universal invariant,not finite tests,and adds no regex,epsilon moves,minimization or pumping lemma."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        0,
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9
      ],
      [
        [
          0,
          1
        ],
        [
          1,
          2
        ],
        [
          2,
          3
        ],
        [
          3,
          4
        ],
        [
          4,
          5
        ],
        [
          5,
          6
        ],
        [
          6,
          7
        ],
        [
          7,
          8
        ],
        [
          8,
          9
        ],
        [
          9,
          0
        ]
      ],
      0,
      [
        0
      ],
      true,
      "state_equals_target_symbol_count_mod_modulus_for_all_prefixes"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 0]], 0, [0], 是, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261102-r07-regular-language-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r07-regular-language-foundations/questions.jsonl",
      "batch_sha256": "6b354df25a379577a6b550ace23ccd63aa6511564e13109eb1a705a45232f831",
      "original_line": 9,
      "original_line_sha256": "d8e4229c3456cd2c7fb0a9f5e5b3bdb73a012d68226658dbc94f689b4f7b71f4",
      "question_compact_sha256": "3c2327ff21e2cbbd4d23446a0c9ac68bc8bda4c5640dfaaefc9145125aafe97a",
      "english_row_compact_sha256": "631b0e81dc61ac457e2062479ecb8e660dfbf4ccc90fbaf9f05b323ac41d4f99",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r07-regular-language-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r07-regular-language-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-regular-language": {
        "sha256": "5b76e4da673cf35d7a7014ce76f103fa504f5f9986498711ed877c15dc35e57a",
        "english_sha256": "b6bfb3d62bc7ec660a984b86c35cabf78b5826bddd2394009ea358db21f4038a",
        "original_path": "public/data/mko/regular-language.json",
        "english_path": "public/data/i18n/editions/mko-regular-language.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5",
    "original_candidate_sha256": "83697cb19c8d6beca446d65b4dcf470a610660193e5cf25ed7298aaa2d07121e",
    "original_candidate_line_sha256": "83697cb19c8d6beca446d65b4dcf470a610660193e5cf25ed7298aaa2d07121e"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p34",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3",
      "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f03e8e8002f45cdbba5d557e024b68d585824798bcd38baf0682e27edf39984b",
      "arithmetic_steps": 0
    },
    "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."
  }
}