{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r04-wilson-theorem-0084",
  "pilot_number": "R0054-27709",
  "number": 27709,
  "family_id": "qf-mod49-wilson-boundary",
  "title": {
    "zh-Hant": "第 27709 題：Wilson定理",
    "en": "Question 27709: Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r04-wilson-theorem-0084",
    "batch_id": "batch-20261015-advance-20261118-r04-wilson-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-wilson-theorem"
    ],
    "title_zh": "Wilson定理",
    "stem_zh": "n>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-boundary；資料={\"mode\":\"finite_testing\",\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(finite_testing, 10, 9, checked_moduli_2_through_10, 否, finite_cutoff_does_not_prove_the_universal_equivalence)",
      "explanation_zh": "Wave6第十二層核對=(finite_testing, 10, 9, checked_moduli_2_through_10, 否, finite_cutoff_does_not_prove_the_universal_equivalence)；依完整明示資料獨立重算；p=2只有一個self-inverse residue 1=-1；composite instances與finite testing不取代converse proof，Wilson不等於Fermat。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Wilson定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184084,
      "parameters": {
        "mode": "finite_testing",
        "n": 10,
        "domain": "Exact_Wilson_congruence",
        "operation": "wilson-boundary",
        "semantic_template": "wilson-boundary|{mode:\"finite_testing\",n:10}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261118-r04-wilson-theorem-0084",
    "batch_id": "batch-20261015-advance-20261118-r04-wilson-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2beedeb9d56404edd43c1156103cc7e40ac9dbe26ce7483b6cb2a49c281a998a",
      "family_id": "qf-mod49-wilson-boundary",
      "seed": 202611184084,
      "semantic_template": "wilson-boundary|{mode:\"finite_testing\",n:10}"
    },
    "fields": {
      "title": "Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate",
      "stem": "Use mode=finite_testing and exact integer n=10. Give (mode, cutoff, tested_modulus_count, tested_range, universal_proof_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(finite_testing, 10, 9, checked_moduli_2_through_10, No, finite_cutoff_does_not_prove_the_universal_equivalence)",
        "explanation": "Reject n≤1 from the theorem domain. Testing moduli2 through a cutoff is not a universal proof. The Fermat comparison supplies named statements but does not recompute powers of a. At p=2 there is one self-inverse nonzero residue. Preserve the mode-specific layouts and distinguish a failed inference from a false congruence. The exact result, retaining the requested order, is (finite_testing, 10, 9, checked_moduli_2_through_10, No, finite_cutoff_does_not_prove_the_universal_equivalence)."
      },
      "skills": [
        "Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate"
      ],
      "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": "n>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-boundary；資料={\"mode\":\"finite_testing\",\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use mode=finite_testing and exact integer n=10. Give (mode, cutoff, tested_modulus_count, tested_range, universal_proof_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [wilson-boundary]",
      "after": "derived = [[mode, cutoff, tested_modulus_count, tested_range, universal_proof_claim, boundary]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接明列此操作與mode完整欄序，不依JavaScript、不由答案形狀猜角色；布林、字串、null與未作聲稱保持不同。",
        "en": "List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [10]",
      "after": "derived = [9, checked_moduli_2_through_10, 否]",
      "relation": "implies",
      "rule_id": "Wilson_finite_cutoff_DESCRIPTION_NOT_performed_all_moduli_or_universal_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "來源分支宣告有限cutoff範圍與個數，未提供逐modulus計算表；本頁不把描述冒稱已執行的全範圍測試。即使有限checks全對，也不代替一般equivalence證明。",
        "en": "This source branch states a finite cutoff/range count,not per-modulus calculation tables.Do not present that description as tests executed for every modulus.Even successful finite checks would not replace a universal equivalence proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。",
          "en": "Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "finite_testing",
      10,
      9,
      "checked_moduli_2_through_10",
      false,
      "finite_cutoff_does_not_prove_the_universal_equivalence"
    ]
  },
  "conclusion": {
    "canonical_value": "(finite_testing, 10, 9, checked_moduli_2_through_10, 否, finite_cutoff_does_not_prove_the_universal_equivalence)",
    "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": {
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      "original_line": 84,
      "original_line_sha256": "7e57ff90bf348a478778db7ae640975ab7a9682945865a5e6459665c6393cf28",
      "question_compact_sha256": "2beedeb9d56404edd43c1156103cc7e40ac9dbe26ce7483b6cb2a49c281a998a",
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        "original_path": "public/data/mko/wilson-theorem.json",
        "english_path": "public/data/i18n/editions/mko-wilson-theorem.en.json"
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    "selected_part_sha256": "ad40cc18f9292c43924a7679f3af46574a5284aa5447f5f7f78a0de98840acb8",
    "original_candidate_sha256": "4a584d83585105a7491be1ec5e17db4c8dc81eecc81822cf5f3495c209185dcd",
    "original_candidate_line_sha256": "4a584d83585105a7491be1ec5e17db4c8dc81eecc81822cf5f3495c209185dcd"
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      "source_question_state": "candidate",
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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      "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-20261026-010-p55",
    "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": {
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    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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}