{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r01-at-most-countable-set-0007", "pilot_number": "R0030-15882", "number": 15882, "family_id": "qf-mod31-countable-finite-certificate", "title": { "zh-Hant": "第 15882 題：至多可數集", "en": "Question 15882: A Complete Finite List Gives an At-Most-Countable Certificate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r01-at-most-countable-set-0007", "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-at-most-countable-set" ], "title_zh": "至多可數集", "stem_zh": "有限完整清單或明示全稱注入證書。操作=countable-finite-certificate；資料={\"elements\":[7,-8,52]}。請依 [elements,injection_pairs,injective,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], 是, finite_implies_at_most_countable)", "explanation_zh": "Wave4收束核對=([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], 是, finite_implies_at_most_countable)；依完整明示資料重算；有限與可數無限不可混同；單一失敗encoding不證不可數。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "至多可數集" ] }, "variant": { "family_id": "qf-mod31-countable-finite-certificate", "seed": 202610311007, "parameters": { "elements": [ 7, -8, 52 ], "domain": "Explicit_at_most_countable_certificate", "operation": "countable-finite-certificate", "semantic_template": "qf-mod31-countable-finite-certificate/{\"elements\":[-8,7,52]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261031-r01-at-most-countable-set-0007", "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "21226e190605f136bbceadb0184bd2ea16764d26245f4fcb73d3298caa09ddec", "family_id": "qf-mod31-countable-finite-certificate", "seed": 202610311007, "semantic_template": "qf-mod31-countable-finite-certificate/{\"elements\":[-8,7,52]}" }, "fields": { "title": "A Complete Finite List Gives an At-Most-Countable Certificate", "stem": "Complete nonempty duplicate-free finite integer list elements=[7, -8, 52] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].", "answer": { "kind": "expression", "value": "([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable)", "explanation": "Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable)." }, "skills": [ "A Complete Finite List Gives an At-Most-Countable Certificate" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "有限完整清單或明示全稱注入證書。操作=countable-finite-certificate；資料={\"elements\":[7,-8,52]}。請依 [elements,injection_pairs,injective,certificate] 欄序回答。", "en": "Complete nonempty duplicate-free finite integer list elements=[7, -8, 52] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position]." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "DECLARED complete finite integer set in original list order=[7, -8, 52];length=3", "after": "nonempty duplicate-free registered family;original order retained;empty-set countability not denied", "relation": "implies", "rule_id": "countable_complete_finite_list_premise_not_sampled_subset", "conditions": [], "explanation": { "zh-Hant": "清單的完整性是題目明示集合前提，不從有限樣本推未知全集；不排序、不刪重複。此題族拒絕空表只是介面限制，空集仍至多可數。", "en": "Completeness is the declared set premise,not inferred for an unknown universe from samples.Keep order and reject duplicates rather than sorting or deduplicating.The family's empty-list rejection is an interface limit,not denial of empty-set countability." }, "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": "original element=7;unique original position=0", "after": "injection pair element_to_position=[7, 0]", "relation": "implies", "rule_id": "finite_countability_injection_direction", "conditions": [], "explanation": { "zh-Hant": "把每個元素送唯一的0起算位置；兩元素位置相同則是清單同一項，故注入到自然數。方向是元素到位置，不是enumeration的反向。", "en": "Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration." }, "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": "original element=-8;unique original position=1", "after": "injection pair element_to_position=[-8, 1]", "relation": "implies", "rule_id": "finite_countability_injection_direction", "conditions": [], "explanation": { "zh-Hant": "把每個元素送唯一的0起算位置；兩元素位置相同則是清單同一項，故注入到自然數。方向是元素到位置，不是enumeration的反向。", "en": "Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration." }, "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": "original element=52;unique original position=2", "after": "injection pair element_to_position=[52, 2]", "relation": "implies", "rule_id": "finite_countability_injection_direction", "conditions": [], "explanation": { "zh-Hant": "把每個元素送唯一的0起算位置；兩元素位置相同則是清單同一項，故注入到自然數。方向是元素到位置，不是enumeration的反向。", "en": "Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration." }, "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": [ [ 7, -8, 52 ], [ [ 7, 0 ], [ -8, 1 ], [ 52, 2 ] ], true, "finite_implies_at_most_countable" ] }, "conclusion": { "canonical_value": "([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], 是, finite_implies_at_most_countable)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261031-r01-at-most-countable-set-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/questions.jsonl", "batch_sha256": "7b382f15a68b0f6e0c1e44dc793ede056d8bd39aff2ca4e878a95654740ee01a", "original_line": 7, "original_line_sha256": "baf514302a99b0c70f032b0d1ea3c07323b3295b1daf7e0980a9dcfd170c13b4", "question_compact_sha256": "21226e190605f136bbceadb0184bd2ea16764d26245f4fcb73d3298caa09ddec", "english_row_compact_sha256": "b6cb8cc3b2add00e5b3695b63d9d33350508962e9d53b7e895b9701be0c89444", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.jsonl" }, "mko_bindings": { "mko-at-most-countable-set": { "sha256": "9c7748686b7974341920a59d07628f20017ef6d65476d751b2feb7424f7b3b57", "english_sha256": "a73faf7f1c04553efad774692eab18a2de59165f0fedaac3ba03ee75c7276def", "original_path": "public/data/mko/at-most-countable-set.json", "english_path": "public/data/i18n/editions/mko-at-most-countable-set.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3", "original_candidate_sha256": "eb527109180b7a7d96428708eb7882fe7e7703e8c81dcc07b7c733d47ac42800", "original_candidate_line_sha256": "eb527109180b7a7d96428708eb7882fe7e7703e8c81dcc07b7c733d47ac42800" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-007-p31", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc", "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c92ef58b9c181e7952db42f62ef5ab0187df791a94254cacb4c669b3f19574c1", "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." } }