{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261011-r05-subset-0084", "pilot_number": "R0009-05109", "number": 5109, "family_id": "qf-density11-subset-candidate-witness-mask", "title": { "zh-Hant": "第 5109 題：三個候選的包含反例資格", "en": "Question 5109: Candidate Inclusion-Counterexample Mask" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261011-r05-subset-0084", "batch_id": "batch-20261008-advance-20261011-r05-subset-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-subset" ], "title_zh": "三個候選的包含反例資格", "stem_zh": "有限整數集合；同一整數型別；A=[-3, -2, -1, 1, 2, 3]、B=[1, 2, 3]、有序候選=[1, 101, 3]。逐項答是否c∈A且c∉B，域外A的候選不算阻擋見證。", "answer": { "kind": "expression", "value": "[否, 否, 否]", "explanation_zh": "包含反例候選資格=[否, 否, 否]。先要求屬A，再排除屬B；共有元素與A外元素都不是反例，不只看候選未進B。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "三個候選的包含反例資格" ] }, "variant": { "family_id": "qf-density11-subset-candidate-witness-mask", "seed": 202610115084, "parameters": { "A": [ -3, -2, -1, 1, 2, 3 ], "B": [ 1, 2, 3 ], "candidates": [ 1, 101, 3 ], "operation": "density11_subset_candidate_witness_mask", "domain": "finite_integer_set_slice", "semantic_template": "qf-density11-subset-candidate-witness-mask/[\"shared\",\"outside_A\",\"shared\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-08", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261008-advance-20261011-r05-subset-0084", "batch_id": "batch-20261008-advance-20261011-r05-subset-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "23c9c877d1ca5b39645a342e3914c0ad0b232862f2dbaa9bd63e6bbb1c6cee31", "family_id": "qf-density11-subset-candidate-witness-mask", "seed": 202610115084, "semantic_template": "qf-density11-subset-candidate-witness-mask/[\"shared\",\"outside_A\",\"shared\"]" }, "fields": { "title": "Candidate Inclusion-Counterexample Mask", "stem": "Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 101, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction.", "answer": { "kind": "expression", "value": "[No, No, No]", "explanation": "Require membership in A before excluding membership in B; shared elements and outside-A elements are not counterexamples. The exact result, retaining the requested order, is [No, No, No]." }, "skills": [ "Candidate Inclusion-Counterexample Mask" ], "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": "有限整數集合；同一整數型別；A=[-3, -2, -1, 1, 2, 3]、B=[1, 2, 3]、有序候選=[1, 101, 3]。逐項答是否c∈A且c∉B，域外A的候選不算阻擋見證。", "en": "Finite integer sets A=[-3, -2, -1, 1, 2, 3] and B=[1, 2, 3]; repeated elements count once. Ordered candidates=[1, 101, 3]. For each c, answer whether c∈A and c∉B; a candidate outside A cannot be an obstruction." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "A=[-3, -2, -1, 1, 2, 3];B=[1, 2, 3];same integer type", "after": "A⊆B means every A-element belongs to B", "relation": "implies", "rule_id": "set_inclusion_definition", "conditions": [], "explanation": { "zh-Hant": "包含的反例必須屬A且不屬B；域外A的數字不是見證，空A不產生阻擋。", "en": "An inclusion obstruction must belong to A but not B;an element outside A is no witness,and empty A gives no obstruction." }, "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": "candidate=1;in A=是;in B=是", "after": "否", "relation": "implies", "rule_id": "ordered_subset_witness_candidate", "conditions": [], "explanation": { "zh-Hant": "必須同時在A與不在B；候選按原順序逐項判定，不因重複或域外數字改變A。", "en": "Require both membership in A and nonmembership in B;judge candidates in their original order without changing A." }, "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": "candidate=101;in A=否;in B=否", "after": "否", "relation": "implies", "rule_id": "ordered_subset_witness_candidate", "conditions": [], "explanation": { "zh-Hant": "必須同時在A與不在B；候選按原順序逐項判定，不因重複或域外數字改變A。", "en": "Require both membership in A and nonmembership in B;judge candidates in their original order without changing A." }, "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": "candidate=3;in A=是;in B=是", "after": "否", "relation": "implies", "rule_id": "ordered_subset_witness_candidate", "conditions": [], "explanation": { "zh-Hant": "必須同時在A與不在B；候選按原順序逐項判定，不因重複或域外數字改變A。", "en": "Require both membership in A and nonmembership in B;judge candidates in their original order without changing A." }, "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": [ false, false, false ], "candidate_rows": [ [ 1, true, true, false ], [ 101, false, false, false ], [ 3, true, true, false ] ] }, "conclusion": { "canonical_value": "[否, 否, 否]", "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-20261008-advance-20261011-r05-subset-foundations", "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r05-subset-foundations/questions.jsonl", "batch_sha256": "faef8ba0762c4b1138b5f3bb62aed94757b8580fef3c3b9df5518b48ec561db7", "original_line": 84, "original_line_sha256": "8fbbb845d39d64edf5b9a2b3075272d424b21641250ff987ea4c8f30ac069485", "question_compact_sha256": "23c9c877d1ca5b39645a342e3914c0ad0b232862f2dbaa9bd63e6bbb1c6cee31", "english_row_compact_sha256": "b19d0a17a5b7a1dd4e7a39b0532a744a6f23f19d64d914e4758340d07e9e32e4", "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r05-subset-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r05-subset-foundations.en.jsonl" }, "mko_bindings": { "mko-subset": { "sha256": "984a77e67dc9322e36b869843513f1cb76f9527912202782848120c123ce99b3", "english_sha256": "af2446cfbbb6eff0aba71545b87591295e79df30d0992a5085cf44be4170d534", "original_path": "public/data/mko/subset.json", "english_path": "public/data/i18n/editions/mko-subset.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f", "original_candidate_sha256": "14b5907b8bf516ec03d48de2849e7d0fb8ca6d17b7225954fb5e6abafb7c443b", "original_candidate_line_sha256": "14b5907b8bf516ec03d48de2849e7d0fb8ca6d17b7225954fb5e6abafb7c443b" }, "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-20261026-005-p10", "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56", "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462" }, "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": "d3c246c04d17752fe59761d61d507f8f669cfd3ec3cf8fc1dab9ca55445296d4", "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." } }