{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r02-real-number-0029", "pilot_number": "R0012-06854", "number": 6854, "family_id": "qf-density14-real-point-closed-hull", "title": { "zh-Hant": "第 6854 題：有限點列與最小包含閉區間", "en": "Question 6854: Finite Distinct Points and Least Containing Closed Interval" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r02-real-number-0029", "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-number" ], "title_zh": "有限點列與最小包含閉區間", "stem_zh": "有理點嵌入實數的有限切片；points=[-3/2, 0/1, 4/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。", "answer": { "kind": "expression", "value": "([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)", "explanation_zh": "有限點列與最小閉區間=([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)。按有理數實值排序去重，再取兩端；最小包含閉區間非點集拓撲閉包。非實數完備性或無理數分類證明，不用近似小數。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "有限點列與最小包含閉區間" ] }, "variant": { "family_id": "qf-density14-real-point-closed-hull", "seed": 202610142029, "parameters": { "points": [ [ -3, 2 ], [ 0, 1 ], [ 4, 3 ] ], "domain": "finite_rational_real_slice", "operation": "density14_real_point_closed_hull", "semantic_template": "qf-density14-real-point-closed-hull/[\"-3/2\",\"0/1\",\"4/3\"]" } }, "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": "2026-10-10", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261010-advance-20261014-r02-real-number-0029", "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3be5c40be76738a4628e07a260366eefa5e066f4b526ded6e3f2f2b2203fda74", "family_id": "qf-density14-real-point-closed-hull", "seed": 202610142029, "semantic_template": "qf-density14-real-point-closed-hull/[\"-3/2\",\"0/1\",\"4/3\"]" }, "fields": { "title": "Finite Distinct Points and Least Containing Closed Interval", "stem": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-3/2, 0/1, 4/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.", "answer": { "kind": "expression", "value": "([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)", "explanation": "Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)." }, "skills": [ "Finite Distinct Points and Least Containing Closed Interval" ], "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": "有理點嵌入實數的有限切片；points=[-3/2, 0/1, 4/3]。依序列 (遞增去重點列, 最小端點lo, 最大端點hi, hi−lo)，所求包含區間是[lo,hi]。", "en": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-3/2, 0/1, 4/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "explicit rational points embedded in the real line", "after": "exact finite order and distance data,not real completeness or irrational classification", "relation": "implies", "rule_id": "rational_real_slice_scope", "conditions": [], "explanation": { "zh-Hant": "以精確有理值處理這個實數中的有限切片，不用近似小數，也不宣稱已證實數完備性或分類無理數。", "en": "Use exact rational values for this finite slice of the real line,without rounded decimals or claims about real completeness or classification of irrationals." }, "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": "[[-3, 2], [0, 1], [4, 3]]", "after": "[-3/2, 0/1, 4/3]", "relation": "implies", "rule_id": "rational_point_numeric_normalization", "conditions": [], "explanation": { "zh-Hant": "按有理數實值而非分數字串排序去重，0/−2與0/3都表示同一點0。", "en": "Sort and deduplicate by rational value,not fraction strings;0/−2 and0/3 represent the same point0." }, "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": "smallest containing interval width := sub(4/3, -3/2)", "after": "smallest containing interval width = 17/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "最右點減最左點給非負區間寬度。", "en": "Rightmost minus leftmost point gives the nonnegative interval width." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 smallest containing interval width，不以小數近似替代。", "en": "Recompute smallest containing interval width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4/3", "-3/2" ], "output": "17/6", "label": "smallest containing interval width" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "all points between -3/2 and 4/3", "after": "smallest containing closed interval=[-3/2,4/3]", "relation": "implies", "rule_id": "containing_interval_not_topological_closure", "conditions": [], "explanation": { "zh-Hant": "任何包含全部點的閉區間都必含最小與最大點，故此區間最小；這不是把有限點集的拓撲閉包填成整段。", "en": "Every containing closed interval must include the smallest and largest points,hence this interval is minimal.This is not a claim that topological closure fills the gaps of a finite point set." }, "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": [ [ "-3/2", "0", "4/3" ], "-3/2", "4/3", "17/6" ] }, "conclusion": { "canonical_value": "([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)", "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-20261010-advance-20261014-r02-real-number-foundations", "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r02-real-number-foundations/questions.jsonl", "batch_sha256": "a21835c6da0df0833d78ad956806cd71881509cf26981f7af7c4dc492119e05c", "original_line": 29, "original_line_sha256": "122c1b65a496605549b9c37099e27ea91d63c874f677bcb3bc71ffdcddad2996", "question_compact_sha256": "3be5c40be76738a4628e07a260366eefa5e066f4b526ded6e3f2f2b2203fda74", "english_row_compact_sha256": "b5ffeeb7727123e178e620e85e530d06a5b264b048caa9f78adff46297eee5ef", "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r02-real-number-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r02-real-number-foundations.en.jsonl" }, "mko_bindings": { "mko-real-number": { "sha256": "2bc8c0a1defae2276ce50808a0761bf4d62f26892e443fcab79fc8090113d880", "english_sha256": "5dde700ff98e1ae0b35a5924253f8ad22515aa33c9a17deab8b7fa9a35e1ef5c", "original_path": "public/data/mko/real-number.json", "english_path": "public/data/i18n/editions/mko-real-number.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99", "original_candidate_sha256": "85ff472af0fdd5aa97593d0fa8068c89eaf4db174afc6c2b7ae2af01ad051705", "original_candidate_line_sha256": "85ff472af0fdd5aa97593d0fa8068c89eaf4db174afc6c2b7ae2af01ad051705" }, "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-p13", "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": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00", "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9" }, "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": "47b5cbf5a22276ae1554cd0a68e9397aa116ecb7507aec5d66e7b69cf99edefe", "arithmetic_steps": 1 }, "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." } }