{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r02-real-number-0055", "pilot_number": "R0012-06830", "number": 6830, "family_id": "qf-density14-real-overlap-closed-intervals", "title": { "zh-Hant": "第 6830 題：兩個閉區間的交與間隔", "en": "Question 6830: Closed Interval Intersection and Separation Gap" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r02-real-number-0055", "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-number" ], "title_zh": "兩個閉區間的交與間隔", "stem_zh": "有理點嵌入實數的有限切片；閉區間包含端點；I=[-4/2, 4/2]、J=[0/1, 6/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。", "answer": { "kind": "expression", "value": "(是, [0/1, 2/1], 0/1)", "explanation_zh": "閉區間交集與分離距離=(是, [0/1, 2/1], 0/1)。取左端最大值、右端最小值，交時間隔0/1；接觸端點不是空集。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": [ "兩個閉區間的交與間隔" ] }, "variant": { "family_id": "qf-density14-real-overlap-closed-intervals", "seed": 202610142055, "parameters": { "I": [ [ -4, 2 ], [ 4, 2 ] ], "J": [ [ 0, 1 ], [ 6, 2 ] ], "domain": "finite_rational_real_slice", "operation": "density14_real_overlap_closed_intervals", "semantic_template": "qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"2/1\"]" } }, "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-0055", "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "be40abdfdc810b6494d5407a8fa1c76680226fd9be336eab0029e594dbd8a6cc", "family_id": "qf-density14-real-overlap-closed-intervals", "seed": 202610142055, "semantic_template": "qf-density14-real-overlap-closed-intervals/[\"intersection\",\"0/1\",\"2/1\"]" }, "fields": { "title": "Closed Interval Intersection and Separation Gap", "stem": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-4/2, 4/2], J=[0/1, 6/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.", "answer": { "kind": "expression", "value": "(Yes, [0/1, 2/1], 0/1)", "explanation": "Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 2/1], 0/1)." }, "skills": [ "Closed Interval Intersection and Separation Gap" ], "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；閉區間包含端點；I=[-4/2, 4/2]、J=[0/1, 6/2]，各端點有序。依序答 (交非空是否, 交的[lo,hi]或[], 分離間隔)。", "en": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-4/2, 4/2], J=[0/1, 6/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers." } } ], "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": "I=[-2,2];J=[0,3]", "after": "intersection bounds=[max left=0,min right=2]", "relation": "implies", "rule_id": "closed_interval_inner_bounds", "conditions": [], "explanation": { "zh-Hant": "閉區間包含端點，交集取較大左端與較小右端；lo=hi仍有一個共同點。", "en": "Closed intervals include endpoints;the intersection uses the larger left and smaller right bound.Even lo=hi gives a common point." }, "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": "lo=0;hi=2;lo≤hi=是", "after": "intersection=[0/1, 2/1];separation=0/1", "relation": "implies", "rule_id": "closed_contact_not_empty", "conditions": [], "explanation": { "zh-Hant": "接觸端點不是空集，分離間隔為0；只有lo>hi才空，不以嚴格小於代替閉區間判準。", "en": "Endpoint contact is not empty and has gap0;only lo>hi is empty.Do not replace the closed-interval condition with strict inequality." }, "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": [ true, [ "0", "2" ], "0" ] }, "conclusion": { "canonical_value": "(是, [0/1, 2/1], 0/1)", "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": 55, "original_line_sha256": "67617074168c5df88c163c3cfec454f396f9945ab13259cd4d04619f0393cc4c", "question_compact_sha256": "be40abdfdc810b6494d5407a8fa1c76680226fd9be336eab0029e594dbd8a6cc", "english_row_compact_sha256": "7da06b3aa2dee8e85728247902aac4c862b0da546b555cad9f4dc2994a307803", "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": "4a4e07e90cf8c8c5ba9c2b4571f8658212d662ab935d562385e71bbeef58d3a2", "original_candidate_line_sha256": "4a4e07e90cf8c8c5ba9c2b4571f8658212d662ab935d562385e71bbeef58d3a2" }, "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "def98c3682094c1caf72c7bf7ea0b49b762437bd7223a4124b27e89175fcf511", "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." } }