{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261107-r02-supremum-and-infimum-0070", "pilot_number": "R0040-20545", "number": 20545, "family_id": "qf-mod38-supinf-lower-bound-greatestness", "title": { "zh-Hant": "第 20545 題：上確界與下確界", "en": "Question 20545: A Lower Bound Still Needs Greatestness to Be the Infimum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261107-r02-supremum-and-infimum-0070", "batch_id": "batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-supremum-and-infimum" ], "title_zh": "上確界與下確界", "stem_zh": "非空且相應有界的實值集合或明示有界開區間。操作=supinf-lower-bound-greatestness；資料={\"values\":[[20,21],[-21,22],[41,23]],\"candidate\":[-21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-21/22, 是, -21/22, 是, lower_bound_and_greatestness_are_separate)", "explanation_zh": "Wave6根層核對=(-21/22, 是, -21/22, 是, lower_bound_and_greatestness_are_separate)；依完整明示資料獨立重算；sup同時是上界與最小上界，inf對偶；兩項責任不可省略；sup與inf不必屬於集合，只有取到時才是maximum或minimum；空集與無界集另處理。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "上確界與下確界" ] }, "variant": { "family_id": "qf-mod38-supinf-lower-bound-greatestness", "seed": 202611072070, "parameters": { "values": [ [ 20, 21 ], [ -21, 22 ], [ 41, 23 ] ], "candidate": [ -21, 22 ], "domain": "Exact_bounded_real_set_extrema", "operation": "supinf-lower-bound-greatestness", "semantic_template": "qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-21,22],\"values\":[[20,21],[-21,22],[41,23]]}" } }, "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；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或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-20261107-r02-supremum-and-infimum-0070", "batch_id": "batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "311fc4a80ebed541c1f4f1de56289b0bbc63eafb9cabcebed0dfa14d59e3d491", "family_id": "qf-mod38-supinf-lower-bound-greatestness", "seed": 202611072070, "semantic_template": "qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-21,22],\"values\":[[20,21],[-21,22],[41,23]]}" }, "fields": { "title": "A Lower Bound Still Needs Greatestness to Be the Infimum", "stem": "The ordered rational values=[20/21, -21/22, 41/23] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-21/22. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).", "answer": { "kind": "expression", "value": "(-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate)", "explanation": "Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate)." }, "skills": [ "A Lower Bound Still Needs Greatestness to Be the Infimum" ], "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": "非空且相應有界的實值集合或明示有界開區間。操作=supinf-lower-bound-greatestness；資料={\"values\":[[20,21],[-21,22],[41,23]],\"candidate\":[-21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "The ordered rational values=[20/21, -21/22, 41/23] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-21/22. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=supinf-lower-bound-greatestness;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[candidate, is_lower_bound, infimum, equals_infimum, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。", "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope." }, "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 complete ordered nonempty finite real-value list=[20/21, -21/22, 41/23]", "after": "initialize upper/lower extremum candidates at ORIGINAL first value;duplicates retained and do not change set bounds", "relation": "implies", "rule_id": "supinf_complete_nonempty_finite_set_not_samples_or_sorting", "conditions": [], "explanation": { "zh-Hant": "這是完整有限集合的給定列表，不是未知集合的有限取樣。重複值可出現但不改sup/inf，輸出仍按原順序保留，不先排序資料。", "en": "This is the supplied complete finite set list,not a sample of an unspecified set.Duplicates do not alter bounds but remain in original output order;do not sort the data first." }, "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": "supinf original value1 minus previous maximum := sub(-21/22, 20/21)", "after": "supinf original value1 minus previous maximum = -881/462", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf original value1 minus previous maximum，不以小數近似替代。", "en": "Recompute supinf original value1 minus previous maximum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-21/22", "20/21" ], "output": "-881/462", "label": "supinf original value1 minus previous maximum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "supinf original value1 minus previous minimum := sub(-21/22, 20/21)", "after": "supinf original value1 minus previous minimum = -881/462", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf original value1 minus previous minimum，不以小數近似替代。", "en": "Recompute supinf original value1 minus previous minimum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-21/22", "20/21" ], "output": "-881/462", "label": "supinf original value1 minus previous minimum" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "ORIGINAL next index=1;value=-21/22;previous extremum values=[20/21, 20/21];differences=[-881/462, -881/462]", "after": "prefix maximum=20/21 attained at original index0;prefix minimum=-21/22 attained at original index1", "relation": "implies", "rule_id": "supinf_every_value_extremum_prefix_invariant", "conditions": [], "explanation": { "zh-Hant": "逐全部原值比較保prefix最大／最小不變式，只在嚴格變大／變小時改witness；沒有漏掉的原值。", "en": "Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted." }, "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": "S006", "uses": [ "G001", "S005" ], "before": "supinf original value2 minus previous maximum := sub(41/23, 20/21)", "after": "supinf original value2 minus previous maximum = 401/483", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf original value2 minus previous maximum，不以小數近似替代。", "en": "Recompute supinf original value2 minus previous maximum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "41/23", "20/21" ], "output": "401/483", "label": "supinf original value2 minus previous maximum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "supinf original value2 minus previous minimum := sub(41/23, -21/22)", "after": "supinf original value2 minus previous minimum = 1385/506", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf original value2 minus previous minimum，不以小數近似替代。", "en": "Recompute supinf original value2 minus previous minimum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "41/23", "-21/22" ], "output": "1385/506", "label": "supinf original value2 minus previous minimum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL next index=2;value=41/23;previous extremum values=[20/21, -21/22];differences=[401/483, 1385/506]", "after": "prefix maximum=41/23 attained at original index2;prefix minimum=-21/22 attained at original index1", "relation": "implies", "rule_id": "supinf_every_value_extremum_prefix_invariant", "conditions": [], "explanation": { "zh-Hant": "逐全部原值比較保prefix最大／最小不變式，只在嚴格變大／變小時改witness；沒有漏掉的原值。", "en": "Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "ALL original values processed;actual maximum witness=[2, 41/23];actual minimum witness=[1, -21/22]", "after": "EVERY listed value between-21/22 and41/23;ANY upper bound must>=actual maximum in the set;ANY lower bound must<=actual minimum;sup=41/23,inf=-21/22,BOTH attained", "relation": "implies", "rule_id": "supinf_bounds_and_least_greatest_by_actual_attaining_points", "conditions": [], "explanation": { "zh-Hant": "最大點本身屬集合，故任何上界都不能比它小；最小點亦迫任何下界不能更大。這同時證bound與leastness／greatestness，有限非空才有此取到性，不外推一般有界集合。", "en": "Because the maximum is itself in the set,no upper bound can be smaller;the minimum similarly bounds every lower bound above.This proves both bounding and leastness/greatestness.Attainment uses finiteness and nonemptiness and is not inferred for arbitrary bounded sets." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "supinf candidate bound test original value0 := sub(20/21, -21/22)", "after": "supinf candidate bound test original value0 = 881/462", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf candidate bound test original value0，不以小數近似替代。", "en": "Recompute supinf candidate bound test original value0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "20/21", "-21/22" ], "output": "881/462", "label": "supinf candidate bound test original value0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "supinf candidate bound test original value1 := sub(-21/22, -21/22)", "after": "supinf candidate bound test original value1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf candidate bound test original value1，不以小數近似替代。", "en": "Recompute supinf candidate bound test original value1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-21/22", "-21/22" ], "output": "0", "label": "supinf candidate bound test original value1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "supinf candidate bound test original value2 := sub(41/23, -21/22)", "after": "supinf candidate bound test original value2 = 1385/506", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 supinf candidate bound test original value2，不以小數近似替代。", "en": "Recompute supinf candidate bound test original value2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "41/23", "-21/22" ], "output": "1385/506", "label": "supinf candidate bound test original value2" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "ORIGINAL candidate=-21/22;lower-bound flags for EVERY original value=[是, 是, 是];separate actual extremum=-21/22", "after": "candidate is bound=是;candidate equals INF=是;a loose bound and a failed candidate are not the extremum", "relation": "implies", "rule_id": "supinf_candidate_bound_and_least_greatest_separate_flags", "conditions": [], "explanation": { "zh-Hant": "候選bound要逐值全稱核；等於已證sup／inf才是相應最緊界。失敗只否決此candidate，不表示集合無界；鬆界通過也不等於最小上界／最大下界。", "en": "Audit the candidate against every value.Equality with the established sup/inf gives the tight bound.A failed candidate does not make the set unbounded,and a loose passing bound is not the least upper/greatest lower bound." }, "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": [ "-21/22", true, "-21/22", true, "lower_bound_and_greatestness_are_separate" ] }, "conclusion": { "canonical_value": "(-21/22, 是, -21/22, 是, lower_bound_and_greatestness_are_separate)", "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-20261107-r02-supremum-and-infimum-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations/questions.jsonl", "batch_sha256": "047c9f5976d4882d33c3c8da47da945de9d425dce6a7e4baaf5442fe93ffba15", "original_line": 70, "original_line_sha256": "790499771915e1aca9764216e0904ab223b081d97d8f280924e67e0dca93d077", "question_compact_sha256": "311fc4a80ebed541c1f4f1de56289b0bbc63eafb9cabcebed0dfa14d59e3d491", "english_row_compact_sha256": "2e49965da0e6e91d3d40e276fff37466a9bd1ab93123442bcd71132ce9704f17", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations.en.jsonl" }, "mko_bindings": { "mko-supremum-and-infimum": { "sha256": "bb254a6cba1cbd62c80ad2e7af827c64072b05dda798122c533bdf47277f2434", "english_sha256": "bf93ac3e45d1cda3acf7f3835fda337bbe68e94e511b2b2c87995468091928e4", "original_path": "public/data/mko/supremum-and-infimum.json", "english_path": "public/data/i18n/editions/mko-supremum-and-infimum.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b", "original_candidate_sha256": "a9020520a1ac5ddd291c5e7d6fe01e46980a2efb982223d1334e10fe35133a61", "original_candidate_line_sha256": "a9020520a1ac5ddd291c5e7d6fe01e46980a2efb982223d1334e10fe35133a61", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-supremum-and-infimum": "5489eb105408d11dade87bd89fad920ba88cc24df13e7cea9016d35cb54c7c1e" } } }, "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-20261026-008-p41", "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": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d", "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 13, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d7723d53628626f9c69fe75ca49ab873fcbb50404ab9ba48d06f575ce23ae96c", "arithmetic_steps": 7 }, "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." } }