{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r07-0061", "pilot_number": "P1000-0886", "number": 886, "family_id": "qf-mod50-optdeep-lower-bound-certificate", "title": { "zh-Hant": "第 886 題：有限最佳化", "en": "Question 886: A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r07-0061", "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-optimization-problem" ], "title_zh": "有限最佳化", "stem_zh": "完整有限feasible domain、明示neighborhood與exact objective table。操作=optdeep-lower-bound-certificate；明示資料={\"candidate\":\"A\",\"lower_bound\":11,\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[11,22,33,44]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, 是, 是, 是, 是, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)", "explanation_zh": "Wave6第十三層核對=([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, 是, 是, 是, 是, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)；依完整明示資料精確重算；local只比較全部neighbors，global比較全部feasible points；lower bound須完整且attained；strict improvement只是否定證書；empty/incomplete domain與ties不得冒充存在、global或unique optimum。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限最佳化" ] }, "variant": { "family_id": "qf-mod50-optdeep-lower-bound-certificate", "seed": 202611197061, "parameters": { "points": [ "A", "B", "C", "D" ], "values": [ 11, 22, 33, 44 ], "candidate": "A", "lower_bound": 11, "domain": "Exact_finite_optimization_density", "operation": "optdeep-lower-bound-certificate", "semantic_template": "optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[11,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[11,1],[22,1],[33,1],[44,1]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_contract_recomputation", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-pulse-20261119-r07-0061", "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "843c657b79425a3fe894e026d33e93509ce15a16e4d8f026dd20865297159cd8", "family_id": "qf-mod50-optdeep-lower-bound-certificate", "seed": 202611197061, "semantic_template": "optdeep-lower-bound-certificate|{\"candidate\":\"A\",\"lower_bound\":[11,1],\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[[11,1],[22,1],[33,1],[44,1]]}" }, "fields": { "title": "A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness", "stem": "Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[11, 22, 33, 44], candidate=A. The declared global lower bound is=11. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope).", "answer": { "kind": "expression", "value": "([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)", "explanation": "Check the bound against every feasible value and exact attainment at the candidate. Retain all minimizers and set the uniqueness flag only if exactly one attains the bound. The certificate applies to this complete finite feasible domain, not unlisted candidates or continuous problems. The exact result, retaining the requested order, is ([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, Yes, Yes, Yes, Yes, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)." }, "skills": [ "A Complete Attained Lower Bound Certifies Globality but Not Automatic Uniqueness" ], "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": "完整有限feasible domain、明示neighborhood與exact objective table。操作=optdeep-lower-bound-certificate；明示資料={\"candidate\":\"A\",\"lower_bound\":11,\"points\":[\"A\",\"B\",\"C\",\"D\"],\"values\":[11,22,33,44]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use the complete finite feasible carrier=[A, B, C, D] with aligned exact objective values=[11, 22, 33, 44], candidate=A. The declared global lower bound is=11. Give (complete_objective_table, lower_bound, candidate, candidate_value, all_values_above_bound, bound_attained, global_optimum, unique, all_global_minimizers, certificate_scope)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "complete feasible domain", "after": "[[A, 11], [B, 22], [C, 33], [D, 44]]", "relation": "implies", "rule_id": "complete_objective_data", "conditions": [], "explanation": { "zh-Hant": "完整表與最小化方向都先固定，不能只抽樣幾點當全域證書。", "en": "Fix the complete table and minimization direction; sampling a few points is not a global certificate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。", "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "f(A)−L := sub(11, 11)", "after": "f(A)−L = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐點核對這真的是完整域的下界。", "en": "Check pointwise that this bounds the entire feasible domain below." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f(A)−L，不以小數近似替代。", "en": "Recompute f(A)−L with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "11", "11" ], "output": "0", "label": "f(A)−L" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "f(B)−L := sub(22, 11)", "after": "f(B)−L = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐點核對這真的是完整域的下界。", "en": "Check pointwise that this bounds the entire feasible domain below." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f(B)−L，不以小數近似替代。", "en": "Recompute f(B)−L with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "22", "11" ], "output": "11", "label": "f(B)−L" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "f(C)−L := sub(33, 11)", "after": "f(C)−L = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐點核對這真的是完整域的下界。", "en": "Check pointwise that this bounds the entire feasible domain below." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f(C)−L，不以小數近似替代。", "en": "Recompute f(C)−L with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "33", "11" ], "output": "22", "label": "f(C)−L" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "f(D)−L := sub(44, 11)", "after": "f(D)−L = 33", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐點核對這真的是完整域的下界。", "en": "Check pointwise that this bounds the entire feasible domain below." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 f(D)−L，不以小數近似替代。", "en": "Recompute f(D)−L with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "44", "11" ], "output": "33", "label": "f(D)−L" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "all f(x)≥L=11", "after": "f(A)=11=L; attained=是", "relation": "implies", "rule_id": "lower_bound_attainment", "conditions": [], "explanation": { "zh-Hant": "下界本身不保證有點達到，須額外核對指定候選的值。", "en": "A lower bound alone does not guarantee attainment; verify the designated candidate's value separately." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。", "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "all attaining points=[A]", "after": "global=true; unique=是", "relation": "implies", "rule_id": "global_certificate_and_uniqueness", "conditions": [], "explanation": { "zh-Hant": "完整下界加達界證明全域最小；唯一性另由全部達界點的數量決定。", "en": "A complete lower bound plus attainment certifies global minimality; uniqueness separately depends on the complete attaining set." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "完整域、鄰域、達界與唯一性是不同責任，不能從局部資料偷換全域結論。", "en": "Completeness, neighborhoods, attainment and uniqueness are separate obligations; local data do not license a global conclusion." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ "A", "11" ], [ "B", "22" ], [ "C", "33" ], [ "D", "44" ] ], "11", "A", "11", true, true, true, true, [ "A" ], "complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness" ] }, "conclusion": { "canonical_value": "([[A, 11], [B, 22], [C, 33], [D, 44]], 11, A, 11, 是, 是, 是, 是, [A], complete_lower_bound_plus_attainment_certifies_global_but_not_automatic_uniqueness)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "完整下界加上候選點達界證明全域最小；唯一性另由全部達界點決定。", "en": "A complete lower bound attained at the candidate proves global minimality; uniqueness is decided separately by the attaining set." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261119-r07-optimization-problem-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r07-optimization-problem-foundations/questions.jsonl", "batch_sha256": "7ca0a9388bb0b3ed98a619610814bbde72bb9ea5459f6f9f7e636bc2261b7d2d", "original_line": 61, "original_line_sha256": "5d8b43c19b5d6b512880f20f5cd9c9f34cb39102ed60b3fa4c9900ed77ea929e", "question_compact_sha256": "843c657b79425a3fe894e026d33e93509ce15a16e4d8f026dd20865297159cd8", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r07-optimization-problem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r07-optimization-problem-foundations.en.jsonl", "english_row_compact_sha256": "2adede791b497027ac9c1d56cf150ff5af8c9f48caf269b31a2c854e1ea1d5ad" }, "mko_bindings": { "mko-optimization-problem": { "sha256": "55c8accf83a0b3a16cb86f438bdc6575f8dcec45e670f324e863e79a1a233514", "english_sha256": "b6379fa21189b83b931814f4a2acdc266e8f73e8fa2fad6e9d2190456191279e", "original_path": "public/data/mko/optimization-problem.json", "english_path": "public/data/i18n/editions/mko-optimization-problem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "1ea404b45d550a21a20b28ae3c1a73e0e8e5e505639f2019d465fa601128ec35", "original_candidate_line_sha256": "1ea404b45d550a21a20b28ae3c1a73e0e8e5e505639f2019d465fa601128ec35" }, "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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-20261026-003-03", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "936944cda51e0820676afe37171232bc6f877aa2cd05c18471d1498f49821daf", "arithmetic_steps": 4 }, "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." } }