{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261024-r07-convex-set-0019", "pilot_number": "R0021-11194", "number": 11194, "family_id": "qf-mod24-convexset-interval-certificate", "title": { "zh-Hant": "第 11194 題：凸集合證書與反例", "en": "Question 11194: A Universal Closed-Interval Convexity Certificate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261024-r07-convex-set-0019", "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-convex-set" ], "title_zh": "凸集合證書與反例", "stem_zh": "量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-20,\"upper\":21}。請依 [lower,upper,universal_scope,convex] 欄序回答。", "answer": { "kind": "expression", "value": "(-20, 21, all x,y in interval and all lambda in [0,1], 是)", "explanation_zh": "依賴結構核對=(-20, 21, all x,y in interval and all lambda in [0,1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；正例使用完整代數證書而非有限lambda抽樣；單一精確反例足以否證但反例失敗不證凸。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "凸集合證書與反例" ] }, "variant": { "family_id": "qf-mod24-convexset-interval-certificate", "seed": 202610247019, "parameters": { "lower": -20, "upper": 21, "domain": "Exact_algebraic_convex_set_certificates", "operation": "convexset-interval-certificate", "semantic_template": "qf-mod24-convexset-interval-certificate/{\"lower\":-20,\"upper\":21}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261024-r07-convex-set-0019", "batch_id": "batch-20261013-advance-20261024-r07-convex-set-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "53f0551703df6e5994190a8a63c824e5fec797bdd47722ec238548fe04235338", "family_id": "qf-mod24-convexset-interval-certificate", "seed": 202610247019, "semantic_template": "qf-mod24-convexset-interval-certificate/{\"lower\":-20,\"upper\":21}" }, "fields": { "title": "A Universal Closed-Interval Convexity Certificate", "stem": "Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-20, upper=21, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).", "answer": { "kind": "expression", "value": "(-20, 21, all x,y in interval and all lambda in [0,1], Yes)", "explanation": "For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-20, 21, all x,y in interval and all lambda in [0,1], Yes)." }, "skills": [ "A Universal Closed-Interval Convexity Certificate" ], "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": "量化所有端點與所有lambda in [0,1]的凸性語義。操作=convexset-interval-certificate；資料={\"lower\":-20,\"upper\":21}。請依 [lower,upper,universal_scope,convex] 欄序回答。", "en": "Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-20, upper=21, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "interval:closed bounds=[-20,21];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1", "after": "lambda>=0;1-lambda>=0;x--20>=0;y--20>=0;21-x>=0;21-y>=0", "relation": "implies", "rule_id": "universal_interval_arbitrary_members", "conditions": [], "explanation": { "zh-Hant": "任取兩個區間成員與任意容許lambda，四個端點差都非負；上下界相等也合法。", "en": "Choose arbitrary interval members and any permitted lambda.All four endpoint slacks are nonnegative;equal bounds are valid." }, "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": "z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,21]", "after": "z--20=lambda*(x--20)+(1-lambda)*(y--20)>=0", "relation": "implies", "rule_id": "universal_interval_lower_identity", "conditions": [], "explanation": { "zh-Hant": "用權重和為1展開z減下界，成為兩個非負乘積之和；不是抽樣代值。", "en": "Use the weights'unit sum to expand z minus the lower bound as a sum of nonnegative products.This is not sampling." }, "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": "z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,21]", "after": "21-z=lambda*(21-x)+(1-lambda)*(21-y)>=0", "relation": "implies", "rule_id": "universal_interval_upper_identity", "conditions": [], "explanation": { "zh-Hant": "同理上界減z也是非負和，因此同時有下界≤z≤上界。", "en": "Likewise the upper bound minus z is nonnegative,so both bounding inequalities hold." }, "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": "interval:both symbolic slack identities hold for every chosen x,y,lambda", "after": "all x,y in [-20,21] and all lambda in [0,1] give z in [-20,21]", "relation": "implies", "rule_id": "universal_interval_quantifier_discharge", "conditions": [], "explanation": { "zh-Hant": "任選的端點與lambda未加特殊限制，故可釋放所有量詞，得到整個閉區間凸性。", "en": "No special restriction was added to the arbitrary endpoints or lambda.Discharge their universal quantifiers to obtain convexity of the whole closed interval." }, "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": [ -20, 21, "all x,y in interval and all lambda in [0,1]", true ] }, "conclusion": { "canonical_value": "(-20, 21, all x,y in interval and all lambda in [0,1], 是)", "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-20261013-advance-20261024-r07-convex-set-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r07-convex-set-foundations/questions.jsonl", "batch_sha256": "51606fc9d1c6f335d18e50f6488d4d2b08c7343abef8be6c7d1fa99ccc3023b0", "original_line": 19, "original_line_sha256": "310b18dc902211c49421e8491c089323671808236cd2c94e34c5f3336ca45256", "question_compact_sha256": "53f0551703df6e5994190a8a63c824e5fec797bdd47722ec238548fe04235338", "english_row_compact_sha256": "31e3996dd430dff8b0f92e7fa729ff267e2b813c08d10d2d5f4d78c6f53c6c8d", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r07-convex-set-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r07-convex-set-foundations.en.jsonl" }, "mko_bindings": { "mko-convex-set": { "sha256": "0d52be71dca290f856018d176bbb26dcd877c887031f690bd1d656db017ce02d", "english_sha256": "f81d0d2b42596c9a4b95349d75fc98970570af520e6edb9c1af988358ff84f0c", "original_path": "public/data/mko/convex-set.json", "english_path": "public/data/i18n/editions/mko-convex-set.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5", "original_candidate_sha256": "0ef4a1d993e62ac5b1e42e4e2b5fa9ee233e24e21f0dc95d482fb259d24d70c5", "original_candidate_line_sha256": "0ef4a1d993e62ac5b1e42e4e2b5fa9ee233e24e21f0dc95d482fb259d24d70c5" }, "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-20261026-006-p22", "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": "47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258", "question_first_witness_sha256": "a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139" }, "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": "d4ecd480a33749973cff475638cff0fc31b5b814f6ebcf68cef25586d735113b", "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." } }