{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261004-advance-20261005-r05-random-variable-0043", "pilot_number": "R0056-28593", "number": 28593, "family_id": "qf-random-variable-threshold-event", "title": { "zh-Hant": "第 28593 題：閾值條件的原像事件", "en": "Question 28593: Preimage Event of a Threshold Condition" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261004-advance-20261005-r05-random-variable-0043", "batch_id": "batch-20261004-advance-20261005-r05-random-variable-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-random-variable" ], "title_zh": "閾值條件的原像事件", "stem_zh": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 19), (3, -15)}。求事件 {s∈S:X(s)≥20}，答基本結果集合而不是機率數值。", "answer": { "kind": "short_text", "value": "∅", "explanation_zh": "事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "閾值條件的原像事件" ] }, "variant": { "family_id": "qf-random-variable-threshold-event", "seed": 202610055043, "parameters": { "S": [ 0, 1, 2, 3 ], "F": "powerset(S)", "values": [ -17, -17, 19, -15 ], "t": 20, "operation": "rv_threshold", "semantic_template": "qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_threshold\",\"t\":20}" } }, "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": "Codex independent producer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-04", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261004-advance-20261005-r05-random-variable-0043", "batch_id": "batch-20261004-advance-20261005-r05-random-variable-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4114d2931c899a08737373557cd7b3cafcaacd95677b51aef5b6133d6db0582a", "family_id": "qf-random-variable-threshold-event", "seed": 202610055043, "semantic_template": "qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_threshold\",\"t\":20}" }, "fields": { "title": "Preimage Event of a Threshold Condition", "stem": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the event {s∈S:X(s)≥20} in increasing order. Return elementary outcomes, not a numerical probability.", "answer": { "kind": "short_text", "value": "∅", "explanation": "Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted." }, "skills": [ "Preimage Event of a Threshold Condition" ], "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": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 19), (3, -15)}。求事件 {s∈S:X(s)≥20}，答基本結果集合而不是機率數值。", "en": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the event {s∈S:X(s)≥20} in increasing order. Return elementary outcomes, not a numerical probability." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-random-variable-threshold-event, rv_threshold]", "after": "derived = [inclusive_threshold_OUTCOME_set_not_probability]", "relation": "implies", "rule_id": "R56_source_answer_role_and_EXACT_family_interface", "conditions": [], "explanation": { "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。", "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[0, 1, 2, 3], powerset(S), [-17, -17, 19, -15]]", "after": "derived = [[[0, -17], [1, -17], [2, 19], [3, -15]]]", "relation": "implies", "rule_id": "RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment", "conditions": [], "explanation": { "zh-Hant": "values逐slot對齊原S，不能把兩列分開排序。有限full powerset包含每個X的逆像，故這個實值函數可測；求value／range／event不須另造均勻權重。", "en": "Align each value with the ORIGINAL S slot,not independently sorted columns.The full powerset contains every preimage of this real-valued function,hence measurability.No invented uniform weights are needed for a value,range or event." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [[[0, -17], [1, -17], [2, 19], [3, -15]], 20]", "after": "derived = [[[0, -17, 否], [1, -17, 否], [2, 19, 否], [3, -15, 否]], [], ∅]", "relation": "implies", "rule_id": "RV_INCLUSIVE_threshold_ALL_outcome_preimages", "conditions": [], "explanation": { "zh-Hant": "逐點用X(s)>=t（含等號），收集原outcome s。輸出事件集合，不把值集合或其機率混進來；無符合者空集合。", "en": "Test X(s)>=t INCLUSIVELY at EVERY outcome,and collect the original s.Return the event set,not a value set or probability;no matches give the empty set." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null } ], "math_model": { "result": "∅" }, "conclusion": { "canonical_value": "∅", "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-20261004-advance-20261005-r05-random-variable-foundations", "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r05-random-variable-foundations/questions.jsonl", "batch_sha256": "e65fbafe078025619c0c64f1e8c66b8d0c0386bc721517cbf25b94c3d3ba5e1f", "original_line": 43, "original_line_sha256": "760f7365361695572d45c65555fb8152fbb2060fa29fe545ae8f47f8e32a565e", "question_compact_sha256": "4114d2931c899a08737373557cd7b3cafcaacd95677b51aef5b6133d6db0582a", "english_row_compact_sha256": "97bc65fedc444c59369ae9394911d0c673cbdd6134e2a284076846968e7d8278", "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r05-random-variable-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r05-random-variable-foundations.en.jsonl" }, "mko_bindings": { "mko-random-variable": { "sha256": "bba00300a619b37481007904b211059750f93a4a2c4ba2b8209eac472ad33e06", "english_sha256": "456778ef562a2c57a14293522fad0ac209632686518a816ad9409094a4c350e1", "original_path": "public/data/mko/random-variable.json", "english_path": "public/data/i18n/editions/mko-random-variable.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5", "original_candidate_sha256": "35942a1b0edd58a2e6eb2433e227bfffebfba0ddc9073d253f82e39cc8ac2f7c", "original_candidate_line_sha256": "35942a1b0edd58a2e6eb2433e227bfffebfba0ddc9073d253f82e39cc8ac2f7c" }, "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-20261026-010-p57", "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8", "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f" }, "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": "0ad2a3db21080309b4235f495a8d79d027914c1d6740ebcd86d256dba34fce0d", "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." } }