{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r05-measurable-function-0011", "pilot_number": "R0065-33436", "number": 33436, "family_id": "qf-mod25-measdeep-preimage-complement-law", "title": { "zh-Hant": "第 33436 題：可測函數密度", "en": "Question 33436: Complement under a Complete Total-Map Preimage" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r05-measurable-function-0011", "batch_id": "batch-20261013-advance-20261025-r05-measurable-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-measurable-function" ], "title_zh": "可測函數密度", "stem_zh": "完整有限sigma代數與總映射。操作=measdeep-preimage-complement-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,2,0,0],\"event\":[2,3]}。請依 [mapping,event,target_complement,preimage,source_complement,preimage_of_complement,equal] 欄序回答。", "answer": { "kind": "expression", "value": "([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], 是)", "explanation_zh": "收束波核對=([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], 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；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "可測函數密度" ] }, "variant": { "family_id": "qf-mod25-measdeep-preimage-complement-law", "seed": 202610255011, "parameters": { "source_n": 4, "target_n": 4, "mapping": [ 2, 2, 0, 0 ], "event": [ 2, 3 ], "domain": "Finite_preimage_law_density", "operation": "measdeep-preimage-complement-law", "semantic_template": "qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,2,0,0],\"source_n\":4,\"target_n\":4}" } }, "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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r05-measurable-function-0011", "batch_id": "batch-20261013-advance-20261025-r05-measurable-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "dcff5317f841b400fd517de38ab6b1c67664e1dc92b9ec9e1f43a1cffef915b9", "family_id": "qf-mod25-measdeep-preimage-complement-law", "seed": 202610255011, "semantic_template": "qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,2,0,0],\"source_n\":4,\"target_n\":4}" }, "fields": { "title": "Complement under a Complete Total-Map Preimage", "stem": "Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.", "answer": { "kind": "expression", "value": "([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)", "explanation": "Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)." }, "skills": [ "Complement under a Complete Total-Map Preimage" ], "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sigma代數與總映射。操作=measdeep-preimage-complement-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,2,0,0],\"event\":[2,3]}。請依 [mapping,event,target_complement,preimage,source_complement,preimage_of_complement,equal] 欄序回答。", "en": "Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod25-measdeep-preimage-complement-law, measdeep-preimage-complement-law, Finite_preimage_law_density, []]", "after": "derived = [[mapping, event, target_complement, preimage, source_complement, complement_preimage, equal]]", "relation": "implies", "rule_id": "R65_SOURCE_bank_keys_operation_domain_ordered_output_roles", "conditions": [], "explanation": { "zh-Hant": "MOD25用LOCAL hyphen及具名domain，矩陣legacy用underscore/reals，BASElogic不補domain/op。輸出tuple/矩陣/數字欄與fraction文字各依來源，不一律保/1；原中英/AST/status不變。", "en": "MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [4, 4, [2, 2, 0, 0]]", "after": "derived = [[2, 2, 0, 0]]", "relation": "implies", "rule_id": "Meas_RAW_complete_total_map_typed_NONEMPTY_carriers", "conditions": [], "explanation": { "zh-Hant": "此來源切片carriers大小1..7，每個原source index恰有typed target。先驗完整表，再取任何逆像；不假設injective或inverse function。", "en": "THIS source slice has carrier sizes1..7 and a typed target at EVERY original source index.Validate the WHOLE table before preimages;neither injectivity nor an inverse function is assumed." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [complement ORIGINAL event, 0, 2, [2, 3]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [complement ORIGINAL event, 1, 2, [2, 3]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [complement ORIGINAL event, 2, 0, [2, 3]]", "after": "derived = [否]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [complement ORIGINAL event, 3, 0, [2, 3]]", "after": "derived = [否]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [complement TARGET complement, 0, 2, [0, 1]]", "after": "derived = [否]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [complement TARGET complement, 1, 2, [0, 1]]", "after": "derived = [否]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [complement TARGET complement, 2, 0, [0, 1]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [complement TARGET complement, 3, 0, [0, 1]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Meas_PREIMAGE_EACH_source_label_target_membership", "conditions": [], "explanation": { "zh-Hant": "source標記i在逆像當且僅當mapping[i]屬目標event；保兩個carrier角色，不把forward image當逆像。", "en": "Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[2, 2, 0, 0], [2, 3], [0, 1]]", "after": "derived = [[0, 1], [2, 3], [2, 3], 是]", "relation": "implies", "rule_id": "Meas_COMPLEMENT_BOTH_preimage_constructions_OWN_carriers_NO_sigma_requirement", "conditions": [], "explanation": { "zh-Hant": "分別在Fin(target_n)取target補集、Fin(source_n)取逆像補集，逐表核兩者相同。此set-law family無另給σ前提，不加measurability或inverse-function要求。", "en": "Take the TARGET complement in Fin(target_n)and the SOURCE preimage complement in Fin(source_n),then compare both constructions.This set-law family supplies no extra sigma premise;do not invent measurability or inverse-function requirements." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ [ 2, 2, 0, 0 ], [ 2, 3 ], [ 0, 1 ], [ 0, 1 ], [ 2, 3 ], [ 2, 3 ], true ] }, "conclusion": { "canonical_value": "([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], 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-20261025-r05-measurable-function-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r05-measurable-function-foundations/questions.jsonl", "batch_sha256": "543be1a1a400d84a28caa099ad4962b01d4e6cf96768f625e3b9f01b33e1e23c", "original_line": 11, "original_line_sha256": "b9af5241ea58f04f064e49ef261690593d08fce5296e99d690a27e5484d7c3f7", "question_compact_sha256": "dcff5317f841b400fd517de38ab6b1c67664e1dc92b9ec9e1f43a1cffef915b9", "english_row_compact_sha256": "465ea7e1d2c26e3dfbac95674913dc8d9f96020c824138fd68e95940e35aa920", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r05-measurable-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r05-measurable-function-foundations.en.jsonl" }, "mko_bindings": { "mko-measurable-function": { "sha256": "0c21c3c35f2f6b0451adf573f920b3dd8e797c631bcae8c14bf6bd959dc1571b", "english_sha256": "92b1662eff5366dc439172033e87ad4948ab0cbef9b4f22c08d2efd3e2310267", "original_path": "public/data/mko/measurable-function.json", "english_path": "public/data/i18n/editions/mko-measurable-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "871e2c05026567a1deae741d6b90b0588100c6482ee167c7f9f45af5b72d8606", "original_candidate_sha256": "b3050f9313d953f7c1e022eff220d9ca273434e8b0724a6080f1fe251a69d1e2", "original_candidate_line_sha256": "b3050f9313d953f7c1e022eff220d9ca273434e8b0724a6080f1fe251a69d1e2" }, "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-20261026-012-p66", "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": "92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6", "question_first_witness_sha256": "012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 11, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "5147271a64854129a9e99ddba9b1b688a779e7626eef6ef34ca1808248ee7ab2", "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." } }