{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261029-r06-iid-sequence-0009", "pilot_number": "R0027-14409", "number": 14409, "family_id": "qf-mod29-iid-bernoulli-cylinder", "title": { "zh-Hant": "第 14409 題：獨立同分布序列", "en": "Question 14409: Exact Bernoulli Cylinder Probability under a Declared Product Law" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261029-r06-iid-sequence-0009", "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-iid-sequence" ], "title_zh": "獨立同分布序列", "stem_zh": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[9,29],\"word\":[0,0,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)", "explanation_zh": "Wave4第三層核對=([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "獨立同分布序列" ] }, "variant": { "family_id": "qf-mod29-iid-bernoulli-cylinder", "seed": 202610296009, "parameters": { "p": [ 9, 29 ], "word": [ 0, 0, 0, 1 ], "domain": "Symbolic_Bernoulli_product_law", "operation": "iid-bernoulli-cylinder", "semantic_template": "qf-mod29-iid-bernoulli-cylinder/{\"p\":[9,29],\"word\":[0,0,0,1]}" } }, "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、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式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-20261029-r06-iid-sequence-0009", "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5573b78fe3c54f6b91b0a8ec8c95f2e847831074ea83eb3f7ed274d68f477133", "family_id": "qf-mod29-iid-bernoulli-cylinder", "seed": 202610296009, "semantic_template": "qf-mod29-iid-bernoulli-cylinder/{\"p\":[9,29],\"word\":[0,0,0,1]}" }, "fields": { "title": "Exact Bernoulli Cylinder Probability under a Declared Product Law", "stem": "Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law).", "answer": { "kind": "expression", "value": "([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)", "explanation": "Count ones and zeros and calculate p^ones*(1−p)^zeros exactly. Empty products equal 1, including degenerate p=0 or p=1. The arbitrary-word formula is the declared symbolic product-law certificate; this finite calculation alone is not a new proof of infinite IID. The exact result, retaining the requested order, is ([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)." }, "skills": [ "Exact Bernoulli Cylinder Probability under a Declared Product Law" ], "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": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[9,29],\"word\":[0,0,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。", "en": "Use the explicitly declared Bernoulli product-law model with exact p=9/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 1]. Give (word,ones,zeros,p,probability,symbolic_product_law)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Bernoulli zero-outcome mass := sub(1, 9/29)", "after": "Bernoulli zero-outcome mass = 20/29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Bernoulli zero-outcome mass，不以小數近似替代。", "en": "Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "9/29" ], "output": "20/29", "label": "Bernoulli zero-outcome mass" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "raw Bernoulli parameter=[9, 29]", "after": "normalized p=9/29;mass at0=20/29;mass at1=9/29;symbolic product-law model explicitly declared", "relation": "implies", "rule_id": "Bernoulli_complete_marginal_and_product_law_premise", "conditions": [], "explanation": { "zh-Hant": "保留完整{0,1}邊際與宣告的乘積法則前提；不是從一組數據、相同期望或pairwise檢查猜無限IID。", "en": "Retain the complete{0,1} marginal and the explicitly declared product-law premise.Infinite IID is not inferred from a data set,equal means or pairwise checks." }, "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": "original ordered finite binary word=[0, 0, 0, 1];p=9/29;one-minus-p=20/29", "after": "cylinder atom factorization uses the same declared symbolic product law", "relation": "implies", "rule_id": "IID_ordered_word_and_given_model_not_unknown_law_test", "conditions": [], "explanation": { "zh-Hant": "原字保留順序，1用p、0用1−p；此分解是模型前提，不是測一個未知相依模型後推得獨立。空字與p=0或1都保留。", "en": "Preserve word order,use p for1 and1−p for0.Factorization is a model premise,not a test proving independence of an unknown dependent law.Retain empty words and p=0 or1 boundaries." }, "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": "binary word number of ones := sum(0, 0, 0, 1)", "after": "binary word number of ones = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 binary word number of ones，不以小數近似替代。", "en": "Recompute binary word number of ones with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0", "0", "1" ], "output": "1", "label": "binary word number of ones" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "binary word number of zeros := sub(4, 1)", "after": "binary word number of zeros = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 binary word number of zeros，不以小數近似替代。", "en": "Recompute binary word number of zeros with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "1" ], "output": "3", "label": "binary word number of zeros" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "(9/29)^1", "after": "Bernoulli one factors power = 9/29", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "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": { "operation": "integer_power", "inputs": [ "9/29", "1" ], "output": "9/29", "label": "Bernoulli one factors power" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "(20/29)^3", "after": "Bernoulli zero factors power = 8000/24389", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "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": { "operation": "integer_power", "inputs": [ "20/29", "3" ], "output": "8000/24389", "label": "Bernoulli zero factors power" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Bernoulli cylinder product := mul(9/29, 8000/24389)", "after": "Bernoulli cylinder product = 72000/707281", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Bernoulli cylinder product，不以小數近似替代。", "en": "Recompute Bernoulli cylinder product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9/29", "8000/24389" ], "output": "72000/707281", "label": "Bernoulli cylinder product" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "word length=4;ones=1;zeros=3;cylinder probability=72000/707281", "after": "arbitrary-word formula p^ones*(1-p)^zeros;empty product1 including degenerate parameters", "relation": "implies", "rule_id": "Bernoulli_cylinder_formula_empty_product_boundary", "conditions": [], "explanation": { "zh-Hant": "乘積可按相同邊際因子收集次數，但沒有排序或改寫原字。零次冪代表空乘積1，包含p=0、1；這個有限代值不另證未知無限序列IID。", "en": "Collect identical marginal factors by counts without sorting or changing the original word.Exponent0 denotes empty-product1,including p=0 or1.This finite evaluation is not a new proof of IID for an unknown infinite sequence." }, "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": [ [ 0, 0, 0, 1 ], 1, 3, "9/29", "72000/707281", "symbolic_product_law" ] }, "conclusion": { "canonical_value": "([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)", "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-20261029-r06-iid-sequence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r06-iid-sequence-foundations/questions.jsonl", "batch_sha256": "eb1e6ee54e11c0710a74de69c6e79201e0d20149f0261f850a24316a257e8115", "original_line": 9, "original_line_sha256": "ffb7bc6b9c0af336a50df3a3cf3ebb622e4249cee4c5f88c66ffda820335fe8d", "question_compact_sha256": "5573b78fe3c54f6b91b0a8ec8c95f2e847831074ea83eb3f7ed274d68f477133", "english_row_compact_sha256": "a58eef938a32ac2a6d5213d56ee97c374136f4433a74f27e4bcdcc448967c636", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r06-iid-sequence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r06-iid-sequence-foundations.en.jsonl" }, "mko_bindings": { "mko-iid-sequence": { "sha256": "bac79da4634be201a10f08ec4489946a636db5ebc2e628c0b1830b743f4ddbd0", "english_sha256": "9fc20d5d2102056dfc01584c16ff07f2a621a4acee305e717a7c68822a5dfa48", "original_path": "public/data/mko/iid-sequence.json", "english_path": "public/data/i18n/editions/mko-iid-sequence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77", "original_candidate_sha256": "ca17934b232740e99a6c4779bb02a456d84580d6b0a3663ead261950d1c3b31a", "original_candidate_line_sha256": "ca17934b232740e99a6c4779bb02a456d84580d6b0a3663ead261950d1c3b31a" }, "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-20261026-007-p28", "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": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe", "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "7edbc448901da809b6263c7ae2d48c550ddf9786899c365a7da3c14d2ea1be06", "arithmetic_steps": 6 }, "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." } }